Characteristics of Quadratic Functions and Graphs

Transcription

1 Characteristics of Quadratic Functions and Graphs Skill: Compare characteristics of a given family of quadratic functions. Skill: Determine the domain and range of quadratic equations algebraically and graphically. F.IF.B.4 Interpret functions that arise in applications in terms of the context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F.IF. B.5 Interpret functions that arise in applications in terms of the context. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes F.IF.C.7a Analyze functions using different representations. 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. F.IF.C.8a Analyze functions using different representations. 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. F.IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Note to teachers: The complete notes for the characteristics Page of 7 0/4/04

2 Notes Skill: solve quadratic equations using graphic techniques. Skill: graph quadratic equations and find possible solutions to those equations using coordinate geometry. F.IF.C.7 Analyze functions using different representations. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. F.IF.B.4 Interpret functions that arise in applications in terms of the context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Solving a Quadratic Equation by Graphing 0. y ax bx c. Step Three: Find the zero(s) or root(s) of the function. These are the solution(s) to the equation. Note: The words x-intercept, zero, root, and solution can be used interchangeably for the above value. Ex 7: Solve the equation x 8 by graphing. Then check by solving algebraically x y ax bx c. y x 8 b 0 x 0 a y , 8 Page of 7 0/4/04

3 The zeros are at 4 and 4, so the solutions are x 4,4. Solve algebraically: x 8 x 6 x x 4 x 4 6 Ex 8: Solve the equation x 3x by graphing. x x y ax bx c. 3 b 3 3 x a y , 4 The zeros are at and, so the solutions are x,. Ex 9: Solve the equation x 4 4x by graphing. 0. x 4x 4 0 Page 3 of 7 0/4/04

4 4 4 y ax bx c. b 4 4 x a y ,0 The root is at, so the solution is x. Check: x x Ex 0: Solve the equation x 3 0 graphically. 0. x 3 0 y x 3 y ax bx c. b 0 0 x 0 a 3 6 y ,3 There is no zero, so this equation has no real solution. Using a Graphing Calculator to Solve Quadratic Equations Ex : Approximate the solution(s) of x 4x using a graphing calculator. 0. x 4x 0 y ax bx c. 4 Step Three: Find the zero(s) of the function. x 4.36,0.36 Page 4 of 7 0/4/04

5 You Try: Solve the quadratic equation 6 3 x graphically. Then check your answer algebraically. QOD: How can you tell from the graph of a quadratic function if the equation has one, two, or no solution? Sample Practice Question(s): The graph of has how many x-intercepts? A. B. C. D. 0 Sample Exam Questions Fiona is designing a skateboard park. One skating area in the park will be shaped like the parabola that is described by the equation below A sketch of Fiona s design for the skating area is shown below. What is the distance across the top of the skating area? A units B 4 units C 5 units D 8 units Page 5 of 7 0/4/04

6 Zero(s) of Quadratic Functions: the x-value(s) where the function intersects the x-axis To find the zero(s), factor the quadratic and set each factor equal to 0. Note: We can graph quadratic functions by plotting the zeros. The vertex is halfway between the zeros. Ex : Find the zero(s) of the quadratic function 3and graph the parabola. Step One: Factor the quadratic polynomial. Step Two: Set each factor equal to 0 and solve. Step Three: Find the coordinates of the vertex. 3 3 y x x x 3 0 x 0 x 3 x 3 x y Step Four: Plot the points and sketch the parabola. Determining the Number of x-intercepts of a Quadratic Function Using the Discriminant Because the x-intercepts of y ax bx c are the same as the zeros of the equation ax bx c 0, we can use the discriminant to determine the number of x-intercepts that a quadratic function has. Ex 3: Sketch the graph of a quadratic function with a negative discriminant. Because the discriminant, b 4ac 0, the function will have no x-intercept. A sample answer is shown in the graph. Note: Any parabola which does not intersect the x-axis is an acceptable answer. Page 6 of 7 0/4/04

7 Application Problem Ex 4: A baton twirler tosses a baton into the air. The baton leaves the twirler s hand 6 feet above the ground and has an initial vertical velocity of 45 feet per second. This can be modeled by the equation h 6t 45t 6, where h is the height (in feet) and t is the time (in seconds). The twirler wants her baton to reach at least 40 feet. Will the baton reach that height? Substitute h. Write in standard form. 40 6t 45t 6 0 6t 45t 34 a 6, b 45, c 34 Find the discriminant. b ac Since the discriminant is less than 0, this equation has no real solution. Therefore, the baton could not reach 40 feet. How high will the baton reach? Graph the function h t t Find the maximum (vertex). The baton will reach approximately ft. You Try: Find values for c so that the equation will have no real solution, one real solution, and two real solutions. x x c 3 0 QOD: Write a quadratic equation which can be factored. Find its discriminant. Teacher Note: Have students share their answers to the QOD and allow students to make a conjecture for how to determine if a quadratic polynomial is factorable using the discriminant. (It must be a perfect square.) Page 7 of 7 0/4/04