9-2 Solving Quadratic Equations by Graphing Algebra 1 Glencoe McGraw-Hill Linda Stamper and JoAnn Evans

Factoring can be used to determine whether the graph of a quadratic functions intersects the x-axis in one or two points. The graph intersects the x-axis when f(x) equals 0. function Factor. related equation –12 Identify the roots. –3 4 1 The graph of the function intersects the x-axis two times.

y y-intercept matchy, matchy! x y -4 • • x -3 -6 2 -6 3 –4, 3

Use factoring to determine how many times the graph of each function intersects the x-axis. Identify each root. Remember: The graph intersects the x-axis when f(x) equals 0. Example 1 Two roots; -5, 2 Example 2 One root; 5 Example 3 Two roots; -1, Example 4 Two roots; -11,

In previous graphing to find solutions, the roots of the equations were integers. Usually the roots of a quadratic equation are not integers. In these cases, use estimation to approximate the roots of the equation. Solve x2 + 6x + 7 by graphing. If integral roots cannot be found, estimate the roots by stating the consecutive integers between which the roots lie. Integral roots are roots that are integers (positive and negative whole numbers).

• • • • • –5<x<-4, -2<x<-1 y x matchy, matchy! -5 2 • • -4 -1 x • • • -2 -1 –5<x<-4, -2<x<-1 -1 2 Notice that the value of the function changes from positive to negative between x values of -5 and -4 and between -2 and -1. The x-intercepts are between -5 and -4 and between -2 and -1. So one root is between -5 and -4 and the other between -2 and -1.

Solve by graphing. If integral roots cannot be found, estimate the roots by stating the consecutive integers between which the roots lie. Example 5 Example 6 Example 7 Example 8 Example 9

–4 < x < -3, 0 < x < 1 Example 5 matchy, matchy! x y -1 -1.5 -2 -3 -3 -7 -7.5 –4 < x < -3, 0 < x < 1

• • • • • Example 6 no real solution y x -4 -2 1 2 3 x y matchy, -4 -2 x 1 • • • • • 2 3

• • • • • Example 7 –5 < x < -4, 1 < x < 2 y x -3 -2 -5 -2 matchy, matchy! x y -3 -2 -5 x -2 • • -1 -6 • • • 1 –5 < x < -4, 1 < x < 2

y Example 8 -1, 5 matchy, matchy! x x y -5 1 -8 • • • • • 3 -8 4 -5

• • • • • Example 9 0 < x < 1, 4 < x < 5 y x 1 -3 -7 2 3 4 matchy, matchy! x x y • • 1 -3 -7 2 • • • 0 < x < 1, 4 < x < 5 3 4

Solve by graphing. If integral roots cannot be found, estimate the roots by stating the consecutive integers between which the roots lie. Example 5 –4 < x < -3, 0 < x < 1 Example 6 no real solution Example 7 –5 < x < -4, 1 < x < 2 Example 8 -1, 5 Example 9 0 < x < 1, 4 < x < 5

Homework 9-A4 Handout A4. All graphs must be completed on graph paper – check out the Colina website to download coordinate planes.