Lesson 1: Analyzing Quadratic Functions

Transcription

1 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 of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. 1. How is a quadratic equation similar to a linear equation? How is it different?. How is a quadratic equation similar to an exponential equation? How is it different? 3. In what situations is it appropriate to use a quadratic model? WORDS TO KNOW axis of symmetry of a parabola extrema factored form of a quadratic function intercept the line through the vertex of a parabola about which the parabola is symmetric. The equation of the axis of b symmetry is x =. a the minima or maxima of a function the intercept form of a quadratic equation, written as f(x) = a(x p)(x q), where p and q are the x-intercepts of the function; also known as intercept form of a quadratic function the point at which a line intercepts the x- or y-axis U-1.1

2 intercept form maximum minimum parabola the factored form of a quadratic equation, written as f(x) = a(x p)(x q), where p and q are the x-intercepts of the function the largest y-value of a quadratic equation the smallest y-value of a quadratic equation the U-shaped graph of a quadratic equation quadratic function a function that can be written in the form f(x) = ax + bx + c, where a 0. The graph of any quadratic function is a parabola. standard form of a quadratic function vertex form vertex of a parabola x-intercept y-intercept a quadratic function written as f(x) = ax + bx + c, where a is the coefficient of the quadratic term, b is the coefficient of the linear term, and c is the constant term a quadratic function written as f(x) = a(x h) + k, where the vertex of the parabola is the point (h, k); the form of a quadratic equation where the vertex can be read directly from the equation the point on a parabola that is the maximum or minimum the point at which the graph crosses the x-axis; written as (x, 0) the point at which the graph crosses the y-axis; written as (0, y) U- Unit : Quadratic Functions and Modeling.1

3 Recommended Resources IXL Learning. Solve an Equation Using the Zero Product Property. This interactive website gives a series of problems and scores them immediately. If the user submits a wrong answer, a description and process for arriving at the correct answer are provided. Users solve quadratic equations by setting factors equal to 0. This activity is meant as a review. PhET Interactive Simulations. Equation Grapher.0. This website allows users to compare the graphs of various self-created equations. West Texas A&M University Virtual Math Lab. Graphs of Quadratic Functions. This tutorial offers a review and worked examples for writing and graphing quadratic functions in different forms, as well as practice problems with worked solutions for reference. U-3.1

4 Lesson.1.1: Graphing Quadratic Functions Introduction You may recall that a line is the graph of a linear function and that all linear functions can be written in the form f(x) = mx + b, in which m is the slope and b is the y-intercept. The solutions to a linear function are the infinite set of points on the line. In this lesson, you will learn about a second type of function known as a quadratic function. Key Concepts A quadratic function is a function that can be written in the form f(x) = ax + bx + c, where x is the variable, a, b, and c are constants, and a 0. This form is also known as the standard form of a quadratic function, where a is the coefficient of the quadratic term, b is the coefficient of the linear term, and c is the constant term. Quadratic functions can be graphed on a coordinate plane. One method of graphing a quadratic function is to create a table of at least five x-values and calculate the corresponding y-values. Once graphed, all quadratic functions will have a U-shape called a parabola. Distinguishing characteristics can be used to describe, draw, and compare quadratic functions. These characteristics include the y-intercept, x-intercepts, the maximum or minimum of the function, and the axis of symmetry. The intercept of a graph is the point at which a line intercepts the x- or y-axis. The x-intercept is the point at which a graph crosses the x-axis. It is written as (x, 0). The x-intercepts of a quadratic function occur when the parabola intersects the x-axis at (x, 0). U-4 Unit : Quadratic Functions and Modeling.1.1

5 The following graph of a quadratic function, f(x) = x x 3, shows the location of the parabola s x-intercepts x-intercepts ( 1, 0) 1 (3, 0) Note that the x-intercepts of this function are ( 1, 0) and (3, 0). The equation of the x-axis is y = 0; therefore, the x-intercepts can also be found in a table by identifying which values of x have a corresponding y-value that is 0. The table of values below corresponds to the function f(x) = x x 3. Notice that the same x-intercepts noted in the graph can be found where the table shows y is equal to 0. x y U-5.1.1

6 The y-intercept of a quadratic function is the point at which the graph intersects the y-axis. It is written as (0, y). The y-intercept of a quadratic is the c value of the quadratic equation when written in standard form. The following graph of a quadratic function, f(x) = x x 3, shows the location of the parabola s y-intercept y-intercept (0, 3) Note that the y-intercept of this equation is (0, 3). The c value of the function is also 3. The axis of symmetry of a parabola is the line through the parabola about which the parabola is symmetric. b The equation of the axis of symmetry is x =. a U-6 Unit : Quadratic Functions and Modeling.1.1

7 The equation of the axis of symmetry for the function f(x) = x x 3 is x = 1 because the vertical line through 1 is the line that cuts the parabola in half Axis of symmetry The vertex of a parabola is the point on a parabola that is the maximum or minimum of the function. The maximum is the largest y-value of a quadratic equation and the minimum is the smallest y-value. The extrema of a graph are the minima or maxima of a function. In other words, an extremum is the function value that achieves either a minimum or maximum. The vertex of a quadratic lies on the axis of symmetry. The vertex is often written as (h, k). The formula b x = is also used to find the x-coordinate of the vertex. a To find the y-coordinate, substitute the value of x into the original function, b ( hk, ) a, f b = a. U-7.1.1

8 The graph that follows shows the relationship between the vertex and the axis of symmetry of a parabola Vertex (1, 4) -5-6 Axis of symmetry Notice that the vertex of the function f(x) = x x 3 is (1, 4). If you know the x-intercepts of the graph, or any two points on the graph with the same y-value, the x-coordinate of the vertex is the point halfway between the values of the x-coordinates. For x-intercepts (r, 0) and (s, 0), the x-coordinate of the vertex is r + s. From the equation of a function in standard form, you can determine if the function has a maximum or a minimum based on the sign of the coefficient of the quadratic term, a. If a > 0, then the parabola opens up and therefore has a minimum value. If a < 0, the parabola opens down and therefore has a maximum value. The value of a of the function f(x) = x x 3 is 1; therefore, the vertex is a minimum. U-8 Unit : Quadratic Functions and Modeling.1.1

9 To graph a function using a graphing calculator, follow these general steps for your calculator model. On a TI-83/84: Step 1: Press the [Y=] button. Step : Type the function into Y1, or any available equation. Use the [X, T, θ, n] button for the variable x. Use the [x ] button for a square. Step 3: Press [WINDOW]. Enter values for Xmin, Xmax, Ymin, and Ymax. The Xscl and Yscl are arbitrary. Leave Xres = 1. Step 4: Press [GRAPH]. On a TI-Nspire: Step 1: Press the [home] key. Step : Arrow over to the graphing icon and press [enter]. Step 3: Type the function next to f1(x), or any available equation, and press [enter]. Use the [X] button for the variable x. Use the [x ] button for a square. Step 4: To change the viewing window, press [menu]. Select 4: Window/ Zoom and select A: Zoom Fit. U-9.1.1

10 Guided Practice.1.1 Example 1 Given the function f(x) = x, identify the key features of the graph: the extremum, vertex, and y-intercept. Then sketch the graph. 1. Determine the extremum of the graph. The extreme value is a minimum when a > 0. It is a maximum when a < 0. Because a = 1, the graph opens upward and the quadratic has a minimum.. Determine the vertex of the graph. The minimum value occurs at the vertex. The vertex is of the form b a, f b a. Use the original function f(x) = x to find the values of a and b in order to find the x-value of the vertex. b x = Formula to find the x-coordinate of a the vertex of a quadratic (0) x = Substitute 1 for a and 0 for b. (1) x = 0 The x-coordinate of the vertex is 0. Substitute 0 into the original equation to find the y-coordinate. f(x) = x Original equation f(0) = (0) Substitute 0 for x. f(0) = 0 The y-coordinate of the vertex is 0. The vertex is located at (0, 0). U-10 Unit : Quadratic Functions and Modeling.1.1

11 3. Determine the y-intercept of the graph. The y-intercept occurs when x = 0. The y-intercept of the function f(x) = x is the same as the vertex, (0, 0). When the equation is written in standard form, the y-intercept is c. 4. Graph the function. Create a table of values and axis of symmetry to identify points on the graph. The axis of symmetry goes through the vertex, so the axis of symmetry is x = 0. For each point to the left of the axis of symmetry, there is another point the same distance on the right side of the axis and vice versa. Choose at least two values of x that are to the right and left of 0. Let s start with x =. f(x) = x Original equation f() = () Substitute for x. f() = 4 An additional point is (, 4). (, 4) is units to the right of the vertex. The point (, 4) is units to the left of the vertex, so (, 4) is also on the graph. To find another set of points on the graph, let s evaluate the original equation for x = 3. f(x) = x Original equation f(3) = (3) Substitute 3 for x. f(3) = 9 An additional point is (3, 9). (3, 9) is 3 units to the right of the vertex. The point ( 3, 9) is 3 units to the left of the vertex, so ( 3, 9) is also on the graph. (continued) U

12 Plot the points and join them with a smooth curve. ( 3, 9) (, 4) (, 4) (3, 9) f(x) = x (0, 0) Example Given the function f(x) = x + 16x 30, identify the key features of the graph: the extremum, vertex, and y-intercept. Then sketch the graph. 1. Determine the extremum of the graph. The extreme value is a minimum when a > 0. It is a maximum when a < 0. Because a =, the graph opens downward and the quadratic has a maximum. U-1 Unit : Quadratic Functions and Modeling.1.1

13 . Determine the vertex of the graph. The maximum value occurs at the vertex. b The vertex is of the form a f b, a. Use the original equation f (x) = x + 16x 30 to find the values of a and b in order to find the x-value of the vertex. b x = Formula to find the x-coordinate of a the vertex of a quadratic ( 16) x = ( Substitute for a and 16 for b. ) x = 4 The x-coordinate of the vertex is 4. Substitute 4 into the original equation to find the y-coordinate. f(x) = x + 16x 30 Original equation f(4) = (4) + 16(4) 30 Substitute 4 for x. f(4) = The y-coordinate of the vertex is. The vertex is located at (4, ). 3. Determine the y-intercept of the graph. The y-intercept occurs when x = 0. Substitute 0 for x in the original equation. f(x) = x + 16x 30 Original equation f(0) = (0) + 16(0) 30 Substitute 0 for x. f(0) = 30 The y-intercept is (0, 30). When the quadratic equation is written in standard form, the y-intercept is c. U

14 4. Graph the function. Use symmetry to identify additional points on the graph. The axis of symmetry goes through the vertex, so the axis of symmetry is x = 4. For each point to the left of the axis of symmetry, there is another point the same distance on the right side of the axis and vice versa. The point (0, 30) is on the graph, and 0 is 4 units to the left of the axis of symmetry. The point that is 4 units to the right of the axis is 8, so the point (8, 30) is also on the graph. Determine two additional points on the graph. Choose an x-value to the left or right of the vertex and find the corresponding y-value. f(x) = x + 16x 30 Original equation f(1) = (1) + 16(1) 30 Substitute 1 for x. f(1) = 16 An additional point is (1, 16). (1, 16) is 3 units to the left of the axis of symmetry. The point that is 3 units to the right of the axis is 7, so the point (7, 16) is also on the graph. Plot the points and join them with a smooth curve. 5 0 f(x) = x + 16x 30 (4, ) (3, 0) (5, 0) (1, 16) (7, 16) (0, 30) (8, 30) U-14 Unit : Quadratic Functions and Modeling.1.1

15 Example 3 Given the function f(x) = x + 6x + 9, identify the key features of its graph: the extremum, vertex, and y-intercept. Then sketch the graph. 1. Determine the extremum of the graph. The extreme value is a minimum when a > 0. It is a or maximum when a < 0. Because a = 1, the graph opens upward and the quadratic has a minimum.. Determine the vertex of the graph. The minimum value occurs at the vertex. b The vertex is of the form a, f b a. Use the original function f(x) = x + 6x + 9 to find the values of a and b in order to find the x-value of the vertex. b Formula to find the x-coordinate of x = a the vertex of a quadratic (6) x = Substitute 1 for a and 6 for b. (1) x = 3 The x-coordinate of the vertex is 3. Substitute 3 into the original equation to find the y-coordinate. f(x) = x + 6x + 9 Original equation f( 3) = ( 3) + 6( 3) + 9 Substitute 3 for x. f( 3) = 0 The y-coordinate of the vertex is 0. The vertex is located at ( 3, 0). U

16 3. Determine the y-intercept of the graph. The y-intercept occurs when x = 0. Substitute 0 for x in the original equation. f(x) = x + 6x + 9 Original equation f(0) = (0) + 6(0) + 9 Substitute 0 for x. f(0) = 9 The y-intercept is (0, 9). 4. Graph the function. Use symmetry to identify an additional point on the graph. The axis of symmetry goes through the vertex, so the axis of symmetry is x = 3. For each point to the left of the axis of symmetry, there is another point the same distance on the right side of the axis and vice versa. The point (0, 9) is on the graph, and 0 is 3 units to the right of the axis of symmetry. The point that is 3 units to the left of the axis is 6, so the point ( 6, 9) is also on the graph. Determine two additional points on the graph. Choose an x-value to the left or right of the vertex and find the corresponding y-value. f(x) = x + 6x + 9 Original equation f( 1) = ( 1) + 6( 1) + 9 Substitute 1 for x. f( 1) = 4 An additional point is ( 1, 4). ( 1, 4) is units to right of the axis of symmetry. The point that is units to the left of the axis is 5, so the point ( 5, 4) is also on the graph. (continued) U-16 Unit : Quadratic Functions and Modeling.1.1

17 Plot the points and join them with a smooth curve ( 6, 9) (0, 9) 8 f(x) = x + 6x ( 5, 4) ( 1, 4) 4 3 ( 3, 0) Example 4 Given the function f(x) = x 1x 10, identify the key features of its graph: the extremum, vertex, and y-intercept. Then sketch the graph. 1. Determine the extremum of the graph. The extreme value is either a minimum, when a > 0, or a maximum, when a < 0. Because a =, the graph opens down and the quadratic has a maximum. U

18 . Determine the vertex of the graph. b The vertex is of the form a, f b a. Use the original function f(x) = x 1x 10 to find the values of a and b in order to find the x-value of the vertex. b Formula to find the x-coordinate of x = a the vertex of a quadratic ( 1) x = ( ) x = 3 Substitute for a and 1 for b. The x-coordinate of the vertex is 3. Substitute 3 into the original equation to find the y-coordinate. f(x) = x 1x 10 Original equation f( 3) = ( 3) 1( 3) 10 Substitute 3 for x. f( 3) = 8 The y-coordinate of the vertex is 8. The vertex is ( 3, 8). 3. Determine the y-intercept of the graph. The y-intercept occurs when x = 0. Substitute 0 for x in the original equation. f(x) = x 1x 10 Original equation f(0) = (0) 1(0) 10 Substitute 0 for x. f(0) = 10 The y-intercept is (0, 10). U-18 Unit : Quadratic Functions and Modeling.1.1

19 4. Graph the function. Use symmetry to identify another point on the graph. Because 0 is 3 units to the right of the axis of symmetry, the point 3 units to the left of the axis will have the same value, so ( 6, 10) is also on the graph. Determine two additional points on the graph. Choose an x-value to the left or right of the vertex and find the corresponding y-value. f(x) = x 1x 10 Original equation f(0) = ( ) 1( ) 10 Substitute for x. f( ) = 6 An additional point is (, 6). (, 6) is 1 unit to right of the axis of symmetry. The point that is 1 unit to the left of the axis is 4, so the point ( 4, 6) is also on the graph. Plot the points and join them with a smooth curve. f(x) = x 1x 10 ( 5, 0) ( 1, 0) ( 6, 10) ( 3, 8) ( 4, 6) (, 6) (0, 10) U

20 PRACTICE UNIT QUADRATIC FUNCTIONS AND MODELING Practice.1.1: Graphing Quadratic Functions For each function that follows, identify the intercepts, vertex, and maximum or minimum. Then, sketch the graph of the function. 1. y = x + 6x 7. y = x 8x 1 3. y = x + 4x y= x x 5. y = x + 10x y = 3x + 6x y = x 1x 16 For each problem that follows, determine whether the function has a minimum or maximum, identify the maximum or minimum, and identify the intercepts. 8. A golfer s ball lands in a sand trap 4 feet below the playing green. The path of the ball on her next shot is given by the equation y = 16x + 0x 4, where y represents the height of the ball after x seconds. 9. The revenue, R(x), generated by an increase in price of x dollars for an item is represented by the equation R(x) = 10x + 100x The flight of a rubber band follows the quadratic equation H(x) = x + 6x + 7, where H(x) represents the height of the rubber band in inches and x is the horizontal distance the rubber band travels in inches after launch. U-0 Unit : Quadratic Functions and Modeling.1.1

21 Lesson.1.: Interpreting Various Forms of Quadratic Functions Introduction Quadratic equations can be written in several forms, including standard form, vertex form, and factored form. While each form is equivalent, certain forms easily reveal different features of the graph of the quadratic function. In this lesson, you will learn to use the various forms of quadratic functions to show the key features of the graph and determine how these key features relate to the characteristics of a real-world situation. Key Concepts Standard Form Recall that the standard form, or general form, of a quadratic function is written as f(x) = ax + bx + c, where a is the coefficient of the quadratic term, b is the coefficient of the linear term, and c is the constant term. When a function is written in standard form, the y-intercept is the value of c. The vertex of the function can be found by first determining the value of x, b x = b, and then finding the corresponding y-value, y= f a a. b The vertex is often written as a, f b a. If a > 0, the function has a minimum and the graph opens up. If a < 0, the function has a maximum and the graph opens down. Vertex Form The vertex form of a quadratic function is written as f(x) = a(x h) + k. In vertex form, the maximum or minimum of the function is identified using the vertex of the parabola, the point (h, k). If a > 0, the function has a minimum, where k is the y-coordinate of the minimum and h is the x-coordinate of the minimum. If a < 0, the function has a maximum, where k is the y-coordinate of the maximum and h is the x-coordinate of the maximum. U-1.1.

22 Because the axis of symmetry goes through the vertex, the axis of symmetry can be identified from vertex form as x = h. The graph of a quadratic function is symmetric about the axis of symmetry. Factored Form The factored form, or intercept form, of a quadratic function is written as f(x) = a(x p)(x q). Recall that the x-intercepts of a function are the x-values where the function is 0. In factored form, the x-intercepts of the function are identified as p and q. Recall that the y-intercept of a function is the point at which the function intersects the y-axis. To determine the y-intercept, substitute 0 for x and simplify. The axis of symmetry can be identified from the factored form since it occurs at the midpoint between the x-intercepts. Therefore, the axis of symmetry is p q x = +. To determine the vertex of the function, calculate the y-value that corresponds to the x-value of the axis of symmetry. If a > 0, the function has a minimum and the graph opens up. If a < 0, the function has a maximum and the graph opens down. U- Unit : Quadratic Functions and Modeling.1.

23 Guided Practice.1. Example 1 Suppose that the flight of a launched bottle rocket can be modeled by the function f(x) = (x 1)(x 6), where f(x) measures the height above the ground in meters and x represents the horizontal distance in meters from the launching spot at x = 1. How far does the bottle rocket travel in the horizontal direction from launch to landing? What is the maximum height the bottle rocket reaches? How far has the bottle rocket traveled horizontally when it reaches its maximum height? Graph the function. 1. Identify the x-intercepts of the function. In the function, f(x) represents the height of the bottle rocket. At launch and landing, the height of the bottle rocket is 0. The function f(x) = (x 1)(x 6) is of the form f(x) = a(x p)(x q), where p and q are the x-intercepts. The x-intercepts of the function are at x = 1 and x = 6. Find the distance between the two points to determine how far the bottle rocket traveled in the horizontal direction. 6 1 = 5 The bottle rocket traveled 5 meters in the horizontal direction from launch to landing.. Determine the maximum height of the bottle rocket. The maximum height occurs at the vertex. p q Find the axis of symmetry using the formula x = +. p q x = + Formula to determine the axis of symmetry 6 1 x = + Substitute 6 for p and 1 for q. x = 3.5 The axis of symmetry is x = 3.5. (continued) U-3.1.

24 Use the axis of symmetry to determine the vertex of the function. f(x) = (x 1)(x 6) Original function f(3.5) = [(3.5) 1][(3.5) 6] Substitute 3.5 for x. f(3.5) = (.5)(.5) f(3.5) = 6.5 The y-coordinate of the vertex is 6.5. The maximum height reached by the bottle rocket is 6.5 meters. 3. Determine the horizontal distance from the launch point to the maximum height of the bottle rocket. We know that the bottle rocket is launched from the point (1, 0) and reaches a maximum height at (3.5, 6.5). Subtract the x-value of the two points to find the distance traveled horizontally =.5 Another method is to take the total distance traveled horizontally from launch to landing and divide by to find the same answer. This is because the maximum value occurs halfway between the x-intercepts of the function. 5.5 = The bottle rocket travels.5 meters horizontally when it reaches its maximum. U-4 Unit : Quadratic Functions and Modeling.1.

25 4. Graph the function. Use a graphing calculator or complete a table of values. Use the x-intercepts and vertex as three of the known points. Choose x-values on either side of the vertex for two additional x-values. x y To determine the y-coordinates of the additional points, substitute each x-value into the original function and solve. f(x) = (x 1)(x 6) Original function f() = [() 1][() 6] Substitute for x. f() = (1)( 4) f() = 4 f(x) = (x 1)(x 6) Original function f(5) = [(5) 1][(5) 6] Substitute 5 for x. f(5) = (4)( 1) f(5) = 4 Fill in the missing table values. x y (continued) U-5.1.

26 Notice that the points (, 4) and (5, 4) are the same number of units from the vertex. Plot the points on a coordinate plane and connect them using a smooth curve. Since the function models the flight of a bottle rocket, it is important to only show the portion of the graph where both time and height are positive U-6 Unit : Quadratic Functions and Modeling.1.

27 Example Reducing the cost of an item can result in a greater number of sales. The revenue function that predicts the revenue in dollars, R(x), for each $1 change in price, x, for a particular item is R(x) = 100(x 7) + 8,900. What is the maximum value of the function? What does the maximum value mean in the context of the problem? What price increase maximizes the revenue and what does it mean in the context of the problem? Graph the function. 1. Determine the maximum value of the function. The function R(x) = 100(x 7) + 8,900 is written in vertex form, f(x) = a(x h) + k, where (h, k) is the vertex. The vertex of the function is (7, 8,900); therefore, the maximum value is 8,900.. Determine what the maximum value means in the context of the problem. The maximum value of 8,900 means that the maximum revenue resulting from increasing the price by x dollars is $8, Determine the price increase that will maximize the revenue and what it means in the context of the problem. The maximum value occurs at the vertex (7, 8,900). This means an increase in price of $7 will result in the maximum revenue. U-7.1.

28 4. Graph the function. Use a graphing calculator or complete a table of coordinates. Use the vertex as one known point. Choose x-values on either side of the vertex to have four additional x-values. x 0 5 y 7 8, To determine the y-coordinates of the additional points, substitute each x-value into the original function and solve. R(x) = 100(x 7) + 8,900 Original function R(0) = 100[(0) 7] + 8,900 Substitute 0 for x. R(0) = 4,000 R(x) = 100(x 7) + 8,900 Original function R(5) = 100[(5) 7] + 8,900 Substitute 5 for x. R(5) = 8,500 R(x) = 100(x 7) + 8,900 Original function R(9) = 100[(9) 7] + 8,900 Substitute 9 for x. R(9) = 8,500 R(x) = 100(x 7) + 8,900 Original function R(14) = 100[(14) 7] + 8,900 Substitute 14 for x. R(14) = 4,000 (continued) U-8 Unit : Quadratic Functions and Modeling.1.

29 Fill in the missing table values. x y 0 4, , , , ,000 Notice that the points (0, 4,000) and (14, 4,000) are the same number of units from the vertex. The same is true for (5, 8,500) and (9, 8,500). Plot the points on a coordinate plane and connect using a smooth curve. Since the function models revenue, it is important to only graph the portion of the graph where both the x- and y-values are positive. Revenue 9,000 8,000 7,000 6,000 5,000 4,000 3,000,000 1,000 0,000 19,000 18,000 17,000 16,000 15,000 14,000 13,000 1,000 11,000 10,000 9,000 8,000 7,000 6,000 5,000 4,000 3,000,000 1,000 (5, 8,500) (0, 4,000) (7, 8,900) (9, 8,500) (14, 4,000) $1 change in price U-9.1.

30 Example 3 A football is kicked and follows a path given by f(x) = 0.03x + 1.8x, where f(x) represents the height of the ball in feet and x represents the horizontal distance in feet. What is the maximum height the ball reaches? What horizontal distance maximizes the height? Graph the function. 1. Determine the maximum height of the ball. The function f(x) = 0.03x + 1.8x is written in standard form, f(x) = ax + bx + c, where a = 0.03, b = 1.8, and c = 0. b The maximum occurs at the vertex, a, f b a. Determine the x-value of the vertex. b x = Formula to find the x-coordinate for the a vertex of a quadratic equation (1.8) x = Substitute values for a and b. ( 0.03) x = 30 Determine the y-value of the vertex. f(x) = 0.03x + 1.8x Original function f(30) = 0.03(30) + 1.8(30) Substitute 30 for x. f(30) = 7 The vertex is (30, 7) and the maximum value is 7 feet. The maximum height the ball reaches is 7 feet.. Determine the horizontal distance of the ball when it reaches its maximum height. The x-coordinate of the vertex maximizes the quadratic. The vertex is (30, 7). The ball will have traveled 30 feet in the horizontal direction when it reaches its maximum height. U-30 Unit : Quadratic Functions and Modeling.1.

31 3. Graph the function. Use a graphing calculator or complete a table of coordinates. Use the vertex as one known point. Choose x-values on either side of the vertex to have four additional x-values. x 5 0 y To determine the y-coordinates of the additional points, substitute each x-value into the original function and solve. f(x) = 0.03x + 1.8x Original function f(5) = 0.03(5) + 1.8(5) Substitute 5 for x. f(5) = 8.5 f(x) = 0.03x + 1.8x Original function f(0) = 0.03(0) + 1.8(0) Substitute 0 for x. f(0) = 4 f(x) = 0.03x + 1.8x Original function f(40) = 0.03(40) + 1.8(40) Substitute 40 for x. f(40) = 4 f(x) = 0.03x + 1.8x Original function f(55) = 0.03(55) + 1.8(55) Substitute 55 for x. f(55) = 8.5 (continued) U-31.1.

32 Fill in the missing table values. x y Notice that the points (5, 8.5) and (55, 8.5) are the same number of units from the vertex. The same is true for (0, 4) and (40, 4). Plot the points on a coordinate plane and connect them using a smooth curve. Since the function models the path of a kicked football, it is important to only show the portion of the graph where both height and horizontal distance are positive (0, 4) (30, 7) (40, 4) Height (5, 8.5) (55, 8.5) Horizontal distance U-3 Unit : Quadratic Functions and Modeling.1.

33 PRACTICE UNIT QUADRATIC FUNCTIONS AND MODELING Practice.1.: Interpreting Various Forms of Quadratic Functions Use the given functions to complete all parts of problems f(x) = x 8x + 1 a. Identify the y-intercept. b. Identify the vertex. c. Identify whether the function has a maximum or minimum.. f(x) = (x 3)(x + 5) a. Identify the x-intercepts. b. Determine the y-intercept. c. Determine the axis of symmetry. d. Determine the vertex. 3. f(x) = 16(x 3) a. Identify the vertex. b. Identify whether the function has a maximum or minimum. Use the given information in each scenario that follows to complete the remaining problems. 4. A butterfly descends toward the ground and then flies back up. The butterfly s descent can be modeled by the equation h(t) = t 10t + 6, where h(t) is the butterfly s height above the ground in feet and t is the time in seconds since you saw the butterfly. Graph the function and identify the vertex. What is the meaning of the vertex in the context of the problem? 5. A cliff diver jumps upward from the edge of a cliff then begins to descend, so that his path follows a parabola. The diver s height, h(t), above the water in feet is given by h(t) = (t 1) + 5, where t represents the time in seconds. Graph the function. What is the vertex and what does it represent in the context of the problem? How many seconds after the start of the dive does the diver reach the initial height? continued U-33.1.

34 PRACTICE UNIT QUADRATIC FUNCTIONS AND MODELING 6. The revenue of producing and selling widgets is given by the function R(w) = 8(w 50)(w + ), where w is the number of widgets produced and R(w) is the amount of revenue in dollars. Graph the function. What are the x-intercepts and what do they represent in the context of the problem? What number of widgets maximizes the revenue? 7. A football is kicked and follows a path given by y = 0.03x + 1.x, where y represents the height of the ball in feet and x represents the horizontal distance in feet. Graph the function. What is the vertex and what does it mean in the context of the problem? How far does the ball travel in the horizontal direction? 8. A frog hops from the bank of a creek onto a lily pad. The path of the jump 1 can be modeled by the equation ( ) ( ) h x= x + 4, where h(x) is the frog s height in feet above the water and x is the number of seconds since the frog jumped. Graph the function. What does the vertex represent in the context of the problem? What is the axis of symmetry? After how many seconds does the height of the frog reach the initial height? 9. The revenue, R(x), generated by an increase in price of x dollars for an item is represented by the equation R(x) = x + 0x Graph the function and identify the vertex. What does the vertex represent in the context of the problem? What is the axis of symmetry? What increase in price results in the same revenue as not increasing the price at all? 10. Decreasing the cost of an item can result in a greater number of sales. The revenue function that predicts the revenue in dollars, R(x), for each $1 decrease in price, x, for a certain item is R(x) = (x 6)(x + 10). Graph the function. Identify the x-intercepts. What do the x-intercepts represent in the context of the problem? What is the axis of symmetry? What increase in price results in the same revenue as not increasing the price at all? U-34 Unit : Quadratic Functions and Modeling.1.