9-4
Solving Quadratic Equations by Graphing
Preview
Warm Up
California Standards
Lesson Presentation
9-4
Solving Quadratic Equations by Graphing
Warm Up
1. Graph y = x2 + 4x + 3.
2. Identify the vertex and zeros of the
function above.
vertex:(–2 , –1);
zeros:–3, –1
9-4
Solving Quadratic Equations by Graphing
California
Standards
21.0 Students graph quadratic
functions and know that their roots are
the x-intercepts. Also covered:
23.0
9-4
Solving Quadratic Equations by Graphing
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.
y = ax2 + bx + c
0 = ax2 + bx + c
9-4
Solving Quadratic Equations by Graphing
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 of the related function. Recall that a
quadratic function may have two, one, or no zeros.
9-4
Solving Quadratic Equations by Graphing
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)
9-4
Solving Quadratic Equations by Graphing
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.
Check 2x2 – 18 = 0
2(3)2 – 18 0
2(9) – 18 0
18 – 18 0
0
0
Substitute 3
and –3 for x in
the original
equation.
2x2 – 18 = 0
2(–3)2 – 18
2(9) – 18
18 – 18
0
0
0
0
0
9-4
Solving Quadratic Equations by Graphing
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.
2x2 – 12x + 18 = 0
y = 2x2 – 12x + 18
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.
9-4
Solving Quadratic Equations by Graphing
Additional Example 1C: Solving Quadratic
Equations by Graphing
Solve the equation by graphing the related function.
2x2 + 4x = –3
Step 1 Write the related function.
y = 2x2 + 4x + 3
(–3, 9) 
Step 2 Graph the function.
 (1, 9)
• The axis of symmetry is x = –1.
• The vertex is (–1, 1).
(–2, 3) (0, 3)
• Two other points (0, 3) and
 (–1, 1)
(1, 9).
• Graph the points and reflect them
across the axis of symmetry.
9-4
Solving Quadratic Equations by Graphing
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.
9-4
Solving Quadratic Equations by Graphing
Check It Out! Example 1a
Solve the equation by graphing the related function.
x2 – 8x – 16 = 2x2
Step 1 Write the related function.
y = x2 + 8x + 16
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.
x = –4
●(–2 , 4)
●
●
●
● (–3, 1)
(–4, 0)
9-4
Solving Quadratic Equations by Graphing
Check It Out! 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.
Check y = x2 + 8x + 16
0
0
0
(–4)2 + 8(–4) + 16
16 – 32 + 16
0
Substitute –4 for x
in the quadratic
equation.
9-4
Solving Quadratic Equations by Graphing
Check It Out! Example 1b
Solve the equation by graphing the related
function.
6x + 10 = –x2
Step 1 Write the related function.
x = –3
y = x2 + 6x + 10
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)
9-4
Solving Quadratic Equations by Graphing
Check It Out! Example 1b Continued
Solve the equation by graphing the related
function.
x2 + 6x + 10 = 0
Step 3 Find the zeros.
The function appears
to have no zeros
The equation has no real-number solutions.
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Solving Quadratic Equations by Graphing
Check It Out! Example 1c
Solve the equation by graphing the related
function.
–x2 + 4 = 0
Step 1 Write the related function.
y = –x2 + 4
Step 2 Graph the function.
Use a graphing calculator.
Step 3 Find the zeros.
The function appears to have
zeros at (2, 0) and (–2, 0).
9-4
Solving Quadratic Equations by Graphing
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.
9-4
Solving Quadratic Equations by Graphing
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.
(–3, 0)
• The vertex is (–2, –1).
 
(–2, –1)
• Two other points are (–3, 0)
and (–4, 3)
• Graph the points and reflect them
across the axis of symmetry.
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Solving Quadratic Equations by Graphing
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.
Check 0 = x2 + 4x + 3
0
0
0
(–3)2 + 4(–3) + 3
9 – 12 + 3
0
0 = x2 + 4x + 3
0
0
0
(–1)2 + 4(–1) + 3
1–4+3
0
9-4
Solving Quadratic Equations by Graphing
Additional Example 2B: Finding Roots of Quadratic
Polynomials
Find the roots of x2 + x – 20
Step 1 Write the related equation.
2 + 4x – 20
y
=
x
2
0 = x + 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.
 (2, –15)



 (1, –18)
(–0.5, –20.25).
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Solving Quadratic Equations by Graphing
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.
Check 0 = x2 + x – 20
0 (–5)2 – 5 – 20
0
0
25 – 5 – 20
0
0 = x2 + x – 20
0
42 + 4 – 20
0
16 + 4 – 20
0
0
9-4
Solving Quadratic Equations by Graphing
Additional Example 2C: Finding Roots of Quadratic
Polynomials
Find the roots of x2 – 12x + 35
Step 1 Write the related equation.
2 – 12x + 35
y
=
x
2
0 = x – 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.
(4, 3)

(5, 0)  
 (6, –1).
9-4
Solving Quadratic Equations by Graphing
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.
Check 0 = x2 – 12x + 35
0 = x2 – 12x + 35
0
52 – 12(5) + 35
0
0
25 – 60 + 35
0
0
0
0
72 – 12(7) + 35
49 – 84 + 35
0
9-4
Solving Quadratic Equations by Graphing
Check It Out! 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

(–2, 0)
Step 3 Graph the related function.
(–1, –2) (–0.5, –2.25).
• 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.
9-4
Solving Quadratic Equations by Graphing
Check It Out! 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.
Check 0 = x2 + x – 2
0
(–2)2 + (–2) – 2
0 = x2 + x – 2
0
0
4–2–2
0
0
0
0
12 + (1) – 2
1+1–2
0
9-4
Solving Quadratic Equations by Graphing
Check It Out! 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

(
, 4) 
Step 2 Write the related function.
y = 9x2 – 6x + 1
Step 3 Graph the related function.
(0, 1)  
• The axis of symmetry is x = .
 ( , 0).
• The vertex is ( , 0).
• Two other points (0, 1) and
( , 4)
• Graph the points and reflect them
across the axis of symmetry.
9-4
Solving Quadratic Equations by Graphing
Check It Out! Example 2b Continued
Find the roots of each quadratic polynomial.
Step 4 Find the zeros.
There appears to be one zero at
is the root of 9x2 – 6x + 1.
Check 0 = 9x2 – 6x + 1
0
9(
)2 – 6(
0
1–2+1
0
0
)+1
. This means that
9-4
Solving Quadratic Equations by Graphing
Check It Out! 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.
  (1, 6)
y = 3x2 – 2x + 5

Step 3 Graph the related function.
• The axis of symmetry is x = .
• The vertex is ( ,
).
• Two other points (1, 6) and
(
,
)
• Graph the points and reflect them
across the axis of symmetry.
9-4
Solving Quadratic Equations by Graphing
Check It Out! 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.
9-4
Solving Quadratic Equations by Graphing
Additional Example 3: Application
A frog jumps straight up from the ground.
The quadratic function f(t) = –16t2 + 12t
models the frog’s 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.
Step 1 Write the related function.
0 = –16t2 + 12t
y = –16t2 + 12t
9-4
Solving Quadratic Equations by Graphing
Additional Example 3 Continued
Step 2 Graph the function.
Use a graphing calculator.
Step 3 Use
to estimate the
zeros.
The zeros appear to be 0 and 0.75.
The frog leaves the ground at 0
seconds and lands at 0.75
seconds.
The frog is off the ground for
about 0.75 seconds.
9-4
Solving Quadratic Equations by Graphing
Additional Example 3 Continued
Check 0 = –16t2 + 12t
0 –16(0.75)2 + 12(0.75)
Substitute 0.75 for t in
0 –16(0.5625) + 9
the quadratic
0
–9 + 9
equation.
0
0
9-4
Solving Quadratic Equations by Graphing
Check It Out! Example 3
What if…? A dolphin jumps out of the water.
The quadratic function y = –16x2 + 32x
models the dolphin’s 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.
Step 1 Write the related function
0 = –16x2 + 32x
y = –16x2 + 32x
9-4
Solving Quadratic Equations by Graphing
Check It Out! Example 3 Continued
Step 2 Graph the function.
Use a graphing calculator.
Step 3 Use
to estimate the
zeros.
The zeros appear to be 0 and 2.
The dolphin leaves the water
at 0 seconds and reenters at
2 seconds.
The dolphin is out of the
water for about 2 seconds.
9-4
Solving Quadratic Equations by Graphing
Check It Out! Example 3 Continued
Check 0 = –16x2 + 32x
0 –16(2)2 + 32(2)
0 –16(4) + 64
0
–64 + 64
0
0
Substitute 2 for x in
the quadratic
equation.
9-4
Solving Quadratic Equations by Graphing
Lesson Quiz
Solve each equation by graphing the related
function.
1. 3x2 – 12 = 0 2, –2
2. x2 + 2x = 8 –4, 2
3. 3x – 5 = x2
ø
4. 3x2 + 3 = 6x 1
5. A rocket is shot straight up from the ground.
The quadratic function f(t) = –16t2 + 96t
models the rocket’s height above the ground
after t seconds. How long does it take for the
rocket to return to the ground? 6 s