1. Graphing Quadratic Functions

3. Math Maintenance Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3

4. 4 Standard: MGSE9-12.F.IF.7a Graph quadratic functions and show intercepts, maxima, and minima (as determined by the function or the context). Essential Question: How can you use the graph of a quadratic function to solve its related quadratic equation?

5. Quadratic Function y = ax 2 + bx + c Quadratic TermLinear TermConstant Term What is the linear term of y = 4x 2 – 3? 0x What is the linear term of y = x 2 - 5x ? -5x What is the constant term of y = x 2 – 5x? 0 Can the quadratic term be zero? No!

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Quadratic function Let a, b, and c be real numbers a  0. The function f (x) = ax 2 + bx + c is called a quadratic function. The graph of a quadratic function is a parabola. Every parabola is symmetrical about a line called the axis (of symmetry). The intersection point of the parabola and the axis is called the vertex of the parabola. x y axis f (x) = ax 2 + bx + c vertex

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Leading Coefficient The leading coefficient of ax 2 + bx + c is a. When the leading coefficient is positive, the parabola opens upward and the vertex is a minimum. When the leading coefficient is negative, the parabola opens downward and the vertex is a maximum. x y f(x) = ax 2 + bx + c a > 0 opens upward vertex minimum x y f(x) = ax 2 + bx + c a < 0 opens downward vertex maximum NOTE: if the parabola opens left or right it is not a function!

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 5 y x 5-5-5 Simple Quadratic Functions The simplest quadratic functions are of the form f (x) = ax 2 (a  0) These are most easily graphed by comparing them with the graph of y = x 2.

9. Identify the Vertex and Axis of Symmetry of a Quadratic Function Vertex =(x, y). thus Vertex = Axis of Symmetry: the line x = This is best read as … ‘the opposite of b divided by the quantity of 2 times a.’ Vertex is minimum point if parabola opens up Vertex is maximum point if parabola opens down

10. y x Axis of Symmetry Parabolas are symmetric. If we drew a line down the middle of the parabola, we could fold the parabola in half. We call this line the Axis of symmetry. The Axis of symmetry ALWAYS passes through the vertex. If we graph one side of the parabola, we could REFLECT it over the Axis of symmetry to graph the other side.

11. Identify the Vertex and Axis of Symmetry Vertex x = y = Vertex = (-1, -3) Axis of Symmetry is x = = = -1 -3 = -1 x y vertex

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Vertex of a Parabola Example: Find the vertex of the graph of f (x) = x 2 – 10x + 22. f (x) = x 2 – 10x + 22 original equation a = 1, b = –10, c = 22 The vertex of the graph of f (x) = ax 2 + bx + c (a  0) At the vertex, So, the vertex is (5, -3).

13. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Example: Basketball Example: A basketball is thrown from the free throw line from a height of six feet. What is the maximum height of the ball if the path of the ball is: The path is a parabola opening downward. The maximum height occurs at the vertex. At the vertex, So, the vertex is (9, 15). The maximum height of the ball is 15 feet.

14. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Example: Maximum Area Example: A fence is to be built to form a rectangular corral along the side of a barn 65 feet long. If 120 feet of fencing are available, what are the dimensions of the corral of maximum area? barn corral x x 120 – 2x Let x represent the width of the corral and 120 – 2x the length. Area = A(x) = (120 – 2x) x = –2x 2 + 120 x The graph is a parabola and opens downward. The maximum occurs at the vertex where a = –2 and b = 120 120 – 2x = 120 – 2(30) = 60 The maximum area occurs when the width is 30 feet and the length is 60 feet.

15. Graphing a Quadratic Function There are 3 steps to graphing a parabola in standard form. STEP 1: Find the Axis of symmetry using: STEP 2: Find the vertex STEP 3: Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve. MAKE A TABLE using x – values close to the Axis of symmetry.

16. STEP 1: Find the Axis of symmetry Graphing a Quadratic Function STEP 2: Find the vertex Substitute in x = 1 to find the y – value of the vertex.

17. 5 –1 STEP 3: Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve. 3 2 yx Graphing a Quadratic Function

18. Y-intercept of a Quadratic Function Y-axis The y-intercept of a Quadratic function can Be found when x = 0. The constant term is always the y- intercept

19. The number of real solutions is at most two. Solving a Quadratic No solutions One solution X = 3 Two solutions X= -2 or X = 2 The x-intercepts (when y = 0) of a quadratic function are the solutions to the related quadratic equation.

20. Identifying Solutions X = 0 or X = 2 Find the solutions of 2x - x 2 = 0 The solutions of this quadratic equation can be found by looking at the graph of f(x) = 2x – x 2 The x- intercepts(or Zero’s) of f(x)= 2x – x 2 are the solutions to 2x - x 2 = 0

21. The Axis of symmetry always goes through the _Vertex_. Thus, the Axis of symmetry gives us the _x-coordinate of the vertex. 21