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

2. 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.

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

4. 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,

5. 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).

6. • • • • • –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.

7. 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

8. –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

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

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

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

12. • • • • • 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

13. 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

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