The tables show values of the cubic function g(x)

Question

after a transformation of the cubic function f(x)
.

x
f(x)

−2
−6

−1
1

0
2

1
3

2
10

x
g(x)

−2
6

−1
−1

0
−2

1
−3

2
−10

What transformation was applied to f(x)
to obtain g(x)
?(1 point)
Responses

Horizontal dilation by a scale factor of 2
Horizontal dilation by a scale factor of 2

Reflection across the y-axis
Reflection across the y -axis

Reflection across the x-axis
Reflection across the x -axis

Vertical dilation by a scale factor of 2
Vertical dilation by a scale factor of 2
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Answer 1

To determine the transformation applied to \( f(x) \) to obtain \( g(x) \), we can analyze the given values from the tables.

The values for \( f(x) \) and \( g(x) \) at corresponding \( x \):

• \( f(-2) = -6 \) and \( g(-2) = 6 \)
• \( f(-1) = 1 \) and \( g(-1) = -1 \)
• \( f(0) = 2 \) and \( g(0) = -2 \)
• \( f(1) = 3 \) and \( g(1) = -3 \)
• \( f(2) = 10 \) and \( g(2) = -10 \)

Looking at the values, we notice:

1. For the input \( x = -2 \), \( g(-2) = -f(-2) \).
2. For the input \( x = -1 \), \( g(-1) = -f(-1) \).
3. For the input \( x = 0 \), \( g(0) = -f(0) \).
4. For the input \( x = 1 \), \( g(1) = -f(1) \).
5. For the input \( x = 2 \), \( g(2) = -f(2) \).

Every value of \( g(x) \) is the negative of the corresponding value \( f(x) \). This indicates that there has been a reflection across the x-axis.

Thus, the transformation applied to \( f(x) \) to obtain \( g(x) \) is:

Reflection across the x-axis.