Bellwork
1. Write in standard form:
y = 3(x – 4)2 + 3
2. Write in intercept form:
2
Y = 3x – 18x + 24
3. Write in vertex form:
y = –2x2 – 12x + 5
Section 5.7
Graphing and Solving Quadratic Inequalities
Quadratic Inequalities
A quadratic function with an inequality
symbol.
Graphing Quadratic Inequalities
1. Graph the Parabola as the boundary line.
<,>: dotted boundary
<,>: solid boundary
2. Choose a test point and shade
appropriately.
Graph each Quadratic Inequality
1. y < x2 – 4x + 1
vertex: x = 4/2 = 2
y = 22 – 4(2) + 1 = –3
(2, –3)
x
y
3 –2
4
1
Test Point: (2, 0)
0 < 22 – 4(2) + 1
0 < –3
FALSE! Don't
shade this point!
Graph each Quadratic Inequality
2
2. y < –2x + 8x – 2
vertex: x = –8/–4 = 2
y = –2(2)2 + 8(2) – 2 = 6
(2, 6)
x
y
3 4
Test Point: (2, 0)
0 < –2(2)2 + 8(2) – 2
0<6
4 –2 TRUE! Shade this
point!
Graph each Quadratic Inequality
2
3. y < 2x – 8x + 5
vertex: x = 8/4 = 2
2
y =2(2) – 4(2) + 1 = –3
(2, –3)
x
y
Test Point: (2, 0)
3 –1 0 < 2(2)2 – 8(2) + 5
4 5 0 < -3
FALSE! Don't
shade this point!
Graph each Quadratic Inequality
2
4. y > -(x – 3) + 1
vertex: (3, 1)
x
y
4 0
5 -3
Test Point: (0,0)
0 > -(0-3)2 + 1
0>-8
True! Shade this
point!
Graph each Quadratic Inequality
5. y < (x – 4)(x + 2)
X Int: (4, 0), (-2, 0)
Vertex:
X = (4 + -2)/2 = 1
Y = (1 – 4)(1 + 2) = – 9
Test Point: (0,0)
0 < (0 – 4)(0 + 2)
0 < -8
False! Don't
shade this point!
Solving Quadratic Inequalities
The solution to a quadratic inequality is
always an interval of numbers that will
work
Always write as ax2 + bx + c symbol 0
There are two methods to solve quadratic
inequalities:
Algebraically
Graphically
Solving Quadratic Inequalities:
Algebraically
Solve the Quadratic like its an equation.
These are the critical points that split the
number line into three parts.
Test a point in each section.
Write the intervals in which the test points
work.
Solve algebraically.
1. 0 < x2 – 3x + 2
Solution:
(-∞, 1) U (2, ∞)
2
x – 3x + 2 > 0
x2 – 3x + 2 = 0
(x – 2)(x – 1) = 0
x=2 x=1
0
0 < 02 – 3(0) + 2
0<2
True
1
1.5
2
3
0 < (1.5)2 – 3(1.5) + 2 0 < 32 – 3(3) + 2
0 < -.25
False
0<2
True
Solve algebraically.
2. x2 – 3x – 10 < 0
Solution:
[-2, 5]
2
x – 3x – 10 = 0
(x – 5)(x + 2) = 0
X=5
x = -2
-3
-2
(-3)2 – 3(-3) – 10 < 0
8<0
False
0
5
6
(0)2 – 3(0) – 10 < 0
62 – 3(6) – 10 < 0
-10 < 0
True
8<0
False
Solve algebraically.
3. 2x2 + 5x < 12
Solution:
[-4, 3/2]
2
2x +5x – 12 < 0
(2x – 3)(x + 4) = 0
X = 3/2 x = -4
-5
2(-5)2 + 5(-5) < 12
25 < 12
False
-4
0
3/2
2(0)2 + 5(0) < 12
0 < 12
True
2
2(2)2 + 5(2) < 12
18 < 12
False
Solve algebraically.
4. 3x2 – x - 8 > -4
Solution:
(-∞, -1) U (4/3, ∞)
2
3x – x - 4 = 0
(3x – 4)(x + 1) = 0
X = 4/3 x = -1
-2
-1
3(-2)2 – (-2) - 8 > -4
6>-4
True
0
4/3
2
3(0)2 – (0) - 8 > -4
3(2)2 – (2) – 8 > -4
-8 > -4
False
2 > -4
True
Solve Graphically.
1. y < 2x2 – 6x – 20
2
2x – 6x – 20 > 0
2x2 – 6x – 20 = 0
2(x2 – 3x – 10) = 0
2(x – 5)(x + 2) = 0
X = 5 x = -2
Solution:
(-∞,-2)U(5,∞)
Solve Graphically.
2. y < -x2 + 2x + 8
2
-x + 2x + 8 > 0
-1(x2 – 2x – 8)= 0
-1(x – 4)(x + 2) = 0
X = 4 x = -2
Solution:
[-2,4]
Solve Graphically.
3. y > 2x2 – 6x – 20
2
2x – 6x – 20 < 0
2(x2 – 3x – 10) = 0
2(x – 5)(x + 2) = 0
X = 5 x = -2
Solution:
(-2,5)
Solve Graphically.
4. y < 2x2 + x – 3
2
2X + x – 3 > 0
2x2 + x – 3 = 0
(2x + 3)(x – 1) = 0
X = -3/2 x = 1
Solution:
(-∞, -3/2)U(1, ∞)
Solve Graphically.
5. y > 3x2 – 4x – 4
3x2 – 4x – 4 < 0
2
3x – 4x – 4 = 0
(3x – 2)(x – 2) = 0
X = 2/3 x = 2
Solution:
[2/3, 2]