Question

The tables show values of the quadratic function g(x)
after a transformation of the quadratic function f(x)
.

x
f(x)

−1
2

0
−1

1
−2

2
−1

3
2

x
g(x)

−12
2

0
−1

12
−2

1
−1

32
2

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

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

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

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

Answer 1

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

Examining the values for \( f(x) \):

- \( f(-1) = 2 \)
- \( f(0) = -1 \)
- \( f(1) = -2 \)
- \( f(2) = -1 \)
- \( f(3) = 2 \)

Now looking at \( g(x) \):

- \( g(-12) = 2 \)
- \( g(0) = -1 \)
- \( g(12) = -2 \)
- \( g(1) = -1 \)
- \( g(32) = 2 \)

We can see that the \( y \)-values of \( g(x) \) match \( f(x) \) inversely around the x-axis when considering the corresponding \( x \)-values.

Next, let's analyze the relationship in terms of \( x \):
- The \( x \)-values are scaled by a factor of 12: \( -12, 0, 12, 1, 32 \) does not maintain the symmetry.
- Noticing that while \( f(-1) = 2 \) matches \( g(-12) = 2 \) and \( f(3) = 2 \) matches \( g(32) = 2 \), the coordinates appear transformed substantially.

But observing further:
- The x-values of \( g(x) \) are adjusted. If we check y-values, \( g(-12) = 2 \) matches \( f(-1) \) with increment in x-area.
- \( g(0) = -1 \) matches \( f(0) = -1 \).
- \( g(12) = -2 \) matches \( f(1) = -2 \).

Thus it appears that the \( x \)-values of the corresponding function \( g(x) \) are actually reflections of the \( f(x) \) values due to the steep change across the middle which outlines a reflection horizontally.

To summarize:
The transformation applied to obtain \( g(x) \) from \( f(x) \) appears to be a **horizontal dilation by a scale factor of 12**, as every \( x \) in \( g(x) \) represents \( f \) evaluated at different scaled intervals \( (every 12 units) \).

Given the provided responses, the closest and most fitting transformation is **Horizontal dilation by a scale factor of 2**.