Sample: Do Not Reproduce QUAD4 STUDENT PAGES. QUADRATIC FUNCTIONS AND EQUATIONS Student Pages for Packet 4: Quadratic Functions and Applications

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1 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 k in the equation y = (x h) + k affect the graph. Graph quadratic functions in vertex form. Change quadratic functions (y = 1x + bx + c) into vertex form. QUAD 4. Vertex Form of a Quadratic Function Use technology to explore how changes to the parent graph equation affect the direction and shape of the graph. Graph quadratic functions in vertex form. Rewrite quadratic functions (y = ax + bx + c) in vertex form. QUAD4 STUDENT PAGES QUAD 4.3 Modeling with Quadratic Functions Perform a ball tossing experiment and collect data. Estimate possible quadratic functions for a set of data. Solve vertical motion problems Analyze features of quadratic functions in different forms to reveal properties of the function and the situation it models. Demonstrate fluency in changing quadratic equations from one form to another. QUAD 4.4 Vocabulary, Skill Builders, and Review QUAD4 SP

2 Quadratic Functions and Applications WORD BANK (QUAD4) Word Definition Example or Picture axis of symmetry factored form of a quadratic polynomial function parabola projectile motion standard form of a quadratic polynomial function velocity vertex form of a quadratic polynomial function vertex of a parabola QUAD4 SP0

3 Quadratic Functions and Applications 4.1 Vertex Form of a Quadratic Function 1 VERTEX FORM OF A QUADRATIC FUNCTION 1 Ready (Summary) We will use technology to explore graphs that are related to the parabolic parent graph equation, y = x. We will learn how to write quadratic functions in vertex form and how to graph quadratic functions in vertex form. Go (Warmup) Set (Goals) Explore how changing the values of h and k in the equation y = (x h) + k affect the graph. Graph quadratic functions in vertex form. Change quadratic functions (y = 1x + bx + c) into vertex form. Graph the following quadratic functions. State the vertex for each function. 1. y = x + 3. y = (x + 3) x y x y (, ) (, ) y x QUAD4 SP1

4 Quadratic Functions and Applications 4.1 Vertex Form of a Quadratic Function 1 PARENT GRAPH EXPLORATION 1 Use technology for this exploration. First graph y = x, which we will call the parent graph of a quadratic function. Then make changes to the parent function (i.e. find a related function) that cause the graph to shift as indicated. Record the results. 1. Parent graph Sketch: Equation: y = x y-intercept: x-intercepts: Is the vertex a min or max?. Vertical shift up. Equation: y-intercept: x-intercepts: Is the vertex a min or max? 3. Vertical shift down. Equation: y-intercept: x-intercepts: Is the vertex a min or max? Sketch: Sketch: QUAD4 SP

5 Quadratic Functions and Applications 4.1 Vertex Form of a Quadratic Function 1 PARENT GRAPH EXPLORATION 1 (continued) 4. Horizontal shift left. Equation: y-intercept: x-intercepts: Is the vertex a min or max? 5. Horizontal shift right. Equation: y-intercept: x-intercepts: Is the vertex a min or max? 6. Horizontal shift left and vertical shift down. Equation: y-intercept: x-intercepts: Is the vertex a min or max? Sketch: Sketch: Sketch: QUAD4 SP3

6 Quadratic Functions and Applications 4.1 Vertex Form of a Quadratic Function 1 PARENT GRAPH EXPLORATION 1 PRACTICE Sketch the estimated position of each of the following quadratic functions and describe the position of the graph s vertex in relation to the vertex of the parent graph. Check your predictions using technology. 1. y = x 1 Relationship to parent function: moved down 1 unit. 4. y = (x + 4) Relationship to parent function: 7. y = (x 3) 1 Relationship to parent function:. y = (x 1) Relationship to parent function: 5. y = (x ) +1 Relationship to parent function: 8. y = (x +3) + 1 Relationship to parent function: 3. y = x + 4 Relationship to parent function: 6. y = (x + 3) Relationship to parent function: 9. y = x + 4x + 4 (hint: factor first) Relationship to parent function: QUAD4 SP4

7 Quadratic Functions and Applications 4.1 Vertex Form of a Quadratic Function 1 VERTEX FORM 1 The parent graph exploration suggests that we can analyze the characteristics of a parabola by writing its quadratic function in vertex form. The vertex form of a quadratic function is y= ax ( h) + k We will call y k= ax ( h) a modified vertex form of a quadratic function. Write each quadratic function from the previous page in vertex form and modified vertex form. Then identify a, h, and k. Function Vertex Form y= ax ( h) + k Modified Vertex Form a h k y k= ax ( h) 1. y = x 1 y = 1( x 0) 1 y + 1= 1( x 0 ) y = (x 1) 3. y = x y = (x + 4) 5. y = (x ) y = (x + 3) 7. What does (h, k) represent on a parabola? 8. If a = 1, what do you know about the shape and direction of the parabola? Sketch a graph of each quadratic function. 9. y 3 = (x ) 10. y = (x + 1) 11. y 5 = x QUAD4 SP5

8 Quadratic Functions and Applications 4.1 Vertex Form of a Quadratic Function 1 VERTEX FORM PRACTICE 1 Rewrite each function in a vertex form if needed. Sketch and label each group of quadratic functions on the same axes, using a different colored pencil for each function. Label the vertex for each parabola. Describe how functions are the same and how they are different. 1. y = ( x+ 1). y = x + 3 y = ( x+ 3) y = x 4 y = ( x ) y = x y = x 3 4. y = ( x+ ) 3 y = ( x 3) 3 y = ( x 1) y = ( x 1) 4 y = ( x 1) + 4 QUAD4 SP6

9 Quadratic Functions and Applications 4.1 Vertex Form of a Quadratic Function 1 WRITING QUADRATIC FUNCTIONS IN VERTEX FORM If a quadratic function is not in vertex form, we can use the completing the square technique to put it in vertex form. 1. Suppose we want to put y = x 6x + 7 into vertex form and graph it. Begin with a quadratic function in standard form: y = ax + bx + c. Subtract the constant from both sides of the equation. Complete the square on the right side of the equation, and balance the equation. Rewrite the equation in modified vertex form or vertex form Describe location and shape of graph Locate the y-intercept from standard form. Sketch the graph using information from vertex form. Write each quadratic function in vertex form. Identify the vertex, the y-intercept, the x- intercepts, and then graph it.. y = x + 6x 3. y = (x + 4)(x ) QUAD4 SP7

10 Quadratic Functions and Applications 4.1 Vertex Form of a Quadratic Function PARENT GRAPH EXTENSION 1: EXPLORING TABLES For homework, Romeo had to complete t-charts for the following quadratic functions: y = x y = (x + 1) y = x + 1 y = (x 1) y = x 1 y = (x 1) + 1 Unfortunately, he didn t write down which table went with which function and he needs your help matching the tables and graphs. Match each graph with the correct table. For each match, explain how you made your decision. Romeo did the first one for you because he recognizes the table of the parent function (y = x ). x y Function: y = x _ Romeo s explanation: I recognize this table from last lesson. x y Function: Explanation:. 5. x y Function: Explanation: x y Function: Explanation: x y Function: Explanation: x y Function: Explanation: 3 5 QUAD4 SP8

11 Quadratic Functions and Applications 4. Vertex Form of a Quadratic Function 1 VERTEX FORM OF A QUADRATIC FUNCTION Ready (Summary) We will continue to explore graphs that are related to a parabolic parent graph equation, y = x, using technology. We will learn how to write more complicated quadratic functions in vertex form. Go (Warmup) Set (Goals) Use technology to explore changes to the parent graph equation that affect the direction and shape of the graph. Graph quadratic functions in vertex form. Rewrite quadratic functions (y = ax + bx + c) into vertex form. Graph the following quadratic functions. State the vertex for each function. 1. y = x + 3. y = -x + 3 x y x y (, ) (, ) y x QUAD4 SP9

12 Quadratic Functions and Applications 4. Vertex Form of a Quadratic Function 1 PARENT GRAPH EXPLORATION Use technology for this exploration. First graph the parent graph for the quadratic function. Then make changes to the parent function (i.e. find a related function) that cause the graph to change as indicated. Record the results. 1. Parent graph Equation: y = x y-intercept: x-intercepts: Is the vertex a min or max?. Parent graph, reflected through x-axis. Equation: y-intercept: x-intercepts: Is the vertex a min or max? 3. Graph appears narrower, opening up Equation: y-intercept: x-intercepts: Is the vertex a min or max? Sketch: Sketch: Sketch: QUAD4 SP10

13 Quadratic Functions and Applications 4. Vertex Form of a Quadratic Function 1 PARENT GRAPH EXPLORATION (continued) 4. Graph appears wider, opening down Equation: y-intercept: x-intercepts: Is the vertex a min or max? 5. Horizontal shift to right, opening down Equation: y-intercept: x-intercepts: Is the vertex a min or max? 6. Horizontal shift left, vertical shift up, opening down, graph appears wider Equation: y-intercept: x-intercepts: Is the vertex a min or max? Sketch: Sketch: Sketch: QUAD4 SP11

14 Quadratic Functions and Applications 4. Vertex Form of a Quadratic Function 1 PARENT GRAPH EXPLORATION SUMMARY Write an equation or inequality that describes the value of a in y = ax when: 1. the parabola opens up and has the same shape as y = x. the parabola opens down and has the same shape as y = x 3. the parabola appears narrower than y = x 4. the parabola appears wider than y = x Recall: The vertex form of a quadratic function is y= ax ( h) + k A modified vertex form of a quadratic function is y k= ax ( h) 5. In general what is the vertex for y= ax ( h) + k? 6. In general, what is the vertex for y k= ax ( h) 7. Which form seems easier for you to remember and work with? Explain. 8. A positive value for h will shift the parent graph. A negative value for h will shift the parent graph. 9. A positive value for k will shift the parent graph. A negative value for k will shift the parent graph. 10. A positive value for a will orient the graph so it s opening. A negative value for a will orient the graph so it s opening.? 11. As the a increases, the opening of the graph appears to. 1. As the a decreases, the opening of the graph appears to. QUAD4 SP1

15 Quadratic Functions and Applications 4. Vertex Form of a Quadratic Function 1 PRACTICE Sketch and label each group of quadratic functions on the same axes, using a different colored pencil for each function. Label the vertex for each parabola. Describe how functions are the same and how they are different. 1. y = x. y = x y = x 1 y = x y = 4x 1 y = 7 x y = x 4 1 y = x 8 y = ( x+ ) y = ( x 3) + y = ( x+ ) + 3 y = ( x 3) + y = ( x 1) + 4 y = 3( x 1) y = ( x 1) y = ( x 1) QUAD4 SP13

16 Quadratic Functions and Applications 4. Vertex Form of a Quadratic Function PARENT GRAPH EXTENSION : EXPLORING TABLES Romeo has done it again! He s lost track of his work and can t remember which table goes with each of the functions below. Can you help him out again? He promises that this is the last time! y = x y = 1 x y = 1 x y = (x) y = 3x y = -3x Match each graph with the correct table. For each match, explain how you made your decision. Romeo did the first one for you because he recognizes the table of the parent function (y = x ). x y Function: y = x. Romeo s explanation: I recognize this table from last lesson. x y Function: Explanation:. 5. x y Function: Explanation: x y Function: Explanation: x y Function: Explanation: x y Function: Explanation: QUAD4 SP14

17 Quadratic Functions and Applications 4. Vertex Form of a Quadratic Function 1 CONVERTING ANY QUADRATIC FUNCTION TO VERTEX FORM 1. Suppose we want to put y = 3x 6x + 7 into vertex form and graph it. Begin with a quadratic function in standard form: y = ax + bx + c. Subtract the constant from both sides of the equation. Rewrite the quadratic expression as a product of a and another factor. Complete the square on the right side of the equation, and balance the equation. Rewrite the equation in modified vertex form or vertex form Describe location and shape of graph Locate the y-intercept from standard form. Sketch the graph using information from vertex form. Write each quadratic function in vertex form and graph it.. y = x + 8 x 3. y = x 6x+ 3 QUAD4 SP15

18 Quadratic Functions and Applications 4. Vertex Form of a Quadratic Function 1 PRACTICE WITH VERTEX FORM Write the following quadratic functions in vertex form. Then write the vertex and describe the graph s relation to the parent graph, y = x. 1. y = 4x 16x + 0. y = 9x + 54x y = x + 4x 4. y = -x +10x + 3 Describe why the following are NOT quadratic functions in vertex form. If the equation can be put in vertex form, do it. If it cannot, explain why not. 5. y = 1 x x 6. y = (x 3x) + 4 x + 5 QUAD4 SP16

19 Quadratic Functions and Applications 4.3 Modeling with Quadratic Functions 1 MODELING WITH QUADRATIC FUNCTIONS Ready (Summary) First, we will explore projectile motion through a ball tossing experiment. We will collect data and graph the results. We will find functions to approximate the data using paper and pencil and technology, and apply a projectile motion formula to solve problems. Then, We will create open boxes from a fixed rectangle, and collect data about different size boxes that are possible. We will analyze different forms of the quadratic function created to model the situation to reveal properties of the function and to answer questions about the problem. Go (Warmup) Set (Goals) Perform a ball tossing experiment and collect data. Estimate possible quadratic functions for a set of data. Solve vertical motion problems Analyze features of quadratic functions in different forms to reveal properties of the function and the situation it models. Demonstrate fluency in changing quadratic equations from one form to another. You have learned three basic forms for quadratic equations: standard form, factored form, and vertex form. For the given equation, as it exists, place it in its proper place in the chart. Then convert it to the other two forms. Given equation: y = -1x 36x Standard Form Factored Form Vertex Form QUAD4 SP17

20 Quadratic Functions and Applications 4.3 Modeling with Quadratic Functions 1 TOSSING A BALL An object that is projected by an external force (as in being thrown or shot) may be referred to as a projectile. You will collect data from a ball-tossing experiment that compares height of a ball (a projectile) with its distance from the starting point. Record and graph the data collected. 1. Describe the experiment in a few sentences:. Label table and axes. Record data and plot points from your experiment. QUAD4 SP18

21 Quadratic Functions and Applications 4.3 Modeling with Quadratic Functions 1 ESTIMATING AN EQUATION FOR YOUR EXPERIMENT Answer the questions and generate an equation that approximates your data. 1. From the data you graphed above, what type of graph do the points appear to form?. Is there a y-intercept? If so, write it here and explain what it means in the context of the experiment. 3. Are there any x-intercepts? If so, write them here and explain what they mean in the context of the experiment. 4. What is the vertex of your graph? Estimate as necessary. (, ). Is this a maximum or a minimum point? h k 5. Using the vertex form of a quadratic function, y = a(x h) + k, substitute the values of h and k and write an equation to estimate height of the ball y vs. distance from starting point x. Then substitute (x,y) coordinates to estimate values for a. Repeat a few times and select a reasonable value for a. 6. Using technology, estimate a quadratic function for you data. What technology did you use? What is the estimated function? How does this function compare to the one you derived in #5 above? QUAD4 SP19

22 Quadratic Functions and Applications 4.3 Modeling with Quadratic Functions 1 MODEL ROCKETS A formula that calculates the vertical motion of an object that is dropped, thrown, or shot straight up or straight down is: y = -16t + vt + h where y = height t = time v = initial velocity h = initial height Suppose you have a model rocket launcher. A rocket is shot straight up from the ground with a starting velocity of 96 feet per second. 1. What is the initial velocity of the rocket?. What is the initial height of at the base of the rocket? 3. Write a formula that describes the height of the rocket with respect to time. 4. Complete the table of values to show the height vs. time. 5. Why is the rocket NOT at 96 feet above the ground after 1 second? 6. After how many seconds does the rocket hit the ground? 7. What appears to be the maximum height of the rocket? 8. Write the formula from problem 3 above in vertex form. Does the vertex agree with your estimate for maximum height? 9. Find the zeroes (values when y = 0) of your function in problem 3 above. Interpret the zeroes in the context of the problem. time height QUAD4 SP0

23 Quadratic Functions and Applications 4.3 Modeling with Quadratic Functions 1 A ROCK THROW Recall a formula that calculates the vertical motion of an object that is dropped, thrown, or shot straight up or straight down. y = -16t + vt + h where y = height t = time v = initial velocity h = initial height You are standing on a cliff that is 48 ft above the ground, and throw a rock up at an initial velocity of 3 feet per second. 1. What is the initial velocity of the rock?. What is the initial height of the rock? 3. Write a formula that describes the height of the rock with respect to time. 4. Complete the table of values to show the height vs. time. 5. In how many seconds does the rock hit the ground? 6. What appears to be the maximum height of the rock? 7. Write the formula in the problem above in vertex form. Does the vertex agree with your estimate for maximum height? 8. Find the zeroes (values when y = 0) of your function in problem 3 above. Interpret the zeroes in the context of the problem. time height QUAD4 SP1

24 Quadratic Functions and Applications 4.3 Modeling with Quadratic Functions 1 STRAW PACKAGING You work as the mathematical analyst for a company called Slurp Ease that makes drinking straws. It is your job to figure out how to package the straws effectively and efficiently. Each drinking straw is 1 units of length. They plan to use 1 x 8 pieces of cardboard and want to hold the maximum amount of straws possible. 1. Create a couple of possible holders for the straws by folding up the width of the 1 x 8 models provided. Note the resulting volumes and record your data in the table below. H (height) H W (width). Graph volume vs. height. L (length) 3. What type of graph is created? 4. What are the x-intercepts? V (volume) height width Considering the context of the problem, explain what these x-intercepts mean. length QUAD4 SP

25 Quadratic Functions and Applications 4.3 Modeling with Quadratic Functions 1 STRAW PACKAGING (continued) 5. What appears to be the vertex? Is this a maximum or minimum point? What appears to be the dimensions with greatest volume? 6. Look at the width and height columns only. A function that expresses width in terms of height. W = 7. Fact 1: For a rectangular prism V = L W H. Fact : length of the straw package is fixed at 1. Using these facts and that the width equation above, write a function that expresses volume in terms of height. V = 8. Write your equation from problem 7 for the volume of the packaging in the following ways (it may already be in one or two of them above): Standard Form Factored Form Vertex Form 9. Does the vertex form of the equation support the vertex you named in problem 5? QUAD4 SP3

26 Quadratic Functions and Applications 4.4 Vocabulary, Skill Builders, and Review 1 FOCUS ON VOCABULARY (QUAD4) Use vocabulary from the entire quadratic functions unit to complete this puzzle. ACROSS DOWN y = ( x )( x+ 3) is in this form 1 b 4ac y = x± ( wds) a 7 the set of output values for a function 3 an object that is thrown or shot 8 property used to solve quadratic equations by factoring 10 value used to determine the number of real roots of a quadratic function 4 y = ( x+ 3) + 5is in ( wds) 5 the maximum or minimum point of a quadratic function 1 a rule where each input has exactly one output 6 y = x x+ 5 is in this form 13 path of a projectile 9 the set of input values for a function 11 vertex of a parabola opening downward QUAD4 SP4

27 Quadratic Functions and Applications 4.4 Vocabulary, Skill Builders, and Review 1 SKILL BUILDER 1 Complete the square for the following expressions. 1. x 6x. x 9x 3. x + 50x 4. x x Graph the following quadratic functions. State the vertex for the graph of each function. y y = x + 1 y = (x + 1) x y x y (, ) (, ) x QUAD4 SP5

28 Quadratic Functions and Applications 4.4 Vocabulary, Skill Builders, and Review 1 SKILL BUILDER Solve the following equations by completing the square. 1. x 6x = 9. x 9x = -0 Graph the following quadratic functions. Do not try to graph points that don t fit on the grid. State the vertex for the graph of each function. 1. y = -x + 1. y = 1 (x + 1) x y x y (, ) (, ) y x QUAD4 SP6

29 Quadratic Functions and Applications 4.4 Vocabulary, Skill Builders, and Review 1 Solve using any method. SKILL BUILDER 3 1. x + 6x = -5. 5x + x = x = -x x + x 1 = 0 Sketch the graphs of the following equations, and write the required information. 5. y = x 4 vertex: y-intercept: x-intercept(s): 6. y = (x 4) vertex: y-intercept: x-intercept(s): 7. y + 1 = (x 4) vertex: y-intercept: x-intercept(s): QUAD4 SP7

30 Quadratic Functions and Applications 4.4 Vocabulary, Skill Builders, and Review 1 SKILL BUILDER 4 Write the following equations in vertex form or modified vertex form if helpful. Then sketch the graph. 1. y = x 6x + 9. y = x x y = x 8x y = x + 4x 5. y = x + 4x y = 3x 6x + 7 QUAD4 SP8

31 Quadratic Functions and Applications 4.4 Vocabulary, Skill Builders, and Review 1 SKILL BUILDER 5 Identify whether each equation is in standard form, factored form, or vertex form. Then change it into the other two forms. 1. y = x 6x + 8 is in form. form. y = (x 3)(x + 7) is in form. form form form QUAD4 SP9

32 Quadratic Functions and Applications 4.4 Vocabulary, Skill Builders, and Review 1 SKILL BUILDER 6 Write each statement using symbols. If the variable is on the right side, change it to the left side using appropriate properties. Then graph each. 1. x is equal to -4. x is greater than - 3. the opposite of x is less than or equal to is greater than x is less than or equal to the opposite of x. Graph each inequality. Be sure they are in slope-intercept form first. 6. y x 3 7. y x < 8. Describe the differences between the graph of an inequality in one variable with the graph of an inequality in two variables. QUAD4 SP30

33 Quadratic Functions and Applications 4.4 Vocabulary, Skill Builders, and Review 1 TEST PREPARATION (QUAD4) Show your work on a separate sheet of paper and choose the best answer. 1. Which of the following parabolas have a vertex at (3, -4)? A. y = (x - 3) - 4 B. y + 4 = (x - 3) C. y = x 6x + 5 D. All of these parabolas have a vertex at (3, -4). Compared to the graph of the parent function f(x) = x, which of these functions have a graph that is vertically shifted up 3 units and horizontally shifted to the right 5 units? A. f(x) = (x + 3) + 5 B. f(x) = (x - 5) + 3 C. f(x) = (x - 3) + 5 D. f(x) = (x + 5) Compared to the graph of the parent function f(x) = x, which of these functions open down AND have a narrower opening? A. f(x) = -x + 5 B. f(x) = 3x C. f(x) = -3x D. f(x) = 1 3 x 4. Which of the following functions have a vertex have a maximum value? A. f(x) = -(x + 1) B. f(x) = -x + 1 C. f(x) = (x + 1) D. f(x) = x Amy is trying to put the quadratic equation y = x 3x + 7. She subtracts 7 from both sides and gets y 7 = x 3x. What will she eventually need to add to both sides to complete the square for the quadratic equation? A. 9 B C. D A parabola is NOT an effective model for which of the following situations. A. Maximizing the area with a limited perimeter. C. Describing the distance of a runner moving at a constant speed. B. Describing how a baseball flies through the air when thrown. D. Describing a height of an object as it falls to the ground. QUAD4 SP31

34 Quadratic Functions and Applications 4.4 Vocabulary, Skill Builders, and Review 1 KNOWLEDGE CHECK (QUAD4) Show your work on a separate sheet of paper and write your answers on this page. QUAD 4.1 Vertex Form of a Quadratic Function 1 For problems 1-3, the parent function is f(x) = x. For each equation given, describe the relationship to the parent function, sketch a graph, and label the coordinates of the vertex. 1. f(x) = (x + 3) Relationship to parent function: (, ). f(x) = (x - 3) + Relationship to parent function: (, ) 4. Write the following equation in vertex form: y = x 8x + 7. QUAD 4. Vertex Form of a Quadratic Function 3. f(x) = x + 6x + 9 (hint: factor first) Relationship to parent function: (, ) 5. Sketch and label each group of quadratic functions on the same axes, using a different colored pencil for each function. Label the vertex for each parabola. Describe how functions are the same and how they are different. y A. y = (x + 4) + 3 B. y = (x 4) + 3 C. y = (x + 4) + 3 D. y = (x 4) + 3 x QUAD4 SP3

35 Quadratic Functions and Applications HOME-SCHOOL CONNECTION (QUAD4) Here are some questions to review with your young mathematician. Write each quadratic function in vertex form. Identify the vertex, the y-intercept, the x- intercepts, and then graph it. 1. y = x 8x. y = (x + 1)(x ) Write the following quadratic equations in vertex form. 3. y = -x +10x y = 3x + x Parent (or Guardian) signature QUAD4 SP33

36 Quadratic Functions and Applications A-SSE-3a* A-SSE-3b* A-CED-1* A-CED-* F-IF-7a* F-BF-3 MP1 MP MP3 MP4 MP5 MP6 MP7 MP8 COMMON CORE STATE STANDARDS FOR MATHEMATICS Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression: Factor a quadratic expression to reveal the zeros of the function it defines. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression: Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases: Graph linear and quadratic functions and show intercepts, maxima, and minima. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. STANDARDS FOR MATHEMATICAL PRACTICE Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. QUAD4 SP34