Transcript Slide 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 rocket’s height above the ground
after t seconds. How long does it take for the rocket
6 seconds
to return to the ground?
Warm Up
1. Graph y = x2 + 4x + 3.
2. Identify the vertex and zeros of the
function above.
vertex:(–2 , –1);
zeros:–3, –1
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:
Equation:
y = ax2 + bx + c
0 = ax2 + bx + c
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.
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)
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
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.
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.
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
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)
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.
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.
Partner Share! 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)
Partner Share! 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.
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.
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.
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
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).
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
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).
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
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

(–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.
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.
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
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

(
, 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.
Partner Share! 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
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.
  (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.
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.
Lesson Review!
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
6 seconds
to return to the ground?