WHAT ARE THE PARTS OF A QUADRATIC?

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1 4.1 Introduction to Quadratics and their Graphs Standard Form of a Quadratic: y ax bx c or f x ax bx c. ex. y x. Every function/graph in the Quadratic family originates from the parent function: While we will not use a table to graph quadratics, let s start off using the table to see what the quadratic parent looks like: y x x y What does the parent graph look like? What do we call these types of graphs? WHAT ARE THE PARTS OF A QUADRATIC? a In standard form for a quadratic, its a value will tell you whether your graph will open up (think smiley face) or down (think frown). If a > 0, then the quadratic will open up. If a < 0, then the quadratic will open down. The Vertex is the lowest or highest point (x, y) on a parabola/quadratic. If the vertex is the highest point on the graph, we call it a maximum. If the vertex is the lowest point on the graph, we call it a minimum. Vertex Because it is a point, we need to know how to find the x and y coordinates. To find the x b coordinate, we use the formula x (memorize) a Once you know the x coordinate of the vertex, simply plug it back into the original equation and you will have the y coordinate of the vertex. The axis of symmetry is an imaginary vertical line that divides the parabola into two equal parts. It only passes through one point on the entire graph, the vertex. Axis of Symmetry Since it is a vertical line, its equation is of the form x =, just like every vertical line. b The equation is x, the same equation we just saw for the x coordinate of the vertex. So a the x coordinate of the vertex and the axis of symmetry will always be the same number!!

2 c When looking at the standard form for a quadratic, its c value is the y-intercept. Remember the y-intercept is the point where the parabola or quadratic crosses the y-axis, which takes the form (0, c). The purpose of finding c and writing it as a point is to help us find additional points on the graph! Directions: Fill in the following values to graph each quadratic. You will have to show all work for credit on tests and quizzes! 1. f x x 8x. f x x 5 a = Opens x y a = Opens x y c = c = Y-intercept, Y-intercept, Vertex, Vertex, Equation for Axis of Symmetry Maximum or Minimum Max/Min Value Equation for Axis of Symmetry Maximum or Minimum Max/Min Value

3 4. Graph Quadratics in Vertex and Intercept Forms Vertex Form of a Quadratic: ex. y a x h k Benefits to VERTEX FORM: It s much easier to find the vertex (h, k). For h, take the opposite of what you see in the ( ). For k, look at what s being added or subtracted from the outside of the ( ). a allows us to see direction of opening, in the same way that a did for standard form. Once you know your vertex and a value, everything else is the same as yesterday!! 1. y 4x 3 1. y x 3 5 a = Opens x y a = Opens x y Vertex, Vertex, Equation for Axis of Symmetry Equation for Axis of Symmetry Maximum or Minimum Maximum or Minimum Max/Min Value Max/Min Value A POINT TO PONDER: We know that standard form and vertex form are two different forms that create the same thing. That being stated, how would you take an equation in vertex form and turn it into an equation in standard form? Can you? y 4 x 3

4 Intercept Form of a Quadratic: y a x px q ex. x-intercepts (roots, zeros) We will start creating these graphs by finding the x-intercepts/roots/zeros. To do so, we will take each, set it equal to 0 and solve. X-intercepts always take the form x, 0. Basically, the intercepts provide us with two points on our parabola. They are special because they are always on the x-axis. Vertex 1. y x 5x 1 The x-coordinate of the vertex is always halfway between the two x-intercepts. So to p q find it, we will take the midpoint of our two intercepts. x (memorize) Once you know the x coordinate of the vertex, simply plug it back into the original equation and you will have the y coordinate of the vertex. a = Opens Intercepts,, Vertex, Equation for Axis of Symmetry Maximum or Minimum Max/Min Value

5 QUADRATIC EQUATION 4.3 Solving Quadratic Equations by Factoring A quadratic equation is when we take a quadratic in standard form y ax bx c and make the y-value 0, in other words: 0 ax bx c Our goal also changes. The purpose changes from simply graphing to solving algebraically. Because it is impossible to combine like terms and solve a quadratic like we would a linear equation, we must find another method. x 5x 6 0 GOAL GREATEST COMMON FACTOR BINOMIALS: DIFFERENCE OF PERFECT SQUARES Factor the expression: Factor and solve to find the solutions: Factor the expression: Factor and solve to find the solutions: x 19x x 6x 0 5x 1 9x 64 TRINOMIALS: Factor the expression: x 3x 54 Solve the equation to find the solutions: x 8x 4 44 Find the roots of the equation: Factors: Solutions:

6 Find the zeros of the function by rewriting the function in intercept form: y x 6x 7 Zeros: If the Area of a rectangle = 84, find the value of x. Problem Solving: A city s skate park is a rectangle 100 feet long by 50 feet wide. The city wants to triple the area of the skate park by adding the same distance x to the length and width. What are the new dimensions of the skate park?

7 4.4 Solve by Factoring (continued) BINOMIALS: GREATEST COMMON FACTOR AND DIFFERENCE OF PERFECT SQUARES Factor the expression: 75x 3 Solve the equation: 3y 18y TRINOMIALS: with coefficients on TRINOMIALS x that may have a GCF attached!! Factor the expression: 3x 7x 0 Factor the expression: 10x 6x 1 Solve the equation: x 4x 30 0 Solve the equation: z 13z 1 5z 4

8 Find the zeros of the function by rewriting the function in intercept form. y x 11x 15 f ( x) x 8. 5x

9 4.6 Square Roots, Laundry, and Imaginary Numbers It s Friday night and you really want to go out with your friends. Your mom says you have to fold all the socks in the laundry basket before you can go. As you fold the socks, the pairs come out of the basket and the singles socks stay inside. When you look in the basket you find: 5 white socks, 6 blue socks, 4 green socks, 3 pink socks, 1 red sock, and yellow socks. Let s write it like this: w b g p r y So you get organized: Remember We are pulling out the number or pairs or each type (what type, how many pairs). Pairs will be pulled to the outside and the number of singles is left on the inside. So our organized laundry basket would look like this: Use the root to draw arrows to the vocabulary term: 1. Index. Radical 3. Radicand 4. Number in front 4 64 Square Roots: What are we being asked to do with the radicand?

10 x y 3. 4 Multiplying a Radical by another Radical You can multiply a radical by another radical as long as they both have the same index. To multiply a radical by another radical, multiply the numbers in front, multiply the radicands, and keep the index the same. Then check your answer to see if anything can be pulled out. If not, it is in simplified form Multiplying a Radical by a Number You can multiply any radical by any number. To multiply a radical by a number, simply write the two next to each other, as being next to implies multiplication. For example, in this distributing problem, the first part is multiplying a radical and a number

11 Radicals in the Denominator One of the fundamental rules of radicals is that a radical cannot be left in the denominator of a fraction. To rationalize a radical is to multiply both the numerator and denominator by a quantity that will, with simplification, convert the denominator from a radical to an integer. For example, when given root 6 in the denominator, we are looking for what we can multiply both the numerator and the denominator by in order to create a quantity in the denominator that can be completely pulled out x 3 8 This one looks different!

12 4.6 (CONTINUED) IMAGINARY NUMBERS IMAGINARY NUMBER Where do they fit in? Are they really imaginary? Redefine the number system COMPLEX NUMBERS a bi Real Numbers Imaginary Numbers 4 5i Basic Properties i 1 i 1 Simplifying: 1. Break down the radicand. Pull out the imaginary part x. 4 7x Pull out any others pairs. 4. Write the simplified answer. Adding and Subtracting with i AND complex numbers: i (1117i) i 14 3i 1. Distribute any negatives first.. Determine the like terms 3. Add or subtract the like terms only. 4. Make sure your answer is written in standard form. Use the Powers of I to write the expressions in Standard Form: 1. Rewrite everything in terms of i i. Insert what you know, i 1 3. Simplify the answer.

13 Multiplying Complex Numbers: 7. 4i 7 15i 8. (3 5 i)(3 5 i) 1. FOIL or distribute everything in first ( ) to everything in the second ( ).. Convert all i 1 3. Combine like terms only. Dividing with i: 9. 5i i 1. i cannot be left in the denominator. 3i 5i. Decide how to get i out of the denominator (You may have to use a conjugate!) 3. Multiply by both the numerator and denominator. 4. Simplify as much as possible i 1. 7 i

14 4.5 Using the Square Root Property GOAL Use square roots to find the zeros/roots of a quadratic equation. QUESTION: WHAT IS THE RELATIONSHIP BETWEEN Method: 1. Isolate the x AND?? HOW CAN THAT HELP US GET A VARIABLE ALONE? TODAY S METHOD: FIND THE ROOTS OR ZEROS USING SQUARE ROOTS. x or x.. Use square roots to cancel out the squaring. 3. Because these are equations, taking the square root of a number can produce both a positive or negative result. So, don t forget to insert the 4. Simplify the root as much as possible. 5. Solve what remains. x c (x 4) 18 = 10 x 5. 3x Look at #-5 with the and make a mental note of how we solved them!!

15 PERFECT SQUARE TRINOMIAL A perfect square trinomial is a trinomial ax bx c 0 that can be factored into a perfect square x 0 or x 0 Factor to create a perfect square trinomial. 6. x 10x 5 7. x 1x 1 4 METHOD If we can create perfect square trinomials, then we can solve for x using what we know about squaring and square rooting. Factor to create the perfect square trinomial. Then solve for x using the Square Root Property. 8. x 10x x 1x 5 Modeling Dropped Objects When an object is dropped, its height h (in feet) above the ground after t seconds can be modeled by the equation: where h 0 is its initial height (in feet). h 16t h0 10. For a science competition, students must design a container that prevents an egg from breaking when dropped from a height of 10 feet. How long does the container take to hit the ground? Use a calculator and round to the nearest tenth.

16 4.7 Solve a Quadratic by Completing the Perfect Square Trinomial Solve for x. x 6x 9 5 WHAT IF IT S NOT A PERFECT SQUARE TRINOMIAL? CAN WE STILL USE THIS METHOD? THE ANSWER: No, well kind of. If it s not a perfect square trinomial, we re going to have to make it into a perfect square trinomial before we can use this method. Find the missing c value that will make it a perfect square trinomial. Then factor it into a perfect square. 1. x 16x c. x 15x c 3. x. 6 x c C = Factored Form: C = Factored Form: C = Factored Form: THERE SEEMS TO BE A RELATIONSHIP BETWEEN B AND C WHAT IS IT? Given any trinomial, as long as we have a b value, we can create a perfect square trinomial. If we take our b value, divide it by, then square it, we will always get our needed c value. To easily factor this, look back at what you got when you divided b by. This is what every trinomial will factor into. HOW CAN THIS HELP US? 4. x 16x 39 0

17 5. x 4x 0 0 Steps: 1. Before we can do any work on the equation, the a value must be 1. If it is not 1, divide everything by that number.. Move the unhelpful c value to the other side. 3. Now create the new c value. Complete the square on b by: a. divide b by b. Square it 4. Balance the equation. 5. Factor the perfect square trinomial. 6. Solve for x. Given the quadratic in Standard Form, convert it to Vertex Form. Then, identify the vertex. 6. x 16x 1 y 7. y x 5x 3 Vertex Form: Vertex:, Vertex Form: Vertex:,

18 Converting from STANDARD FORM to VERTEX FORM Given: y x 8x 3 Steps: y x 8x 3 1. Notice that the given is not a perfect square trinomial. We need a perfect square trinomial for vertex form. y x 8x 3. Check to make sure that a is 1. If a is not 1, factor it from each number in y 3 x 8x 3. Isolate the x s. the ( ). Remember, we used to divide by a. Why is that no longer an option? y 3 x 8x Complete the square. y 3 16 x 8x Balance the equation. ** If you factored out a, mentally redistribute to find out what changed on the right side. Once you determine this, balance the left side. x y 19 4 y x Factor the ( ). Simplify the left side. 8. Isolate y. 4.8 Use the Quadratic Formula and the Discriminant Often, the simplest way to solve or find the zeros/roots of a quadratic is to solve by factoring. BUT, sometimes the quadratic doesn't factor, or you simply cannot figure out the factoring. The Quadratic Formula can always find the value of x (the roots/zeros) and is set up to do so without the issues one can run into with the other methods. For ax + bx + c = 0, the value of x is given by: OR (the mistake proof version) x b b 4a c a Common errors: For the Formula to work, you must first have the "(quadratic) = 0". When identifying a, b, c, make sure to include their sign in the problem. The "a" at the bottom of the Formula is underneath everything in the numerator, not just the square root. Make sure that you are careful not to drop the square root or the "plus/minus" in the middle of your calculations Remember that " b " means "the square of ALL of b, including the sign". So, do not leave b negative, because the square of any real number is positive.

19 Discriminant b 4ac OR b 4a c If D > 0 the quadratic will have two real solutions (x-intercepts, roots, zeros). If D = 0 the quadratic will have one real solution (x-intercept, root, zero) or double root. If D < 0 the quadratic will have two imaginary solutions (x-intercepts, roots, zeros). Find the Discriminant. Use it to determine whether the quadratic has two real solutions, one real solution, or two imaginary b solutions. Then, find those solutions using the quadratic formula. x b 4a c a 11z 10 z 7z 3 4x 1 6x D = 1 Real Real Imaginary Use the quadratic formula to solve the equation. D = 1 Real Real Imaginary 3x 4.5x 3 5x.75

20 When an object is launched or thrown, an extra term is added to the formula to account for the object s Modeling Launched Objects initial vertical velocity. Remember that the height is h (in feet), the time it s in the air is t, h 0 is the object s initial height (in feet), and t v 0 is the object s initial vertical velocity. h 16t v0t h0 A juggler tosses a ball into the air. The ball leaves the juggler s hand 4 feet above the ground and has an initial vertical velocity of 40 feet per second. Write a model for the situation and then answer the questions. Find the maximum height of the ball. Find how long it would take for the ball to hit the ground. The juggler catches the ball when it falls back to a height of 3 feet. After how many seconds the ball to reach a height of 3 feet? 4 feet 3 feet

21 4.10 Write Quadratic Functions and Models We know that quadratics can be written in Standard, Vertex, or Intercept form. We also know the differences between each of the forms. Today, we are going to look at the graph of a quadratic, and use our knowledge about each form to write an equation for that quadratic. STANDARD FORM: y ax bx c VERTEX FORM: y a x h k INTERCEPT FORM: y a x px q ex. ex. ex. Example 1: Write a quadratic equation for the parabola shown. What do you know from the graph? What information is it showing you? Which model (form) would best use this information? Think: With which model could I take this information and plug it in for the variables?? Apply the information to the indicated form: Equation: Example : Write a quadratic equation for the parabola shown. What do you know from the graph? What information is it showing you? Which model (form) would best use this information? Think: With which model could I take this information and plug it in for the variables?? Apply the information to the indicated form: Equation: