4.9: Graph and Solve Quadratic Inequalities

4.9: Graph and Solve Quadratic Inequalities
1.
2.
3.
Objectives:
To graph quadratic
inequalities in two
variables
To graph a system
of quadratic
inequalities
To solve quadratic
inequalities in one
variable
Assignment:
• P. 304-307: 3-5, 6-14
even, 18, 19, 20, 22,
27-66 M3, 69, 72, 76,
78, 79
• Challenge Problems
Warm-Up
Graph each of the following inequalities.
1. y > −(1/2)x
2. 3x – 4y ≤ 12
Objective 1
You will be able
to graph
quadratic
inequalities in
two variables
𝑦 ≤ 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
Quadratic Inequalities (Two)
To graph a quadratic inequality in 2 variables:
1. Graph the boundary parabola: solid or dashed
𝑦 > 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
𝑦 ≥ 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
𝑦 < 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
𝑦 ≤ 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
Quadratic Inequalities (Two)
To graph a quadratic inequality in 2 variables:
2. Shade the appropriate region: inside or outside
𝑦 > 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
𝑦 ≥ 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
𝑦 < 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
𝑦 ≤ 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
Quadratic Inequalities (Two)
To graph a quadratic inequality in 2 variables:
•
Remember: Any point in the shaded region is a solution
𝑦 > 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
𝑦 ≥ 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
𝑦 < 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
𝑦 ≤ 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
Exercise 1
Graph the inequality.
y   x2  4x
Exercise 2
Graph each inequality.
1.
𝑦 > 𝑥 2 + 2𝑥 − 8
2. 2𝑥 2 − 3𝑥 + 1 ≥ 𝑦
Exercise 2
Graph each inequality.
1.
𝑦 > 𝑥 2 + 2𝑥 − 8
2. 2𝑥 2 − 3𝑥 + 1 ≥ 𝑦
Exercise 2
Graph each inequality.
1.
𝑦 > 𝑥 2 + 2𝑥 − 8
2. 2𝑥 2 − 3𝑥 + 1 ≥ 𝑦
Objective 2
You will be able to graph a system of quadratic inequalities
Systems of Quadratic Inequalities
Graphing a system of quadratic inequalities
is as easy as graphing systems of linear
inequalities. You need colored pencils!
1. Graph each inequality and shade each
region in a different color (or in different
directions).
2. The solution is the overlapping region
that gets shaded every color (or all
directions).
Exercise 3
Graph the system of
inequalities.
y  x2  3
y  2 x 2  4 x  2
Exercise 4
Graph the system of
inequalities.
y  x2
y   x2  5
Objective 3
You will be able to solve
quadratic inequalities in one
variable
Less ThAND
GreatOR
Quadratic Inequalities (One)
Solving quadratic inequalities in one variable
is similar to solving a combination of linear
inequalities and absolute value
inequalities.
• Like linear: graphed on a number line
• Like absolute value: involves “and” or “or”
intervals
– “And” = segment; “Or” = 2 rays in opposite
directions
Quadratic Inequalities (One)
Consider the simple quadratic inequality:
x 2  25
Now taking the square root of both sides
doesn’t translate well with inequalities, so
solve the corresponding quadratic
equation:
x 2  25
x  5
These are our critical values. Graph them.
Quadratic Inequalities (One)
Consider the simple quadratic inequality:
x 2  25
Now you have 3 intervals to consider.
Which one(s) make(s) the inequality true?
x  5 or x  5
GreatOR
Quadratic Inequalities (One)
Consider the simple quadratic inequality:
x 2  25
Now you have 3 intervals to consider.
Which one(s) make(s) the inequality true?
5  x  5
Less ThAND
Quadratic Inequalities (One)
Another way to think about solving a
quadratic inequality in one variable is to
relate it to a quadratic inequality in two
variables.
Instead of 𝑥 2 − 4 < 0, consider 𝑥 2 − 4 < 𝑦.
Now graph this inequality, shading the
appropriate region.
Quadratic Inequalities (One)
x2  4  0
x2  4  y
The answer is where
the shading touches
the x-axis:
2 x  2
Quadratic Inequalities (One)
x2  4  0
x2  4  y
The answer is where
the shading touches
the x-axis:
x  2 or x  2
Exercise 5
Solve the given inequality.
2x2  7 x  4
Exercise 6
Solve the given inequality.
2x2  2x  3
Exercise 7
Find the domain of each function.
1. 𝑦 = 𝑥 2 − 121
2. 𝑦 = 121 − 𝑥 2
Exercise 8
Find the value of b so that the quadratic equation
4x2 + bx + 9 = 0 has a) 1 real solution, b) 2 real
solutions, or c) 2 imaginary solutions.
4.9: Graph and Solve Quadratic Inequalities
1.
2.
3.
Objectives:
To graph quadratic
inequalities in two
variables
To graph a system
of quadratic
inequalities
To solve quadratic
inequalities in one
variable
Assignment
• P. 304-307: 3-5,
6-14 even, 18,
19, 20, 22, 27-66
M3, 69, 72, 76,
78, 79
• Challenge
Problems
“Can I see the sandwiches, again?”