Chapter 9 Section 5

Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. More on Graphing Quadratic Equations; Quadratic Functions Graph quadratic equations of the form y = ax 2 + bx + c (a  0). Use a graph to determine the number of real solutions of a quadratic equation

Copyright © 2012, 2008, 2004 Pearson Education, Inc. More on Graphing Quadratic Equations; Quadratic Functions In Section 5.4, we graphed the quadratic equation y = x 2. By plotting points, we obtained the graph of a parabola shown here. Recall the lowest (or highest point if the parabola opens downward) point on the graph is called the vertex of the parabola. The vertical line through the vertex is called the axis, or axis of symmetry. The two halves of the parabola are mirror images of each other across this axis. Slide 9.5-3

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Connect the dots x y

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 1 Graph quadratic equations of the form y = ax 2 + bx + c (a  0). Slide 9.5-4

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Graph quadratic equations of the form y = ax 2 + bx + c (a  0). Every equation of the form y = ax 2 + bx + c with a  0, has a graph that is a parabola. The vertex is the most important point to locate when graphing a quadratic equation. Earlier in the course we solved linear equations in one variable that were of the form Ax + B = C; then graphed linear equations in two variables that were of the form y = mx + b. Now, we are ready to do the same sort of thing for quadratic equations. We know how to solve ax 2 + bx + c = 0; and now we graph y = ax 2 + bx + c. Slide 9.5-5

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Graphing the Parabola y = ax 2 + bx + c Step 1: Find the vertex. Let x = and find the corresponding y-value by substituting for x in the equation. Step 2: Find the y-intercept. Let x = 0 and solve for y. Step 3: Find the x-intercepts (if they exist). Let y = 0 and solve for x. Step 4: Plot the intercepts and the vertex. Step 5: Find and plot additional ordered pairs near the vertex and intercepts as needed, using symmetry about the axis of the parabola. Slide Graph quadratic equations of the form y = ax 2 + bx + c (a  0). (cont’d)

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Graph y = x 2 + 2x – 8. Solution: Slide EXAMPLE 1 Graphing a Parabola by Finding the Vertex and Intercepts Vertex Y-intercept X-intercepts

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Graph y = –x 2 + 2x + 4. Solution: Slide EXAMPLE 2 Graphing a Parabola

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 2 Use a graph to determine the number of real solutions of a quadratic equation. Slide 9.5-9

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use a graph to determine the number of real solutions of a quadratic equation. Using the vertical line test from Section 3.6, we see that the graph of an equation of the form y = ax 2 + bx + c is the graph of a function. A function defined by an equation of the form  (x) = ax 2 + bx + c (a  0) is called a quadratic function. The domain (possible x-values) of a quadratic function is the set of all real numbers, or (– ,  ); the range (the resulting y-values) can be determined after the function is graphed. Slide

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: There are two real solutions, {±2}, that correspond to the graph. Decide from the graph how many real number solutions the corresponding equation f (x) = 0 has. Give the solution set. Slide EXAMPLE 3 Determining the Number of Real Solutions from Graphs

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Homework Book Beginning Algebra Page 585 Exercises 1-18 (even)