Homework

1
Homework
Solve each equation by graphing the related
function. 1. 3x2 12 0 2. x2 2x 8 3. 3x
5 x2 4. 3x2 3 6x 5. A rocket is shot
straight up from the ground. The quadratic
function f(t) 16t2 96t models the rockets
height above the ground after t seconds. How long
does it take for the rocket to return to the
ground?
6 seconds
2
Warm Up 1. Graph y x2 4x 3. 2. Identify
the vertex and zeros of the function above.
vertex(2 , 1) zeros3, 1
3
Every quadratic function has a related quadratic
equation. The standard form of a quadratic
equation is ax2 bx c 0, where a, b, and c
are real numbers and a ? 0.
When writing a quadratic function as its related
quadratic equation, you replace y with 0.
Function y ax2 bx c Equation 0 ax2
bx c
4
One way to solve a quadratic equation in standard
form is to graph the related function and find
the x-values where y 0. In other words, find
the zeros, or roots, of the related function.
Recall that a quadratic function may have two,
one, or no zeros.
5
Additional Example 1A Solving Quadratic
Equations by Graphing
Solve the equation by graphing the related
function.
2x2 18 0
Step 1 Write the related function.
2x2 18 y, or y 2x2 0x 18
Step 2 Graph the function.

• The axis of symmetry is x 0.
• The vertex is (0, 18).
• Two other points (2, 10) and
• (3, 0)
• Graph the points and reflect them
• across the axis of symmetry.

x 0
?
?
(3, 0)
?
?
(2, 10)
?
(0, 18)
6
Additional Example 1A Continued
Solve the equation by graphing the related
function.
2x2 18 0
Step 3 Find the zeros.
The zeros appear to be 3 and 3.
The solutions of 2x2 18 0 are 3 and 3.
Substitute 3 and 3 for x in the original
equation.
7
Additional Example 1B Solving Quadratic
Equations by Graphing
Solve the equation by graphing the related
function.
12x 18 2x2
Step 1 Write the related function.
Step 2 Graph the function.
Use a graphing calculator.
Step 3 Find the zeros.
The only zero appears to be 3. This means 3 is
the only root of 2x2 12x 18.
8
Additional Example 1C Solving Quadratic
Equations by Graphing
Solve the equation by graphing the related
function.
2x2 4x 3
Step 2 Graph the function.

• The axis of symmetry is x 1.
• The vertex is (1, 1).
• Two other points (0, 3) and
• (1, 9).
• Graph the points and reflect them
• across the axis of symmetry.

9
Additional Example 1C Continued
Solve the equation by graphing the related
function.
2x2 4x 3
Step 3 Find the zeros.
The function appears to have no zeros.
The equation has no real-number solutions.
10
Partner Share! Example 1a
Solve the equation by graphing the related
function.
x2 8x 16 2x2
Step 1 Write the related function.
y x2 8x 16
x 4
Step 2 Graph the function.

• The axis of symmetry is x 4.
• The vertex is (4, 0).
• The y-intercept is 16.
• Two other points are (3, 1) and
• (2, 4).
• Graph the points and reflect them
• across the axis of symmetry.

?
?
(2 , 4)
?
(3, 1)
?
?
(4, 0)
11
Partner Share! Example 1a Continued
Solve the equation by graphing the related
function.
x2 8x 16 2x2
Step 3 Find the zeros.
The only zero appears to be 4.
Substitute 4 for x in the quadratic equation.
12
Partner Share! Example 1b
Solve the equation by graphing the related
function.
6x 10 x2
x 3
Step 2 Graph the function.

• The axis of symmetry is x 3 .
• The vertex is (3 , 1).
• The y-intercept is 10.
• Two other points (1, 5) and
• (2, 2)
• Graph the points and reflect them
• across the axis of symmetry.

(1, 5)
?
?
(2, 2)
?
?
?
(3, 1)
13
Partner Share! Example 1b Continued
Solve the equation by graphing the related
function.
x2 6x 10 0
The equation has no real-number solutions.
14
Check It Out! Example 1c
Solve the equation by graphing the related
function.
x2 4 0
Step 1 Write the related function.
Step 3 Find the zeros.
The function appears to have zeros at (2, 0) and
(2, 0).
15
Recall from Chapter 7 that a root of a polynomial
is a value of the variable that makes the
polynomial equal to 0. So, finding the roots of a
quadratic polynomial is the same as solving the
related quadratic equation.
16
Additional Example 2A Finding Roots of Quadratic
Polynomials
Find the roots of x2 4x 3
Step 1 Write the related equation.
0 x2 4x 3
y x2 4x 3
Step 2 Write the related function.
y x2 4x 3
Step 3 Graph the related function.
(4, 3)

• The axis of symmetry is x 2.
• The vertex is (2, 1).
• Two other points are (3, 0)
• and (4, 3)
• Graph the points and reflect them
• across the axis of symmetry.

(3, 0)
?
(2, 1)
17
Additional Example 2A Continued
Find the roots of each quadratic polynomial.
Step 4 Find the zeros.
The zeros appear to be 3 and 1. This means 3
and 1 are the roots of x2 4x 3.
18
Additional Example 2B Finding Roots of Quadratic
Polynomials
Find the roots of x2 x 20
Step 1 Write the related equation.
y x2 4x 20
0 x2 x 20
Step 2 Write the related function.
y x2 4x 20
Step 3 Graph the related function.

• The axis of symmetry is x .
• The vertex is (0.5, 20.25).
• Two other points are (1, 18)
• and (2, 15)
• Graph the points and reflect them
• across the axis of symmetry.

19
Additional Example 2B Continued
Find the roots of x2 x 20
Step 4 Find the zeros.
The zeros appear to be 5 and 4. This means 5
and 4 are the roots of x2 x 20.
20
Additional Example 2C Finding Roots of Quadratic
Polynomials
Find the roots of x2 12x 35
Step 1 Write the related equation.
y x2 12x 35
0 x2 12x 35
Step 2 Write the related function.
y x2 12x 35
Step 3 Graph the related function.

• The axis of symmetry is x 6.
• The vertex is (6, 1).
• Two other points (4, 3) and
• (5, 0)
• Graph the points and reflect them
• across the axis of symmetry.

21
Additional Example 2C Continued
Find the roots of x2 12x 35
Step 4 Find the zeros.
The zeros appear to be 5 and 7. This means 5 and
7 are the roots of x2 12x 35.
22
Partner Share! Example 2a
Find the roots of each quadratic polynomial.
x2 x 2
y x2 x 2
Step 1 Write the related equation.
0 x2 x 2
Step 2 Write the related function.
y x2 x 2
Step 3 Graph the related function.

• The axis of symmetry is x 0.5.
• The vertex is (0.5, 2.25).
• Two other points (1, 2) and
• (2, 0)
• Graph the points and reflect them
• across the axis of symmetry.

23
Partner Share! Example 2a Continued
Find the roots of each quadratic polynomial.
Step 4 Find the zeros.
The zeros appear to be 2 and 1. This means 2
and 1 are the roots of x2 x 2.
24
Partner Share! Example 2b
Find the roots of each quadratic polynomial.
9x2 6x 1
y 9x2 6x 1
Step 1 Write the related equation.
0 9x2 6x 1
?
Step 2 Write the related function.
y 9x2 6x 1
Step 3 Graph the related function.
?
25
Partner Share! Example 2b Continued
Find the roots of each quadratic polynomial.
Step 4 Find the zeros.
There appears to be one zero at . This means
that is the root of 9x2 6x 1.
26
Partner Share! Example 2c
Find the roots of each quadratic polynomial.
3x2 2x 5
y 3x2 2x 5
Step 1 Write the related equation.
0 3x2 2x 5
Step 2 Write the related function.
y 3x2 2x 5
?
Step 3 Graph the related function.
27
Partner Share! Example 2c Continued
Find the roots of each quadratic polynomial.
Step 4 Find the zeros.
There appears to be no zeros. This means that
there are no real roots of 3x2 2x 5.
28
Additional Example 3 Application
A frog jumps straight up from the ground. The
quadratic function f(t) 16t2 12t models the
frogs height above the ground after t seconds.
About how long is the frog in the air?
When the frog leaves the ground, its height is 0,
and when the frog lands, its height is 0. So
solve 0 16t2 12t to find the times when the
frog leaves the ground and lands.
29
Additional Example 3 Continued
Step 2 Graph the function.
Use a graphing calculator.
The frog is off the ground for about 0.75 seconds.
30
Additional Example 3 Continued
Check 0 16t2 12t
Substitute 0.75 for t in the quadratic equation.
?
31
Check It Out! Example 3
What if? A dolphin jumps out of the water. The
quadratic function y 16x2 32x models the
dolphins height above the water after x seconds.
About how long is the dolphin out of the water?
Check your answer.
When the dolphin leaves the water, its height is
0, and when the dolphin reenters the water, its
height is 0. So solve 0 16x2 32x to find the
times when the dolphin leaves and reenters the
water.
32
Check It Out! Example 3 Continued
Step 2 Graph the function.
Use a graphing calculator.
The dolphin is out of the water for about 2
seconds.
33
Check It Out! Example 3 Continued
Check 0 16x2 32x
Substitute 2 for x in the quadratic equation.
?
34
Lesson Review!
Solve each equation by graphing the related
function. 1. 3x2 12 0 2. x2 2x 8 3. 3x
5 x2 4. 3x2 3 6x 5. A rocket is shot
straight up from the ground. The quadratic
function f(t) 16t2 96t models the rockets
height above the ground after t seconds. How long
does it take for the rocket to return to the
ground?
2, 2
4, 2
ø
1
6 seconds