Section 6.1: Quadratic Functions and their Characteristics Vertical Intercept Vertex Axis of Symmetry Domain and Range Horizontal Intercepts
|
|
- Abigail Owen
- 6 years ago
- Views:
Transcription
1 Lesson 6 Quadratic Functions and Equations Lesson 6 Quadratic Functions and Equations We are leaving exponential functions behind and entering an entirely different world. As you work through this lesson, you will learn to identify quadratic functions and their graphs (called parabolas). You will learn the important parts of the parabola including the direction of opening, vertex, intercepts and axis of symmetry. You will use graphs of quadratic functions to solve quadratic equations and, finally, you will learn how to use the characteristics of quadratic functions to solve applications. Even if you are not asked to graph a given quadratic function, doing so is always a good idea so that you can get a visual feel for the problem at hand. Lesson Topics Section 6.1: Quadratic Functions and their Characteristics Vertical Intercept Vertex Axis of Symmetry Domain and Range Horizontal Intercepts Section 6.2: Applications of Quadratic Functions Section 6.3: Quadratic Equations in Standard Form Section 6.4: Solving Quadratic Equations Graphically Section 6.5: Solving Quadratic Inequalities Graphically Section 6.6: Quadratic Regression 219
2 Lesson 6 Quadratic Functions and Equations 220
3 Lesson 6 - MiniLesson Section 6.1 Characteristics of Quadratic Functions A quadratic function is a function which can be written in the following form:! f ( x ) = ax 2 + bx + c! f ( x ) has three distinct terms each with its own coefficient:!ax 2 is the first term and has coefficient!a!bx is the second term and has coefficient! b!c is the third term, called the constant term, and has coefficient!c Note: If any term is missing, the coefficient of that term is 0!a,! b and!c can be any real numbers. Note that!a cannot be 0. The graph of this function is called a parabola, is shaped like a U and opens either up or down.!a determines which direction the parabola opens (!a > 0 opens up,!a < 0 opens down)!c is the vertical intercept with coordinates! (0, c) Each of the following are quadratic functions. Do not let the fractions and missing terms fool you! f (x)= 2x 2 3x +5 h(x)= 2x 2! g(x)= x 2 + 4x k(x)= 1! 3 x Graph of the Basic Quadratic Function Domain:!,! f ( x ) =!x 2 ( ), Range: 0,! ), Vertex: 0, 0! ( ) ( ), Vertical Intercept: 0, 0! ( ) ( ) and increasing on the interval 0,! ( ) Horizontal Intercept:! 0, 0 Function behavior: decreasing on the interval!, 0 221
4 Problem 1 WORKED EXAMPLE GRAPH QUADRATIC FUNCTIONS Given the quadratic function, f x! ( ) = x 2 + 4x 2, complete the table and generate a graph of the function on your calculator using the standard viewing window. Identity the coefficients a, b, c a = 1, b = 4, c = 2 Which direction does the parabola open? a = 1 which is greater than 0 so parabola opens up What is the vertical intercept? c = 2; Vertical intercept: (0, 2) Graph the function on your calculator Enter Y1=x^2+4x 2, Press Zoom: 6 for Standard Window to get the graph below. Problem 2 MEDIA EXAMPLE GRAPH QUADRATIC FUNCTIONS Given the Quadratic Function f x! ( ) = x 2 2x + 3, complete the table and generate a graph of the function on your calculator using the standard viewing window. Identity the coefficients a, b, c Which direction does the parabola open? Why? What is the vertical intercept? Plot and label on the graph. 222
5 Problem 3 YOU TRY GRAPH QUADRATIC FUNCTIONS a) Given the quadratic function f x! ( ) = 2x 2!5, complete the table and generate a graph of the function on your calculator using the standard viewing window. Identity the coefficients a, b, c Which direction does the parabola open? Why? What is the vertical intercept? Plot and label on the graph. b) Given the quadratic function f x! ( ) = 4x 2 +2x 1, complete the table and generate a graph of the function on your calculator using the standard viewing window. Identity the coefficients a, b, c Which direction does the parabola open? Why? What is the vertical intercept? Plot and label on the graph. 223
6 Given a quadratic function,! f x Quadratic Functions: Vertex/Axis Of Symmetry ( ) = ax 2 + bx + c The vertex is the lowest or highest point of the associated parabola and is always written as an ordered pair. b To find the vertex input value, identify coefficients a and b then compute. 2a Substitute this input value into f x! ( ) to determine the corresponding output value. The axis of symmetry equation is the equation of the vertical line that passes through the vertex and divides the parabola in half. Axis of symmetry equation x! = b! 2a Problem 4 WORKED EXAMPLE Quadratic Functions: Vertex/Axis Of Symmetry Given the Quadratic Function f x! ( ) =!x 2 +!4x!2, complete the table below. Identity the coefficients a, b, c a = 1, b = 4, c = 2 Determine the coordinates of the vertex. Input Value b x = 2a (4) = 2(1) = 2 Output Value f 2 = ( 2) ( ) 2 = = 6 + 4( 2) 2 Vertex Ordered Pair: ( 2, 6) Identify the axis of symmetry equation. Axis of symmetry equation:!x! = 2 Graph the function. Plot and label the vertex and axis of symmetry equation on the graph. 224
7 Problem 5 MEDIA EXAMPLE Quadratic Functions: Vertex/Axis Of Symmetry Given the quadratic function! f (x)= x 2 2x +3, complete the table below. Identity the coefficients a, b, c Determine the coordinates of the vertex. Identify the axis of symmetry equation. Graph the function. Plot and label the vertex and axis of symmetry equation on the graph. Problem 6 YOU TRY Quadratic Functions: Vertex/Axis Of Symmetry Given the quadratic function! f x Identity the coefficients a, b, c ( ) = 2x 2!5, complete the table below. Determine the coordinates of the vertex. Identify the axis of symmetry equation. Graph the function. Plot and label the vertex and axis of symmetry equation on the graph. 225
8 Problem 7 WORKED EXAMPLE Quadratic Functions: Domain and Range Determine the domain and range of the quadratic function f x! ( ) =!x 2 +!4x!2 graphed below. The vertex is plotted and labeled. Write your answers in interval notation and inequality notation. Domain!of!f (x): All!Real!Numbers! Inequality notation:!! < x <!!!!!! ( ) Interval notation:!, Range!of!f (x):! Since the parabola opens up, the vertex ( 2, 6) is the lowest point on the graph. The output value of 6 is the lowest output on the graph. The range is therefore: Inequality notation:! f x ( ) 6 Interval notation: 6,!! )!!! Problem 8 MEDIA EXAMPLE Quadratic Functions: Domain and Range Sketch the graph and find and label the vertex ordered pair for f x! ( ) = 2x 2!6. Then, use the graph to help you determine the domain and range. Write your domain and range answers in interval notation and inequality notation. Domain!of!f (x):! Range!of!f (x):! 226
9 Problem 9 MEDIA EXAMPLE Quadratic Functions: Domain and Range Determine the domain and range of the graph given below. Write your answers in interval notation and inequality notation. Domain!of!f (x):! Range!of!f (x):! Problem 10 YOU TRY Quadratic Functions: Domain and Range a) Sketch the graph and find, plot and label the vertex ordered pair for f x! ( ) = 2x 2!5. Use the graph to determine the domain and range. Write your answers in interval notation and inequality notation. Domain!of!f (x):! Range!of!f (x):! b) Determine the domain and range of the graph shown below. Write your answers in interval notation and inequality notation. Domain!of!f (x):! Range!of!f (x):! 227
10 Horizontal Intercepts of a Quadratic Function The quadratic function,! f (x)= ax 2 + bx + c, will have horizontal intercepts if its parabola crosses the x-axis (i.e. if! f x ( ) =!0 ). These points are marked on the graph above as G and H. To find the coordinates of these points, what we are really doing is solving the equation!ax 2 + bx + c =!0. At this point, we will use the steps below to solve this equation. In the next lesson, we will learn other methods for solving these equations. Always begin by graphing the given quadratic function so you can see which of the cases above you are presented with. Using the Graphing/Intersection Method to solve!ax 2 + bx + c =!0: 1. Press Y= then enter f x! ( )into Y1 and enter 0 into Y2. 2. Use Zoom: 6 to graph on the standard window. Inspect the graph adjusting the window as needed to see if there are any places where the graph intersects the horizontal axis. 3. Use the Graphing/Intersection Method once to determine G and again to determine H, if both exist. Problem 11 Given the quadratic function! f x WORKED EXAMPLE Horizontal Intercepts of a Quadratic Function ( ) =!x 2 +!4x! 2 draw a sketch of the graph then use the Graphing/Intersection Method on your TI 83/84 calculator to identify the horizontal intercepts rounded to 2 decimal places. If these exist, label them on the graph. If there are no intercepts, indicate that as well. 1. Press Y= then enter!x 2 +!4x!2! into Y1 and enter 0 into Y2. 2. Graphing on a standard window, the graph clearly shows there are 2 horizontal intercepts. 3. Use the Graphing/Intersection Method once to determine G as ( 4.45, 0). You may have to move your cursor close to G during the First Curve? step. Use the method again to determine H as (0.45, 0). You may have to move your cursor close to H during the First Curve? step. 228
11 Problem 12 MEDIA EXAMPLE Horizontal Intercepts of a Quadratic Function For each of the following functions, draw a sketch of the graph then use the Graphing /Intersection Method on your TI 83/84 calculator to identify the horizontal intercepts rounded to 2 decimal places. If these exist, label them on the graph. If there are no intercepts, indicate that as well. a) f x! ( ) = 2x 2 + 6x + 3 b) g x! ( ) =!x 2 x! +!2 c) h x! ( ) =!x 2 6x! +!9 229
12 Problem 13 YOU TRY Horizontal Intercepts of a Quadratic Function a) Given the quadratic function f x! ( ) = 2x 2 5, sketch the graph then use the Graphing /Intersection Method on your TI 83/84 calculator to identify the horizontal intercepts rounded to 2 decimal places. If these exist, label them on the graph. If there are no intercepts, indicate that as well. b) Given the quadratic function! f x ( ) = 3x 2 + 2x + 5, sketch the graph then use the Graphing /Intersection Method on your TI 83/84 calculator to identify the horizontal intercepts rounded to 2 decimal places. If these exist, label them on the graph. If there are no intercepts, indicate that as well. 230
13 Section 6.2 Applications of Quadratic Functions A large number of quadratic applications involve launching objects into the sky (arrows, baseballs, rockets, etc ) or throwing things off buildings or spanning a distance with an arched shape. While the specifics of each problem are certainly different, the information below will guide you as you decipher the different parts. How to Solve Quadratic Application Problems 1. Draw an accurate graph of the function using first quadrant values only. Label the x-axis with the input quantity and units. Label the y-axis with the output quantity and units. 2. Identify, plot and label the vertical intercept. 3. Identify, plot and label the vertex. 4. Identify, plot and label the positive horizontal intercept(s). Usually, there is only one horizontal intercept that is importat. If both are needed, then plot them both and include negative input values in your graph for part Once you have completed steps 1 4, read the specific questions you are asked to solve. Sample questions that involve the vertical intercept 0, c! ( ): How high was the object at time!t = 0? Answer: c What was the starting height of the object? Answer: c Sample questions that involve the vertex: How high was the object at its highest point? Answer: y-value of the vertex What was the max height of the object? Answer: y-value of the vertex How long did it take the object to get to its max height? Answer: x-value of the vertex What is the practical range of this function? [0, y-value of the vertex] Sample questions that usually involve the positive horizontal intercept x! (,!0 2 ) : When did the object hit the ground? Answer:! x 2 What is the practical domain of this function?! 0,x 2 How far was the object from the center?! x 2 231
14 Problem 14 WORKED EXAMPLE Applications of Quadratic Functions The function! h t ( ) =! 16t 2 +!80t +!130, where! arrow shot into the sky as a function of time (seconds). h(t) is height in feet, models the height of an Before even looking at the specific questions asked, find the items below and plot/label the graph. 1. Identify the vertical intercept. (0, 130) since c = Determine the vertex. b 80 The input value of the vertex is x = = = a 2( 16) b 2 The corresponding output value is f ( ) = f (2.5) = 16(2.5) + 80(2.5) = 230 2a 3. Determine the positive horizontal intercept using the process discussed in earlier examples, we want to solve! 16t 2 +!80t! +!130 = 0. Using the Graphing/Intersection Method, the positive horizontal intercept is (6.29, 0). 4. Draw an accurate graph of the function using first quadrant values only. Label the horizontal axis with the input quantity and units. Label the vertical axis with the output quantity and units. Label the vertex and intercepts. QUESTIONS TO ANSWER NOW: a) After how many seconds does the arrow reach its highest point? The input value of the vertex is 2.5. So, the arrow reaches its highest point after 2.5 seconds. b) How high is the arrow at its highest point? The output value of the vertex is 230. So, the arrow is 230 feet above the ground at its highest point. c) After how many seconds does the arrow hit the ground? The horizontal intercept is (6.29, 0). The arrow will hit the ground after 6.29 seconds. d) What is the practical domain of this function? Time starts at 0 seconds and goes until the arrow hits the ground. So, practical domain is [0, 6.29] seconds. e) What is the practical range of this function? The arrow passes through all height values from 0 (when it hits the ground) to its max height of 230 ft. So, practical range is [0, 230] feet. f) What does the vertical intercept represent? The vertical intercept represents the height of the arrow at time t = 0. Thus, the arrow starts at 130 feet off the ground. 232
15 Problem 15 MEDIA EXAMPLE Applications of Quadratic Functions The height of a train tunnel is modeled by the quadratic function! h x is the distance, in feet, from the center of the tracks and! h x ( ) =! 0.35x 2 +!25, where x ( ) is the height of the tunnel, also in feet. Assume that the high point of the tunnel is directly in line with the center of the train tracks. a) Draw a complete diagram of this situation. Find and label each of the following: vertex, horizontal intercept (positive side), and vertical intercept. b) How wide is the base of the tunnel? c) A train with a flatbed car 6 feet off the ground is carrying a large object that is 15 feet high. How much room will there be between the top of the object and the top of the tunnel? 233
16 Problem 16 YOU TRY Applications of Quadratic Functions 2 A toy rocket is shot straight up into the air. The function H ( t) = 16t + 128t + 3 gives the height (in feet) of the rocket after t seconds. Round answers to two decimal places as needed. All answers must include appropriate units of measure. a) Draw a complete diagram of this situation. Find and label each of the following: vertex, horizontal intercept (positive side), and vertical intercept. b) How long does it take for the rocket to reach its maximum height? Write your answer in a complete sentence. c) What is the maximum height of the rocket? Write your answer in a complete sentence. d) How long does it take for the rocket to hit the ground? Write your answer in a complete sentence. e) Identify the vertical intercept. Write it as an ordered pair and interpret its meaning in a complete sentence. f) Determine the practical domain of! H(t). Use interval notation and include units. g) Determine the practical range of! H(t). Use interval notation and include units. 234
17 Section 6.3 Quadratic Equations in Standard Form We will do a lot of work in this class, and you will work in future classes, to solve quadratic equations. Many of the solution methods require that the quadratic equation be first written in what is called standard form. Quadratic Equation in Standard Form A quadratic equation in standard form is an equation which can be written in the form:!ax 2 + bx + c = 0 a, b and c can be any real numbers and, as with quadratic functions,! a 0. The equation must be set equal to 0 to be considered in standard form. Problem 17 WORKED EXAMPLE Quadratic Equations: Standard Form Write the quadratic equation!4x 2!2x! =!5 in standard form. Identify coefficients a, b and c. 4x 2 2x = 5 Subtract 5 from both sides to set the equation 4x 2 2x 5 = 5 5 equal to 0.! 4x 2 2x 5 = 0 a is the coefficient of the!x 2 a! =!4 term b! =! 2 b is the coefficient of the!x term c is the constant term! c! =! 5 Problem 18 MEDIA EXAMPLE Quadratic Equations: Standard Form Write the quadratic equation! 3x 2 =!8! +!4x in standard form. Identify coefficients a, b and c. Problem 19 YOU TRY Quadratic Equations: Standard Form Write the quadratic equation! 2x! +!4 =! 3x 2 in standard form. Identify coefficients a, b and c. 235
18 Problem 20 WORKED EXAMPLE Quadratic Equations: Standard Form Given the quadratic equation 2 x!!3! identify the coefficients a, b and c. ( ) 2!5 = 0, write the equation in standard form and Original equation as given. ( ). Square the binomial x!!h! Multiply the result by a. Combine with k to get the final result in standard form. Identify the coefficients a, b and c. 2( x!!3) 2!5! =!0! ( )( x 3) 5 = 0 2 x 3 2( x 2 3x 3x + 9) 5 = 0 2( x 2 6x + 9) 5 = 0!! 2x 2 +12x 18 5 = 0! 2x 2 +12x 23= 0! a =! 2, b =!12, c =!%23 Problem 21 MEDIA EXAMPLE Quadratic Equations: Standard Form Given the quadratic equation 4 x! +!1! the coefficients a, b and c. ( ) 2 =!%2, write the equation in standard form and identify Problem 22 YOU TRY Quadratic Equations: Standard Form Given the quadratic equation 2 x! +!3! the coefficients a, b and c. ( ) 2 =!1, write the equation in standard form and identify 236
19 Problem 23 WORKED EXAMPLE Quadratic Equations: Standard Form Given the quadratic equation x 1! ( )( x +2)! =!5, write the equation in standard form and identify coefficients a, b and c. Subtract 5 from both sides to set the equation = 0. Then foil and combined like terms to write in standard form. ( x 1) ( x +2) 5 = 0 (x 2 +2x x 2) 5 = 0 x 2 + x 2 5 = 0! x 2 + x 7 = 0 a is the coefficient of the!x 2 term a! =!1 b is the coefficient of the!x term b! =!1 c is the constant term! c! =! 7 Problem 24 MEDIA EXAMPLE Quadratic Equations: Standard Form Given the quadratic equation 3 2x 1! identify coefficients a, b and c. ( )( x +5) = 9, write the equation in standard form and Problem 25 YOU TRY Quadratic Equations: Standard Form Given the quadratic equation 2 x + 4! ( )( x 3) = 6, write the equation in standard form and identify coefficients a, b and c. 237
20 Section 6.4 Solving Quadratic Equations Graphically Solutions to a Quadratic Equation in Standard Form As stated previously, a quadratic equation in standard form has at most 2 real number solutions. These solutions can be written as x and! 1! x. If the solutions are real numbers, then there are 2 two horizontal intercepts for the graph of f x! (! x,0 1 ) and x! (,0 2 ). ( ) =!ax 2 +!bx! +!c! and these can be written as Note that if a parabola does not have horizontal intercepts, then its solutions lie in the complex number system. The corresponding equation has no real solutions but does have two complex solutions. More on this in the next lesson! There are three possible cases for number of solutions to a quadratic equation in standard form. All of the sample graphs open up. The same cases hold true if the graphs open down. CASE 1: One, repeated, real number solution The parabola touches the x-axis in just one location which is at the vertex. CASE 2: Two unique, real number solutions The parabola crosses the x-axis at two unique locations. CASE 3: No real number solutions exist but two complex number solutions do exist. The parabola does NOT cross the x-axis. 238
21 Problem 26 MEDIA EXAMPLE How Many and What Kind of Solutions? Use your graphing calculator to help you determine the number and type of solutions to each of the quadratic equations below. Begin by putting the equations into standard form. Draw an accurate sketch of the parabola in an appropriate viewing window. If your solutions are real number solutions, use the graphing/intersection method to find them. Use proper notation to write the solutions and the horizontal intercepts of the parabola. Label the intercepts on your graph. a)!x 2 10x +25 = 0 Horizontal Intercepts: Number of Real Solutions: Real Solutions: b)! 2x 2 +!8x!3! =!0 Horizontal Intercepts: Number of Real Solutions: Real Solutions: c)!3x 2!2x =! 5 Horizontal Intercepts: Number of Real Solutions: Real Solutions: 239
22 Problem 27 YOU TRY How Many and What Kind of Solutions? Use your graphing calculator to help you determine the number and type of solutions to each of the quadratic equations below. Begin by putting the equations into standard form. Draw an accurate sketch of the parabola in an appropriate viewing window. If your solutions are real number solutions, use the Graphing/Intersection Method to find them. Use proper notation to write the solutions and the horizontal intercepts of the parabola. Label the intercepts on your graph. Round as needed to one decimal place. a)! x 2!6x 9! =!0 Horizontal Intercepts: Number of Real Solutions: Real Solutions: b)!3x 2 +!5x +!20! =!0 Horizontal Intercepts: Number of Real Solutions: Real Solutions: c)!2x 2 +!5x! = 7 Horizontal Intercepts: Number of Real Solutions: Real Solutions: 240
23 Graphically Solving a Quadratic Equation That is Not in Standard Form A quadratic equation of the form! ax 2 + bx + c! =!d can be solved in the following way using your graphing calculator: 1. Go to!y = 2. Let!Y1 = ax 2 +!bx! +!c 3. Let! Y2! = d 4. Graph the two equations. You may need to adjust your window to be sure the intersection(s) is/are visible. 5. For each intersection, use 2 nd >Calc>Intersect. Follow on-screen directions to designate each graph then determine intersection (pressing ENTER each time). 6. Solution(s) to the equation are the intersecting x-values NOTE: The Graphing/Intersection Method will provide us only with approximate solutions to a quadratic equation when decimal solutions are present. To find exact solution values, you will need to use the Quadratic Formula. This will be covered in a later lesson. Problem 28 WORKED EXAMPLE Solve Quadratic Equations Graphically Solve the equation! 3x 2!2x!4! =! 5 by graphing. There are two intersection points. Follow the process above to find the intersections ( 1, 5) and (0.33, 5). Solutions to the equation are! x 1 = 1, x 2 = Problem 29 MEDIA EXAMPLE Solve Quadratic Equations Graphically Solve!x 2!10x +1! =!4. Plot and label the graphs and intersection points that are part of your solution process. Identify the final solutions clearly. Round to 2 decimal places. Xmin = Xmax = Ymin = Ymax = 241
24 Problem 30 YOU TRY Solve Quadratic Equations Graphically a) Solve!2x 2!5! =!6. Plot and label the graphs and intersection points that are part of your solution process. Round your answers to two decimal places. Identify the final solutions clearly. Xmin = Xmax = Ymin = Ymax = b) Solve!x 2 +!9x!18! =!32. Plot and label the graphs and intersection points that are part of your solution process. Round your answers to two decimal places. Identify the final solutions clearly. Xmin = Xmax = Ymin = Ymax = 242
25 Section 6.5 Solving Quadratic Inequalities by Graphing In this section, we will see how a quadratic inequality written in standard form can be solved graphically. We will study an algebraic solution method in the next lesson. Quadratic inequalities are written just like quadratic equations except the = sign is replaced with an inequality symbol. See the forms below. Quadratic Inequalities Quadratic inequalities can be written in one of the following standard forms:!ax 2 + bx + c > 0!ax 2 + bx + c < 0!ax 2 + bx + c 0!ax 2 + bx + c 0 The example below illustrates how to understand and approach these problems graphically. Problem 31 MEDIA EXAMPLE Solving Quadratic Inequalities Graphically Solve the quadratic inequality,! x 2 + x 2 4, using the Graphing/Intersection Method. Show a sketch of your graph in the box below. Write your final result in interval and inequality notations. 243
26 Problem 32 WORKED EXAMPLE Solving Quadratic Inequalities Graphically Solve the following quadratic inequalities using the Graphing/Intersection Method. Show a sketch of your graph in the box below. Write your final results in interval and inequality notation. Note: For each problem below, press Y= on your graphing calculator and let Y1 =! x 2 + x 2 and Y2 = 4. Press ZOOM 6 to graph the equations on the standard graphing window. Use the Graphing/Intersection Method twice to determine the intersection points of (! 3, 4) and (2, 4). a)! x 2 + x 2 > 4 What we are looking for in the graph are the values of y that are strictly greater than 4. This is the solid part of the graph (see arrows in the graph below). Our solution must be written in terms of the values of x that would generate the solid part of the graph. That would be x-values smaller than 3 or greater than 2. The way we write this is: Inequality Notation:!x < 3!!or!!x > 2 Interval notation:, 3! ( ) ( 2, ) The symbol above means union and should be used when joining two intervals together in a solution. Note also that endpoints are not included in this solution because the inequality is strictly greater than 4. Contrast that to the previous media example. 244
27 b)! x 2 + x 2 4 The graph below shows only the segment of! Y1 = x 2 + x 2 that is less than or equal to 4. The rest of the graph is a dotted line. Our solution to the inequality! x 2 + x 2 4 must indicate the values of x that are used to produce the solids part of the graph. Our solution is: Inequality Notation:! 3 x 2 Interval notation: 3,2! c)! x 2 + x 2 < 4 The graph below shows only the segment of! Y1 = x 2 + x 2 that is strictly less than 4. Note that the only difference from part b) is that our plotted endpoint are not included. Our solution to the inequality! x 2 + x 2 < 4 must indicate the values of x that are used to produce this graph. Our solution is: ( ) Inequality Notation:! 3< x < 2 Interval notation:! 3,2 245
28 Problem 33 YOU TRY Solving Quadratic Inequalities Graphically Solve the following quadratic inequalities using the graphing/intersection method. Show your graph in the blank space below and provide your solutions in both inequality and interval notation. This problem is similar to problem 34 so you should be able to make one graph that will help you solve all of the inequalities. Hint: You may want to let your Ymax = 20 on your calculator window for the graph. a)! x b)! x 2 4 > 12 c)! x d)! x 2 4 <
29 Section 6.6 Quadratic Regression Problem 34 WORKED EXAMPLE Quadratic Regression The table below shows the height, H, in feet, of an arrow t seconds after being shot. t H(t) Use the Quadratic Regression feature of your calculator to generate a mathematical model for this situation. Round to three decimals places. Press STAT>EDIT>ENTER to show data entry area. The STAT button is on the second row, third column. Press STAT > CALC > 5:QuadReg Thus, your quadratic function (with values rounded as the problem indicates) is:! H ( t )! =! t t +!13.6 Enter your function into Y1 to obtain a graph of your data and regression line. Use viewing window xmin=0 xmax=7 ymin=0 ymax=
30 Problem 35 MEDIA EXAMPLE Quadratic Regression The table below shows the height, H, in feet, of a golf ball t seconds after being hit. t H(t) a) Use the Quadratic Regression feature of your calculator to generate a mathematical model for this situation. Use function notation with appropriate variables. Round coefficients to two decimal places. b) Use your model to predict the height of the golf ball at 5 seconds. Round your answer to two decimal places. How does this compare to the value in the data table? c) Use your model to determine the maximum height of the golf ball. Round your answer to two decimal places. d) Use your model to determine how long it will take the golf ball to hit the ground. Round your answer to two decimal places. e) Use your model to determine the practical domain and practical range for this situation. Practical Domain Practical Range 248
31 Problem 36 YOU TRY Quadratic Regression The annual unemployment percentage rate data for the Phoenix-Mesa-Scottsdale area from 2008 to 2014 are shown in the table below (Source: Year Unemployment Rate a) Use the quadratic regression feature of your calculator to write a function, U(t), that represents the given data with t in years since Use function notation and appropriate variables. Round coefficients to two decimal places. b) Use your graphing calculator to generate a scatterplot of the data and regression line on the same screen. You must use an appropriate viewing window. In the space below, draw what you see on your calculator screen, and write down the viewing window you used. Xmin = Xmax = Ymin = Ymax = c) Using your function from part a), identify the vertical intercept of U(t). Write it as an ordered pair and interpret its meaning in a complete sentence. Round answers to two decimals. d) Identify the vertex of the function found in part a) and interpret its meaning in a complete sentence. Round answers to two decimals. 249
Lesson 8 Introduction to Quadratic Functions
Lesson 8 Introduction to Quadratic Functions We are leaving exponential and logarithmic functions behind and entering an entirely different world. As you work through this lesson, you will learn to identify
More informationLesson 6 - Practice Problems
Lesson 6 - Practice Problems Section 6.1: Characteristics of Quadratic Functions 1. For each of the following quadratic functions: Identify the coefficients a, b and c. Determine if the parabola opens
More informationLesson 8 Practice Problems
Name: Date: Lesson 8 Section 8.1: Characteristics of Quadratic Functions 1. For each of the following quadratic functions: Identify the coefficients a, b, c Determine if the parabola opens up or down and
More informationQuadratics Functions: Review
Quadratics Functions: Review Name Per Review outline Quadratic function general form: Quadratic function tables and graphs (parabolas) Important places on the parabola graph [see chart below] vertex (minimum
More informationEXERCISE SET 10.2 MATD 0390 DUE DATE: INSTRUCTOR
EXERCISE SET 10. STUDENT MATD 090 DUE DATE: INSTRUCTOR You have studied the method known as "completing the square" to solve quadratic equations. Another use for this method is in transforming the equation
More information3.1 INTRODUCTION TO THE FAMILY OF QUADRATIC FUNCTIONS
3.1 INTRODUCTION TO THE FAMILY OF QUADRATIC FUNCTIONS Finding the Zeros of a Quadratic Function Examples 1 and and more Find the zeros of f(x) = x x 6. Solution by Factoring f(x) = x x 6 = (x 3)(x + )
More informationUnit: Quadratic Functions
Unit: Quadratic Functions Learning increases when you have a goal to work towards. Use this checklist as guide to track how well you are grasping the material. In the center column, rate your understand
More informationUnit 6 Quadratic Functions
Unit 6 Quadratic Functions 12.1 & 12.2 Introduction to Quadratic Functions What is A Quadratic Function? How do I tell if a Function is Quadratic? From a Graph The shape of a quadratic function is called
More informationQuadratic Functions CHAPTER. 1.1 Lots and Projectiles Introduction to Quadratic Functions p. 31
CHAPTER Quadratic Functions Arches are used to support the weight of walls and ceilings in buildings. Arches were first used in architecture by the Mesopotamians over 4000 years ago. Later, the Romans
More informationMAFS Algebra 1. Quadratic Functions. Day 17 - Student Packet
MAFS Algebra 1 Quadratic Functions Day 17 - Student Packet Day 17: Quadratic Functions MAFS.912.F-IF.3.7a, MAFS.912.F-IF.3.8a I CAN graph a quadratic function using key features identify and interpret
More informationLesson 10 Rational Functions and Equations
Lesson 10 Rational Functions and Equations Lesson 10 Rational Functions and Equations In this lesson, you will embark on a study of rational functions. Rational functions look different because they are
More informationY. Butterworth Lehmann & 9.2 Page 1 of 11
Pre Chapter 9 Coverage Quadratic (2 nd Degree) Form a type of graph called a parabola Form of equation we'll be dealing with in this chapter: y = ax 2 + c Sign of a determines opens up or down "+" opens
More informationSection 7.2 Characteristics of Quadratic Functions
Section 7. Characteristics of Quadratic Functions A QUADRATIC FUNCTION is a function of the form " # $ N# 1 & ;# & 0 Characteristics Include:! Three distinct terms each with its own coefficient:! An x
More informationUnit 2: Functions and Graphs
AMHS Precalculus - Unit 16 Unit : Functions and Graphs Functions A function is a rule that assigns each element in the domain to exactly one element in the range. The domain is the set of all possible
More informationLesson 8 - Practice Problems
Lesson 8 - Practice Problems Section 8.1: A Case for the Quadratic Formula 1. For each quadratic equation below, show a graph in the space provided and circle the number and type of solution(s) to that
More informationLesson 1: Analyzing Quadratic Functions
UNIT QUADRATIC FUNCTIONS AND MODELING Lesson 1: Analyzing Quadratic Functions Common Core State Standards F IF.7 F IF.8 Essential Questions Graph functions expressed symbolically and show key features
More informationLesson 12 Course Review
In this lesson, we will review the topics and applications from Lessons 1-11. We will begin with a review of the different types of functions, and then apply each of them to a set of application problems.
More informationBut a vertex has two coordinates, an x and a y coordinate. So how would you find the corresponding y-value?
We will work with the vertex, orientation, and x- and y-intercepts of these functions. Intermediate algebra Class notes More Graphs of Quadratic Functions (section 11.6) In the previous section, we investigated
More informationNO CALCULATOR ON ANYTHING EXCEPT WHERE NOTED
Algebra II (Wilsen) Midterm Review NO CALCULATOR ON ANYTHING EXCEPT WHERE NOTED Remember: Though the problems in this packet are a good representation of many of the topics that will be on the exam, this
More informationLearning Packet THIS BOX FOR INSTRUCTOR GRADING USE ONLY. Mini-Lesson is complete and information presented is as found on media links (0 5 pts)
Learning Packet Student Name Due Date Class Time/Day Submission Date THIS BOX FOR INSTRUCTOR GRADING USE ONLY Mini-Lesson is complete and information presented is as found on media links (0 5 pts) Comments:
More informationChapter 3 Practice Test
1. Complete parts a c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex.
More informationUNIT 3 EXPRESSIONS AND EQUATIONS Lesson 3: Creating Quadratic Equations in Two or More Variables
Guided Practice Example 1 Find the y-intercept and vertex of the function f(x) = 2x 2 + x + 3. Determine whether the vertex is a minimum or maximum point on the graph. 1. Determine the y-intercept. The
More informationUNIT 5 QUADRATIC FUNCTIONS Lesson 6: Analyzing Quadratic Functions Instruction
Prerequisite Skills This lesson requires the use of the following skills: factoring quadratic expressions finding the vertex of a quadratic function Introduction We have studied the key features of the
More informationLesson 4 Exponential Functions I
Lesson 4 Exponential Functions I Lesson 4 Exponential Functions I Exponential functions play a major role in our lives. Population growth and disease processes are real-world problems that involve exponential
More informationFinal Exam Review Algebra Semester 1
Final Exam Review Algebra 015-016 Semester 1 Name: Module 1 Find the inverse of each function. 1. f x 10 4x. g x 15x 10 Use compositions to check if the two functions are inverses. 3. s x 7 x and t(x)
More information2.3. Graphing Calculators; Solving Equations and Inequalities Graphically
2.3 Graphing Calculators; Solving Equations and Inequalities Graphically Solving Equations and Inequalities Graphically To do this, we must first draw a graph using a graphing device, this is your TI-83/84
More informationSection 9.3 Graphing Quadratic Functions
Section 9.3 Graphing Quadratic Functions A Quadratic Function is an equation that can be written in the following Standard Form., where a 0. Every quadratic function has a U-shaped graph called a. If the
More informationProperties of Graphs of Quadratic Functions
H e i g h t (f t ) Lesson 2 Goal: Properties of Graphs of Quadratic Functions Identify the characteristics of graphs of quadratic functions: Vertex Intercepts Domain and Range Axis of Symmetry and use
More informationQuadratic Functions. *These are all examples of polynomial functions.
Look at: f(x) = 4x-7 f(x) = 3 f(x) = x 2 + 4 Quadratic Functions *These are all examples of polynomial functions. Definition: Let n be a nonnegative integer and let a n, a n 1,..., a 2, a 1, a 0 be real
More informationSection 6.2: Properties of Graphs of Quadratic Functions. Vertex:
Section 6.2: Properties of Graphs of Quadratic Functions determine the vertex of a quadratic in standard form sketch the graph determine the y intercept, x intercept(s), the equation of the axis of symmetry,
More informationAlgebra II Quadratic Functions
1 Algebra II Quadratic Functions 2014-10-14 www.njctl.org 2 Ta b le o f C o n te n t Key Terms click on the topic to go to that section Explain Characteristics of Quadratic Functions Combining Transformations
More information1. a. After inspecting the equation for the path of the winning throw, which way do you expect the parabola to open? Explain.
Name Period Date More Quadratic Functions Shot Put Activity 3 Parabolas are good models for a variety of situations that you encounter in everyday life. Example include the path of a golf ball after it
More informationLesson 3.1 Vertices and Intercepts. Important Features of Parabolas
Lesson 3.1 Vertices and Intercepts Name: _ Learning Objective: Students will be able to identify the vertex and intercepts of a parabola from its equation. CCSS.MATH.CONTENT.HSF.IF.C.7.A Graph linear and
More informationAlgebra II Quadratic Functions and Equations - Extrema Unit 05b
Big Idea: Quadratic Functions can be used to find the maximum or minimum that relates to real world application such as determining the maximum height of a ball thrown into the air or solving problems
More informationEXAMPLE. 1. Enter y = x 2 + 8x + 9.
VI. FINDING INTERCEPTS OF GRAPHS As we have seen, TRACE allows us to find a specific point on the graph. Thus TRACE can be used to solve a number of important problems in algebra. For example, it can be
More informationIt is than the graph of y= x if a > 1.
Chapter 8 Quadratic Functions and Equations Name: Instructor: 8.1 Quadratic Functions and Their Graphs Graphs of Quadratic Functions Basic Transformations of Graphs More About Graphing Quadratic Functions
More informationx 2 + 8x - 12 = 0 Aim: To review for Quadratic Function Exam #1 Homework: Study Review Materials
Aim: To review for Quadratic Function Exam #1 Homework: Study Review Materials Do Now - Solve using any strategy. If irrational, express in simplest radical form x 2 + 8x - 12 = 0 Review Topic Index 1.
More informationLesson 11 Rational Functions
Lesson 11 Rational Functions In this lesson, you will embark on a study of rational functions. These may be unlike any function you have ever seen. Rational functions look different because they are in
More informationMAC Rev.S Learning Objectives. Learning Objectives (Cont.) Module 4 Quadratic Functions and Equations
MAC 1140 Module 4 Quadratic Functions and Equations Learning Objectives Upon completing this module, you should be able to 1. understand basic concepts about quadratic functions and their graphs.. complete
More informationGraphing Absolute Value Functions
Graphing Absolute Value Functions To graph an absolute value equation, make an x/y table and plot the points. Graph y = x (Parent graph) x y -2 2-1 1 0 0 1 1 2 2 Do we see a pattern? Desmos activity: 1.
More informationPR3 & PR4 CBR Activities Using EasyData for CBL/CBR Apps
Summer 2006 I2T2 Process Page 23. PR3 & PR4 CBR Activities Using EasyData for CBL/CBR Apps The TI Exploration Series for CBR or CBL/CBR books, are all written for the old CBL/CBR Application. Now we can
More informationTypes of Functions Here are six common types of functions and examples of each. Linear Quadratic Absolute Value Square Root Exponential Reciprocal
Topic 2.0 Review Concepts What are non linear equations? Student Notes Unit 2 Non linear Equations Types of Functions Here are six common types of functions and examples of each. Linear Quadratic Absolute
More informationAlgebra 1 Notes Quarter
Algebra 1 Notes Quarter 3 2014 2015 Name: ~ 1 ~ Table of Contents Unit 9 Exponent Rules Exponent Rules for Multiplication page 6 Negative and Zero Exponents page 10 Exponent Rules Involving Quotients page
More informationMINI LESSON. Lesson 1a Introduction to Functions
MINI LESSON Lesson 1a Introduction to Functions Lesson Objectives: 1. Define FUNCTION 2. Determine if data sets, graphs, statements, or sets of ordered pairs define functions 3. Use proper function notation
More informationToday is the last day to register for CU Succeed account AND claim your account. Tuesday is the last day to register for my class
Today is the last day to register for CU Succeed account AND claim your account. Tuesday is the last day to register for my class Back board says your name if you are on my roster. I need parent financial
More informationAlgebra II Chapter 4: Quadratic Functions and Factoring Part 1
Algebra II Chapter 4: Quadratic Functions and Factoring Part 1 Chapter 4 Lesson 1 Graph Quadratic Functions in Standard Form Vocabulary 1 Example 1: Graph a Function of the Form y = ax 2 Steps: 1. Make
More informationUnit #3: Quadratic Functions Lesson #13: The Almighty Parabola. Day #1
Algebra I Unit #3: Quadratic Functions Lesson #13: The Almighty Parabola Name Period Date Day #1 There are some important features about the graphs of quadratic functions we are going to explore over the
More informationSample: Do Not Reproduce QUAD4 STUDENT PAGES. QUADRATIC FUNCTIONS AND EQUATIONS Student Pages for Packet 4: Quadratic Functions and Applications
Name Period Date QUADRATIC FUNCTIONS AND EQUATIONS Student Pages for Packet 4: Quadratic Functions and Applications QUAD 4.1 Vertex Form of a Quadratic Function 1 Explore how changing the values of h and
More informationSection 1.1: Functions and Models
Section 1.1: Functions and Models Definition: A function is a rule that assigns to each element of one set (called the domain) exactly one element of a second set (called the range). A function can be
More informationQuadratic Functions Dr. Laura J. Pyzdrowski
1 Names: (8 communication points) About this Laboratory A quadratic function in the variable x is a polynomial where the highest power of x is 2. We will explore the domains, ranges, and graphs of quadratic
More informationSlide 2 / 222. Algebra II. Quadratic Functions
Slide 1 / 222 Slide 2 / 222 Algebra II Quadratic Functions 2014-10-14 www.njctl.org Slide 3 / 222 Table of Contents Key Terms Explain Characteristics of Quadratic Functions Combining Transformations (review)
More informationUNIT 5 QUADRATIC FUNCTIONS Lesson 7: Building Functions Instruction
Prerequisite Skills This lesson requires the use of the following skills: multiplying linear expressions factoring quadratic equations finding the value of a in the vertex form of a quadratic equation
More information1.1 - Functions, Domain, and Range
1.1 - Functions, Domain, and Range Lesson Outline Section 1: Difference between relations and functions Section 2: Use the vertical line test to check if it is a relation or a function Section 3: Domain
More informationSemester 2 Review Problems will be sectioned by chapters. The chapters will be in the order by which we covered them.
Semester 2 Review Problems will be sectioned by chapters. The chapters will be in the order by which we covered them. Chapter 9 and 10: Right Triangles and Trigonometric Ratios 1. The hypotenuse of a right
More informationAlgebra 2B CH 5. WYNTK & TEST Algebra 2B What You Need to Know , Test
Algebra 2B CH 5 NAME: WYNTK 5.1 5.3 & 5.7 5.8 TEST DATE: HOUR: Algebra 2B What You Need to Know 5.1 5.3, 5.7-5.8 Test A2.5.1.2 Be able to use transformations to graph quadratic functions and answer questions.
More informationChpt 1. Functions and Graphs. 1.1 Graphs and Graphing Utilities 1 /19
Chpt 1 Functions and Graphs 1.1 Graphs and Graphing Utilities 1 /19 Chpt 1 Homework 1.1 14, 18, 22, 24, 28, 42, 46, 52, 54, 56, 78, 79, 80, 82 2 /19 Objectives Functions and Graphs Plot points in the rectangular
More informationLesson 8: Graphs and Graphing Linear Equations
A critical skill required for the study of algebra is the ability to construct and interpret graphs. In this lesson we will learn how the Cartesian plane is used for constructing graphs and plotting data.
More informationTest 3 review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Test 3 review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Approximate the coordinates of each turning point by graphing f(x) in the standard viewing
More informationSection 4.4 Quadratic Functions in Standard Form
Section 4.4 Quadratic Functions in Standard Form A quadratic function written in the form y ax bx c or f x ax bx c is written in standard form. It s not right to write a quadratic function in either vertex
More informationSolving Simple Quadratics 1.0 Topic: Solving Quadratics
Ns Solving Simple Quadratics 1.0 Topic: Solving Quadratics Date: Objectives: SWBAT (Solving Simple Quadratics and Application dealing with Quadratics) Main Ideas: Assignment: Square Root Property If x
More informationFalling Balls. Names: Date: About this Laboratory
Falling Balls Names: Date: About this Laboratory In this laboratory,1 we will explore quadratic functions and how they relate to the motion of an object that is dropped from a specified height above ground
More informationQUADRATICS Graphing Quadratic Functions Common Core Standard
H Quadratics, Lesson 6, Graphing Quadratic Functions (r. 2018) QUADRATICS Graphing Quadratic Functions Common Core Standard Next Generation Standard F-IF.B.4 For a function that models a relationship between
More informationTransformations with Quadratic Functions KEY
Algebra Unit: 05 Lesson: 0 TRY THIS! Use a calculator to generate a table of values for the function y = ( x 3) + 4 y = ( x 3) x + y 4 Next, simplify the function by squaring, distributing, and collecting
More informationThings to Know for the Algebra I Regents
Types of Numbers: Real Number: any number you can think of (integers, rational, irrational) Imaginary Number: square root of a negative number Integers: whole numbers (positive, negative, zero) Things
More informationUnit 2-2: Writing and Graphing Quadratics NOTE PACKET. 12. I can use the discriminant to determine the number and type of solutions/zeros.
Unit 2-2: Writing and Graphing Quadratics NOTE PACKET Name: Period Learning Targets: Unit 2-1 12. I can use the discriminant to determine the number and type of solutions/zeros. 1. I can identify a function
More information10.3 vertex and max values with comparing functions 2016 ink.notebook. March 14, Vertex and Max Value & Page 101.
10.3 vertex and max values with comparing functions 2016 ink.notebook Page 101 Page 102 10.3 Vertex and Value and Comparing Functions Algebra: Transformations of Functions Page 103 Page 104 Lesson Objectives
More informationProperties of Quadratic functions
Name Today s Learning Goals: #1 How do we determine the axis of symmetry and vertex of a quadratic function? Properties of Quadratic functions Date 5-1 Properties of a Quadratic Function A quadratic equation
More informationLesson 8: Graphs and Graphing Linear Equations
In this chapter, we will begin looking at the relationships between two variables. Typically one variable is considered to be the input, and the other is called the output. The input is the value that
More informationGraph Quadratic Functions Using Properties *
OpenStax-CNX module: m63466 1 Graph Quadratic Functions Using Properties * OpenStax This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 By the end of this
More informationUnit 1 Quadratic Functions
Unit 1 Quadratic Functions This unit extends the study of quadratic functions to include in-depth analysis of general quadratic functions in both the standard form f ( x) = ax + bx + c and in the vertex
More information9.1: GRAPHING QUADRATICS ALGEBRA 1
9.1: GRAPHING QUADRATICS ALGEBRA 1 OBJECTIVES I will be able to graph quadratics: Given in Standard Form Given in Vertex Form Given in Intercept Form What does the graph of a quadratic look like? https://www.desmos.com/calculator
More informationALGEBRA 1 NOTES. Quarter 3. Name: Block
2016-2017 ALGEBRA 1 NOTES Quarter 3 Name: Block Table of Contents Unit 8 Exponent Rules Exponent Rules for Multiplication page 4 Negative and Zero Exponents page 8 Exponent Rules Involving Quotients page
More information2. The diagram shows part of the graph of y = a (x h) 2 + k. The graph has its vertex at P, and passes through the point A with coordinates (1, 0).
Quadratics Vertex Form 1. Part of the graph of the function y = d (x m) + p is given in the diagram below. The x-intercepts are (1, 0) and (5, 0). The vertex is V(m, ). (a) Write down the value of (i)
More informationLet s review some things we learned earlier about the information we can gather from the graph of a quadratic.
Section 6: Quadratic Equations and Functions Part 2 Section 6 Topic 1 Observations from a Graph of a Quadratic Function Let s review some things we learned earlier about the information we can gather from
More information6.4 Vertex Form of a Quadratic Function
6.4 Vertex Form of a Quadratic Function Recall from 6.1 and 6.2: Standard Form The standard form of a quadratic is: f(x) = ax 2 + bx + c or y = ax 2 + bx + c where a, b, and c are real numbers and a 0.
More informationKEY Algebra: Unit 10 Graphing Quadratic Equations & other Relations
Name: KEY Algebra: Unit 10 Graphing Quadratic Equations & other Relations Date: Test Bank Part I: Answer all 15 questions in this part. Each correct answer will receive credits. No partial credit will
More informationName: Chapter 7 Review: Graphing Quadratic Functions
Name: Chapter Review: Graphing Quadratic Functions A. Intro to Graphs of Quadratic Equations: = ax + bx+ c A is a function that can be written in the form = ax + bx+ c where a, b, and c are real numbers
More informationYou should be able to plot points on the coordinate axis. You should know that the the midpoint of the line segment joining (x, y 1 1
Name GRAPHICAL REPRESENTATION OF DATA: You should be able to plot points on the coordinate axis. You should know that the the midpoint of the line segment joining (x, y 1 1 ) and (x, y ) is x1 x y1 y,.
More informationChapter 6 Practice Test
MPM2D Mr. Jensen Chapter 6 Practice Test Name: Standard Form 2 y= ax + bx+ c Factored Form y= a( x r)( x s) Vertex Form 2 y= a( x h) + k Quadratic Formula ± x = 2 b b 4ac 2a Section 1: Multiply Choice
More informationII. Functions. 61. Find a way to graph the line from the problem 59 on your calculator. Sketch the calculator graph here, including the window values:
II Functions Week 4 Functions: graphs, tables and formulas Problem of the Week: The Farmer s Fence A field bounded on one side by a river is to be fenced on three sides so as to form a rectangular enclosure
More informationCollege Algebra. Quadratic Functions and their Graphs. Dr. Nguyen October 12, Department of Mathematics UK
College Algebra Quadratic Functions and their Graphs Dr. Nguyen nicholas.nguyen@uky.edu Department of Mathematics UK October 12, 2018 Agenda Quadratic functions and their graphs Parabolas and vertices
More informationNOTES Linear Equations
NOTES Linear Equations Linear Parent Function Linear Parent Function the equation that all other linear equations are based upon (y = x) Horizontal and Vertical Lines (HOYY VUXX) V vertical line H horizontal
More informationDo you need a worksheet or a copy of the teacher notes? Go to
Name Period Day Date Assignment (Due the next class meeting) Wednesday Thursday Friday Monday Tuesday Wednesday Thursday Friday Monday Tuesday Wednesday Thursday Friday Monday Tuesday Wednesday Thursday
More informationQuadratic Functions, Part 1
Quadratic Functions, Part 1 A2.F.BF.A.1 Write a function that describes a relationship between two quantities. A2.F.BF.A.1a Determine an explicit expression, a recursive process, or steps for calculation
More informationSpecific Objectives Students will understand that that the family of equation corresponds with the shape of the graph. Students will be able to create a graph of an equation by plotting points. In lesson
More informationFoundations of Math II
Foundations of Math II Unit 6b: Toolkit Functions Academics High School Mathematics 6.6 Warm Up: Review Graphing Linear, Exponential, and Quadratic Functions 2 6.6 Lesson Handout: Linear, Exponential,
More information8-4 Transforming Quadratic Functions
8-4 Transforming Quadratic Functions Warm Up Lesson Presentation Lesson Quiz Algebra 1 Warm Up For each quadratic function, find the axis of symmetry and vertex, and state whether the function opens upward
More informationMore Ways to Solve & Graph Quadratics The Square Root Property If x 2 = a and a R, then x = ± a
More Ways to Solve & Graph Quadratics The Square Root Property If x 2 = a and a R, then x = ± a Example: Solve using the square root property. a) x 2 144 = 0 b) x 2 + 144 = 0 c) (x + 1) 2 = 12 Completing
More informationWHAT YOU SHOULD LEARN
GRAPHS OF EQUATIONS WHAT YOU SHOULD LEARN Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs of equations. Find equations of and sketch graphs of
More information1. Answer: x or x. Explanation Set up the two equations, then solve each equation. x. Check
Thinkwell s Placement Test 5 Answer Key If you answered 7 or more Test 5 questions correctly, we recommend Thinkwell's Algebra. If you answered fewer than 7 Test 5 questions correctly, we recommend Thinkwell's
More information6 Using Technology Wisely
6 Using Technology Wisely Concepts: Advantages and Disadvantages of Graphing Calculators How Do Calculators Sketch Graphs? When Do Calculators Produce Incorrect Graphs? The Greatest Integer Function Graphing
More informationMath 1113 Notes - Quadratic Functions
Math 1113 Notes - Quadratic Functions Philippe B. Laval Kennesaw State University September 7, 000 Abstract This handout is a review of quadratic functions. It includes a review of the following topics:
More informationQuadratic Functions (Section 2-1)
Quadratic Functions (Section 2-1) Section 2.1, Definition of Polynomial Function f(x) = a is the constant function f(x) = mx + b where m 0 is a linear function f(x) = ax 2 + bx + c with a 0 is a quadratic
More informationPolynomial and Rational Functions. Copyright Cengage Learning. All rights reserved.
2 Polynomial and Rational Functions Copyright Cengage Learning. All rights reserved. 2.1 Quadratic Functions Copyright Cengage Learning. All rights reserved. What You Should Learn Analyze graphs of quadratic
More informationMid-Chapter Quiz: Lessons 4-1 through 4-4
1. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex for f (x) = 2x 2 + 8x 3. Then graph the function by making a table of values. 2. Determine whether f (x)
More informationQUESTIONS 1 10 MAY BE DONE WITH A CALCULATOR QUESTIONS ARE TO BE DONE WITHOUT A CALCULATOR. Name
QUESTIONS 1 10 MAY BE DONE WITH A CALCULATOR QUESTIONS 11 5 ARE TO BE DONE WITHOUT A CALCULATOR Name 2 CALCULATOR MAY BE USED FOR 1-10 ONLY Use the table to find the following. x -2 2 5-0 7 2 y 12 15 18
More informationWriting Equivalent Forms of Quadratic Functions Adapted from Walch Education
Writing Equivalent Forms of Quadratic Functions Adapted from Walch Education Recall The standard form, or general form, of a quadratic function is written as f(x) = ax 2 + bx + c, where a is the coefficient
More informationpractice: quadratic functions [102 marks]
practice: quadratic functions [102 marks] A quadratic function, f(x) = a x 2 + bx, is represented by the mapping diagram below. 1a. Use the mapping diagram to write down two equations in terms of a and
More information3. Solve the following. Round to the nearest thousandth.
This review does NOT cover everything! Be sure to go over all notes, homework, and tests that were given throughout the semester. 1. Given g ( x) i, h( x) x 4x x, f ( x) x, evaluate the following: a) f
More informationCourse Number 432/433 Title Algebra II (A & B) H Grade # of Days 120
Whitman-Hanson Regional High School provides all students with a high- quality education in order to develop reflective, concerned citizens and contributing members of the global community. Course Number
More information