Chapter 8 Quadratic Functions

Chapter Sections 8.1 – Solving Quadratic Equations by Completing the Square 8.2 – Solving Quadratic Equations by the Quadratic Formulas 8.3 – Quadratic Equations: Applications and Problem Solving 8.4 – Writing Equations in Quadratic Form 8.5 – Graphing Quadratic Functions 8.6 – Quadratic and Other Inequalities in One Variable Chapter 1 Outline

Graphing Quadratic Functions § 8.5 Graphing Quadratic Functions

Quadratic Functions Quadratic Function A quadratic function is a function that can be written in the form f(x) = ax2 + bx + c For real numbers a, b, and c, with a ≠ 0.

Definitions The graph of every quadratic function is a parabola. The vertex is the lowest point on a parabola that opens upward, or the highest point on a parabola that opens downward.

Definitions Graphs of quadratic equations have symmetry about a line through the vertex. This line is called the axis of symmetry. The sign of a, the numerical coefficient of the squared term, determines whether the parabola will open upward or downward.

Vertex of a Parabola Vertex of a Parabola The parabola represented by the function f(x) = ax2 + bx + c will have vertex Since we often find the y-coordinate of the vertex by substituting the x-coordinate of the vertex into f(x), the vertex may also be designated as

Axis of Symmetry of a Parabola For a quadratic function of the form f(x) = ax2 + bx + c, the equation of the axis of symmetry of the parabola is

x-Intercepts of a Parabola To find the x-intercepts (if there are any) of a quadratic function, solve the equation ax2 + bx + c = 0 for x. This equation may be solved by factoring, by using the quadratic formula, or by completing the square.

Graph Quadratic Functions Example Consider the quadratic function y = x2 + 8x – 12. Determine whether the parabola opens upward or downward. Find the y-intercept. Find the vertex. Find the equation of the axis of symmetry. Find the x-intercepts, if any. Draw the graph. continued

Graph Quadratic Functions Since a is -1, which is less than 0, the parabola opens downward. b. To find the y-intercept, set x = 0 and solve for y. The y-intercept is (0, 12) continued

Graph Quadratic Functions c. First, find the x-coordinate, then find the y-coordinate of the vertex. From the function, a = -1, b = 8, and c = -12. Since the x-coordinate of the vertex is not a fraction, we will substitute x = 4 into the original function to determine the y-coordinate of the vertex. The vertex is (4, 4). continued

Graph Quadratic Functions d. Since the axis of symmetry is a vertical line through the vertex, the equation is found using the same formula used to find the x-coordinate of the vertex (see part c). Thus, the equation of the axis of symmetry is x = 4. continued

Graph Quadratic Functions e. To find the x-intercepts, set y = 0. Thus, the x-intercepts are (2, 0) and (6, 0). These values could also be found by the quadratic formula (or by completing the square). continued

Graph Quadratic Functions f. Draw the graph.

Solve Maximum and Minimum Problems A parabola that opens upward has a minimum value at its vertex, and a parabola that opens downward has a maximum value at its vertex.

Understand Translations of Parabolas Start with the basic graph of f(x) = ax2 and translate, or shift, the position of the graph to obtain the graph of the function you are seeking. - 4 x y 4 Notice that the value of a in the graph f(x) = ax2 determines the width of the parabola. As |a| gets larger, the parabola gets narrower, and as |a| gets smaller, the parabola gets wider.

Understand Translations of Parabolas Start with the basic graph of f(x) = ax2 and translate, or shift, the position of the graph to obtain the graph of the function you are seeking. - 4 x y 4 If h is a positive real number, the graph of g(x) = a(x – h)2 will be shifted h units to the right of the graph g(x) = ax2. If h is a negative real number, the graph of g(x) = a(x – h)2 will be shifted |h| units to the left.

Understand Translations of Parabolas Start with the basic graph of f(x) = ax2 and translate, or shift, the position of the graph to obtain the graph of the function you are seeking. - 4 x y 4 In general, the graph of g(x) = ax2 + k is the graph of f(x) = ax2 shifted k units up if k is a positive real number and |k| units down if k is a negative real number.

Understand Translations of Parabolas Parabola Shifts For any function f(x) = ax2, the graph of g(x) = a(x-h)2 + k will have the same shape as the graph of f(x). The graph of g(x) will be the graph of f(x) shifted as follows: If h is a positive real number, the graph will be shifted h units to the right. If h is a negative real number, the graph will be shifted |h| units to the left. If k is a positive real number, the graph will be shifted k units up. If k is a negative real number, the graph will be shifted |k| units down.

Understand Translations of Parabolas Axis of Symmetry and Vertex of a Parabola The graph of any function of the form f(x) = a(x – h)2 + k will be a parabola with axis of symmetry x = h and vertex at (h, k).

Write Functions in the Form f(x) = a(x – h)2 + k If we wish to graph parabolas using translations, we need to change the form of a function from f(x) = ax2 + bx + c to f(x) = a(x – h)2 + k. To do this we complete the square as we discussed in Section 8.1. Example Given f(x) = x2 – 6x + 10, Write f(x) in the form of f(x) = a(x – h)2 + k. Graph f(x).

Write Functions in the Form f(x) = a(x – h)2 + k We use the x2 and -6x terms to obtain a perfect square trinomial. Now we take half the coefficient of the x-term and square it. We then add this value, 9, within the parentheses. continued

Write Functions in the Form f(x) = a(x – h)2 + k By doing this we have created a perfect square trinomial within the parentheses, plus a constant outside the parentheses. We express the perfect square trinomial as the square of a binomial. The function is now in the form we are seeking. continued

Write Functions in the Form f(x) = a(x – h)2 + k b) Graph f(x).