Use this to answer the question.

Question

A graphed function starting on the x axis has an arrow at the other end passes through three points on a coordinate plane. The x-axis ranges from negative 9 to 9 in unit increments and the y-axis ranges from negative 11 to 11 in unit increments.

Which table shows that y=−f(x)
results in a reflection over the x-axis?

(1 point)
Responses

x −f(x)
2 0
3 −1
6 −2
x f left parenthesis x right parenthesis 2 0 3 negative 1 6 negative 2

x −f(x)
2 0
3 1
6 2 x f left parenthesis x right parenthesis 2 0 3 1 6 2

x −f(x)
−2
0
−3
−1
−6
−2
x f left parenthesis x right parenthesis negative 2 0 negative 3 negative 1 negative 6 negative 2

x −f(x)
−2
0
−3
1
−6
2 x f left parenthesis x right parenthesis negative 2 0 negative 3 1 negative 6 2
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Answer 1

To determine which table shows that $y=-f(x)$ results in a reflection over the x-axis, we need to look for a table where the values of $-f(x)$ are the negative counterparts of the values of $f(x)$.

Let's examine each response:

1. • For $x=2$, $f(2)=0$ and $-f(2)=0$ (not a reflection)
   • For $x=3$, $f(3)=-1$ and $-f(3)=1$ (this shows reflection)
   • For $x=6$, $f(6)=-2$ and $-f(6)=2$ (this shows reflection)
2. • For $x=2$, $f(2)=0$ and $-f(2)=0$ (not a reflection)
   • For $x=3$, $f(3)=1$ and $-f(3)=-1$ (not a reflection)
   • For $x=6$, $f(6)=2$ and $-f(6)=-2$ (not a reflection)
3. • For $x=-2$, $f(-2)=-2$ and $-f(-2)=2$ (this shows reflection)
   • For $x=0$, $f(0)=0$ and $-f(0)=0$ (not a reflection)
   • For $x=-3$, $f(-3)=-1$ and $-f(-3)=1$ (this shows reflection)
   • For $x=-6$, $f(-6)=-2$ and $-f(-6)=2$ (this shows reflection)
4. • For $x=-2$, $f(-2)=-2$ and $-f(-2)=2$ (this shows reflection)
   • For $x=0$, $f(0)=0$ and $-f(0)=0$ (not a reflection)
   • For $x=-3$, $f(-3)=1$ and $-f(-3)=-1$ (not a reflection)
   • For $x=-6$, $f(-6)=2$ and $-f(-6)=-2$ (not a reflection)

From this analysis, the first option shows the necessary reflection for $f(3)$ and $f(6)$, and the third option also demonstrates the reflection for points within $f(-2)$ and $f(-3)$.

However, because the pairs in the first response demonstrate the reflection consistently under positive values, the response that best shows that $y=-f(x)$ corresponds to a reflection over the x-axis is:

First Response (y = -f(x)):

• For $x=2,-f(2)=0$
• For $x=3,-f(3)=1$
• For $x=6,-f(6)=2$

So, I would select the first option as the answer.