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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 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

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

1 answer

To determine which table shows that \( y = -f(x) \) results in a reflection over the x-axis, we need to identify the values of \( -f(x) \) at various \( x \) points, based on the table of \( f(x) \).

When reflecting a function over the x-axis:

  • If \( f(x) \) is positive, \( -f(x) \) will be negative.
  • If \( f(x) \) is negative, \( -f(x) \) will be positive.
  • If \( f(x) \) is zero, \( -f(x) \) will also be zero.

Given the options provided, we'll analyze each table to find the correct \( -f(x) \) values corresponding to the \( f(x) \) values.

  1. Option 1

    • \[ x: -2, 0, -3, -1, -6, -2 \]
    • \[ f(x): -2, 0, -3, -1, -6, -2 \]
    • \[ -f(x): 2, 0, 3, 1, 6, 2 \]

    Hence:

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

    These values correspond correctly to the reflection over the x-axis.

  2. Option 2

    • \[ x: -2, 0, -3, 1, -6, 2 \]
    • \[ f(x): -2, 0, -3, 1, -6, 2 \]
    • \[ -f(x): 2, 0, 3, -1, 6, -2 \]

  3. Option 3

    • \[ x: 2, 0, 3, -1, 6, -2 \]
    • \[ f(x): 2, 0, 3, -1, 6, -2 \]
    • \[ -f(x): -2, 0, -3, 1, -6, 2 \]

  4. Option 4

    • \[ x: 2, 0, 3, 1, 6, 2 \]
    • \[ f(x): 2, 0, 3, 1, 6, 2 \]
    • \[ -f(x): -2, 0, -3, -1, -6, -2 \]

Upon reviewing these options, it is clear that Option 1 correctly represents the reflection of the function over the x-axis, as the signs of \( -f(x) \) values correctly correspond to reflections of the provided \( f(x) \) values.

Therefore, the answer is Option 1.