5.7 Graphs of Quadratic Inequalities

p. 271

Forms of Quadratic InequalitiesForms of Quadratic Inequalitiesy<ax2+bx+c y>ax2+bx+cy≤ax2+bx+c y≥ax2+bx+c

Graphs will look like a parabola with a solid or dotted line and a shaded section.

The graph could be shaded inside the parabola or outside.

Steps for graphingSteps for graphing

1. Sketch the parabola y=ax2+bx+c(dotted line for < or >, solid line for ≤ or ≥)** remember to use 5 points for the graph!2. Choose a test point and see whether it is a

solution of the inequality.3. Shade the appropriate region.

(if the point is a solution, shade where the point is, if it’s not a solution, shade the other region)

Example:Graph y ≤ x2+6x- 4

3)1(2

62

abx

* Vertex: (-3,-13)

* Opens up, solid line

1341894)3(6)3( 2

y 9- 5-

12- 4-13- 3-12- 2-9- 1-

yx

•Test Point: (0,0)

0≤02+6(0)-4

0≤-4 So, shade where the point is NOT!

Test point

Graph: y>-x2+4x-3

* Opens down, dotted line.

* Vertex: (2,1)

2)1(2

42

abx

13843)2(4)2(1 2

yy

* Test point (0,0)

0>-02+4(0)-3

0>-3

x y

0 -3

1 0

2 1

3 0

4 -3

Test Point

Last Example! Sketch the intersection of the given inequalities.1 y≥x2 and 2 y≤-x2+2x+4

Graph both on the same coordinate plane. The place where the shadings overlap is the solution.

Vertex of #1: (0,0)Other points: (-2,4), (-1,1),

(1,1), (2,4)

Vertex of #2: (1,5)Other points: (-1,1), (0,4), (2,4),

(3,1)

* Test point (1,0): doesn’t work in #1, works in #2.

SOLUTION!

Assignment