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Name
Class
Date
20.2 Connecting Intercepts
and Linear Factors
Essential Question: How are x-intercepts of a quadratic function and its linear factors related?
Explore
Resource
Locker
Connecting Factors and x–Intercepts
Use graphs and linear factors to find the x–intercepts of a parabola.
A
B
Graph y = x + 4 and y = x - 2 using a graphing calculator. Then
sketch the graphs on the grid.
8
Identify the x-intercept of each line.
4
The x-intercepts are -4 and 2 .
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C
y
x
-8
-4
The quadratic function y = (x + 4) (x - 2) is the product of the two
linear factors that have been graphed. Use a graphing calculator to
graph the function y = (x + 4) (x - 2). Then sketch a graph of the
quadratic function on the same grid with the linear factors that have
been graphed.
D
Identify the x-intercepts of the parabola.
E
What do you notice about the x–intercepts of the parabola?
0
4
8
-4
-8
The x-intercepts are -4 and 2 .
The x–intercepts of the parabola are the same as those of the two linear factors.
Reflect
1.
2.
y
Use a graph to determine whether 2x 2 + 5x - 12 is the product of
the linear factors 2x - 3 and x + 4.
Yes. The x-intercepts of y = 2x 2 + 5x - 12 are the same
4
as those of y = 2x - 3 and y = x + 4.
0
-8
Discussion Make a conjecture about the linear factors and
x-intercepts of a quadratic function.
The x-intercepts of the parabola are the same as those
of the two linear factors.
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Rewriting from Factored Form to Standard Form
Explain 1
A quadratic function is in factored form when it is written as y = k(x - a)(x - b) where k ≠ 0.
Example 1 Write each function in standard form.

y = 2(x + 1)(x - 4)

Multiply the two linear factors.
Multiply the two linear factors.
(
y = 3(
y = 2(x2 - 4x + x - 4)
y=3
y = 2(x - 3x - 4)
2
Multiply the resulting trinomial by 2.
x-5
)(
x-2
x 2 - 7x + 10
)
)
Multiply the resulting trinomial by 3.
y =2 x 2 - 6x - 8
The standard form of
y =2 x 2 - 6x - 8.
y = 3(x - 5)(x - 2)
y = 3x 2 - 21x + 30
y = 2(x + 1)(x - 4)is
The standard form of
y = 3(x - 5)(x - 2)is
y =3 x 2 - 21x + 30 .
Reflect
3.
How do the signs in the factors affect the sign of the x–term in the resulting trinomial?
If both signs in the factors are negative, then the x-term will be negative. If both signs in
the factors are positive, then the x-term will be positive.
4.
How do the signs in the factors affect the sign of the constant term in the resulting trinomial?
If both signs in the factors are different, then the constant term will be negative. If both
signs in the factors are the same, then the constant term will be positive.
Your Turn
Write each function in standard form.
y = (x - 7)(x - 1)
6.
y = 4(x 2 + 2x - 3)
y = x 2 - 8x + 7
y = x 2 - 8x + 7
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y = 4(x - 1)(x + 3)
y = 4(x 2 + 3x - x - 3)
y = x 2 - x - 7x + 7
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y = 4x 2 + 8x - 12
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Explain 2
Connecting Factors and Zeros
In the Explore you learned that the factors in factored form indicate the x-intercepts of a function. In a previous
lesson you learned that the x-intercepts of a graph are the zeros of the function.
Example 2 Write each function in standard form. Determine
each function.
x-intercepts and zeros of
y = 2(x - 1)(x - 3)

y = 2(x 2 - 3x - x + 3)
Write the function in standard form.
y = 2(x 2 - 4x + 3)
x–intercepts.
The factors indicate the
* Factor
(x – 1) indicates anx–intercept of 1.
* Factor
(x – 3) indicates anx–intercept of 3.
y = 2x2 - 8x + 6
The x-intercepts of a graph are the zeros of the function.
* An
x–intercept of 1 indicates that the function has a zero of 1.
* An
x–intercept of 3 indicates that the function has a zero of 3.
y = 2(x + 4)(x + 2)

(
Write the function in standard form.
The factors indicate the
* Factor
* Factor
y = 2 x+4
x–intercepts.
(x + 4) indicates anx–intercept of -4 .
(x - 3) indicates an x–intercept of –2.
y=2
y=
)( x+2 )
x2 + 6x + 8
2x 2 + 12x + 16
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The x–intercepts of a graph are the zeros of the function.
* An
x–intercept of –4 indicates that the function has a zero of -4 .
* An
x–intercept of
-2 indicates that the function has a zero of –2.
Reflect
7.
Discussion What are the zeros of a function?
A zero of a function is an x–value that makes the y-value equal to zero. The zeros of a
function are the x–intercepts.
8.
How many x-intercepts can quadratic functions have?
0,1, or 2
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Your Turn
Write each function in standard form. Determine x–intercepts and zeros of each function.
9.
y = -2(x + 5)(x + 1)
10. y = 5(x - 3)(x - 1)
y = -2(x + 5)(x + 1)
y = 5(x - 3)(x - 1)
y = -2x 2 - 12x - 10
y = 5x 2 - 20x + 15
The factored form shows that
x–intercepts are -5 and -1.;
the zeros are -5 and -1.
The factored form shows
that x-intercepts are 1 and 3;
the zeros are 1 and 3.
y = -2(x 2 + 6x + 5)
Explain 3
y = 5(x 2 - 4x + 3)
Writing Quadratic Functions Givenx-Intercepts
Given two quadratic functions ƒ(x) = (x - a)(x - b) and g(x) = k(x - a)(x - b), where k is any non-zero real
constant, examine the x–intercepts for each quadratic function.
f(x) = (x - a)(x - b)
0 = (x - a)(x - b)
x-a=0
or
x - b =0
x=a
x=b
g(x) = k(x - a)(x - b)
0 = k(x - a)(x - b)
0 = (x - a)(x - b)
x - a = 0 or x - b = 0
x=a
x=b
Notice that ƒ(x) = (x - a)(x - b) and g(x) = k(x - a)(x - b) have the same x-intercepts. You can use the
factored form to construct a quadratic function given the x–intercepts and the value of k.
Example 3 For the two given intercepts, use the factored form to generate a quadratic
function for each given constant k. Write the function in standard form.

x-intercepts: 2 and 5; k = 1, k = -2, k =3
Write the quadratic function with
k = 1. ƒ(x) = k(x - a)(x - b)
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ƒ(x) = 1(x - 2)(x - 5)
ƒ(x) = (x - 2)(x - 5)
ƒ(x) =x 2 - 7x + 10
Write the quadratic function with
k = -2.
ƒ(x) = -2(x - 2)(x - 5)
ƒ(x) = -2(x 2 - 7x + 10)
ƒ(x) = -2x 2 + 14x - 20
Write the quadratic function with
k = 3.
ƒ(x) = 3(x - 2)(x - 5)
ƒ(x) = 3(x 2 - 7x + 10)
ƒ(x) = 3x 2 - 21x + 30
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B
x-intercepts: -3 and 4; k = 1, k = -3, k = 2
Write the quadratic function with k = 1.
ƒ(x) =
(x + 3)(x - 4)
ƒ(x) =
x 2 - x - 12
Write the quadratic function with k = -3.
ƒ(x) = -3(x + 3)(x - 4)
ƒ(x) =
-3x 2 + 3x + 36
Write the quadratic function with k = 2.
ƒ(x) =
2(x + 3)(x - 4)
ƒ(x) = 2x 2 - 2x - 24
Reflect
11. How are the functions with same intercepts but different constant factors the same? How are they different?
Same intercepts and zeros, same axis of symmetry; They have different vertices.
Your Turn
For the given two intercepts and three values of k generate three quadratic functions.
Write the functions in factored form and standard form.
12. x-intercepts: 1 and 8; k = 1, k = -4, k = 5
13. x–intercepts: -7 and 3; k = 1, k = -5, k = 7
k = 1:
k = 1:
f(x) = (x - 1)(x - 8)
f(x) = (x + 7)(x - 3)
f(x) = (1x + 7)(x - 3)
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f(x) = 1(x - 1)(x - 8)
f(x) = x 2 - 9x + 8
f(x) = x 2 + 4x - 21
k = -4:
k = -5:
f(x) = -4(x 2 -9x + 8)
f(x) = -5(x 2 + 4x - 21)
f(x) = -5(x +7)(x - 3)
f(x) = -4(x - 1)(x - 8)
f(x) = -4x 2 + 36x - 32
f(x) = -5x 2 - 20x + 105
k = 5:
k = 7:
f(x) 5(x - 1)(x - 8)
f(x) = 7x + 7x - 3
f(x) = 5(x 2 - 9x + 8)
f(x) = 7x 2 + 4x - 21
f(x) = 5x 2 - 45x + 40
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Elaborate
14. If the x–intercepts of a quadratic function are 3 and 8, what can be said about the x–intercepts of its
linear factors?
The x–intercepts of its linear factors are 3 and 8.
15. If a quadratic function has only one zero, it has to occur at the vertex of the parabola. Using the graph of
a quadratic function, explain why.
If there is only one zero, then the graph of the function has only one x–intercept.
This can only occur when the vertex of the parabola lies on the x–axis because a
parabola is symmetric.
16. How are x–intercepts and zeros related?
The x–intercepts of a quadratic function are the same as the zeros of a quadratic function.
17. What would the factored form look like if there were only one x–intercept?
f(x) = k(x - a)2 where a is the x–intercept.
18. Essential Question Check-In How can you find x–intercepts of a quadratic function if its
linear factors are known?
The x–intercepts of a quadratic function are the same as the x–intercepts of its
linear factors.
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