Transcript Graphing & Solving Quadratic Inequalities

Graphing & Solving Quadratic
Inequalities 5.7
What is different in the graphing process
of an equality and an inequality?
How can you check the x-intercepts of a
quadratic equation or inequality?
Why is it important to test 3 intervals
after you have found the critical x
values?
There are four types of quadratic
inequalities in two variables.
y > ax2 + bx + c
y < ax2 + bx + c
y ≥ ax2 + bx + c
y ≤ ax2 + bx + c
Graphing a Quadratic Inequality in
Two Variables
y > x2 − 2x − 3
x = −b/2a=
y=
Where to shade…
• Test a point inside the parabola,
such as (0, 0).
• If you get a true statement when you
substitute the point into the inequality,
shade that area.
• If you get a false statement when you
substitute the point into the inequality,
shade the opposite area.
Rappelling
A manila rope used for rappelling down a cliff
can safely support a weight W (in pounds)
provided W ≤ 1480d 2
where d is the rope’s diameter (in inches).
Graph the inequality.
SOLUTION
Graph W = 1480d 2 for nonnegative values of
d. Because the inequality symbol is ≤, make
the parabola solid. Test a point inside the
parabola, such as (1, 2000).
Rappelling continued
W ≤ 1480d 2
2000 ≤ 1480(1)2
2000 ≤ 1480
Because (1, 2000) is not a solution,
shade the region below the parabola.
Solving a Quadratic Inequality by Graphing
(Looking for x intercepts)
x2 − 6x + 5 < 0.
Let y = 0 and
factor to solve.
(x−1)(x−5) = 0
x = 1 or x = 5
Solution 1<x<5
Solving a Quadratic Inequality by Graphing
2x2 + 3x− 3 ≥ 0
0.69 or −2.19
(Use quadratic formula to factor.)
Graphing a System of Quadratic Inequalities
y ≥ x2 −4
y< −x2 −x +2
x = −b/2a =
y=
x=−b/2a =
y=
Graph the system of quadratic inequalities.
y < – x2 + 4
y > x2 – 2x – 3
Inequality 1
Inequality 2
SOLUTION
STEP 1
Graph y ≤ – x2 + 4. The graph is the red region
inside and including the parabola y = – x2 + 4.
STEP 2
Graph y > x2– 2x – 3. The graph is the blue region
inside (but not including) the parabola y = x2 –2x – 3.
STEP 3
Identify the purple region where the two graphs overlap.
This region is the graph of the system.
Robotics
The number T of teams that have
participated in a robot-building
competition for high school
students can be modeled by
T(x) = 7.51x2 –16.4x + 35.0, 0 ≤ x ≤ 9
Where x is the number of years since 1992.
For what years was the number of teams
greater than 100?
You want to find the values of x
for which: T(x) > 100
7.51x2 – 16.4x – 65 > 0
Graph y = 7.51x2 – 16.4x – 65 on the domain 0 ≤ x ≤ 9.
The graph’s x-intercept is about 4.2. The graph lies
above the x-axis when 4.2 < x ≤ 9.
There were more than 100 teams participating in
the years 1997–2001.
Solve a quadratic inequality algebraically
Solve x2 – 2x > 15 algebraically.
SOLUTION
First, write and solve the equation obtained by
replacing > with = .
x2 – 2x = 15
x2 – 2x – 15 = 0
(x + 3)(x – 5) = 0
x = – 3 or x = 5
Write equation that corresponds
to original inequality.
Write in standard form.
Factor.
Zero product property
The numbers – 3 and 5 are the critical x-values of the
inequality x2 – 2x > 15. Plot – 3 and 5 on a number line,
using open dots because the values do not satisfy the
inequality. The critical x-values partition the number
line into three intervals. Test an x-value in each
interval to see if it satisfies the inequality.
Test x = – 4:
Test x = 1:
12 –2(1) 5 –1 >15
ANSWER
The solution is x < – 3 or x > 5.
Test x = 6:
62 –2(6) = 24 >15

What is different in the graphing process
of an equality and an inequality?
•If the equation is > or <, the parabola is
dashed.
•Either the inside or the outside of the
parabola is shaded. Pick a point not on
the parabola and see if it makes a true
statement. If true, shade where the point
is located. If not, shade the other area.
What is different in the graphing
process of an equality and an
inequality?
How can you check the x-intercepts of
a quadratic equation or inequality?
Why is it important to test 3 intervals
after you have found the critical x
values?
Assignment 1.9
Page 70, 3-24 every 3rd problem,
36-57 every 3rd problem
Assignment 1.9