4.9: Graph and Solve Quadratic Inequalities

4.9: Graph and Solve Quadratic Inequalities
Objectives:
1. To graph quadratic
inequalities in two
variables
2. To graph a system of
quadratic inequalities
3. 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  ax2  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  ax2  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  ax2  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.
2
y

x
 2x  8
1.
2
2
x
 3x  1  y
2.
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:
2
x  25
Now taking the square root of both sides
doesn’t translate well with inequalities, so
solve the corresponding quadratic
2
equation:
x  25
x  5
These are our critical values. Graph them.
Quadratic Inequalities (One)
Consider the simple quadratic inequality:
2
x  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:
2
x  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.
2
2
Instead of x  4  0, consider x  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.
2
y

x
 121
1.
2
y

121

x
2.
Exercise 8
Find the value of b so that the quadratic
equation 4x2 + bx + 9 = 0 has a) 2 real
solutions, b) 1 real solution, or c) 2
imaginary solutions.
4.9: Graph and Solve Quadratic Inequalities
Objectives:
1. To graph quadratic
inequalities in two
variables
2. To graph a system of
quadratic inequalities
3. To solve quadratic
inequalities in one
variable
Assignment
• P. 304-307: 3-5, 614 even, 18, 19,
20, 22, 27-66 M3,
69, 72, 76, 78, 79
• Challenge
Problems
“Can I see the sandwiches?”