To determine which table shows that \( y = -f(x) \) results in a reflection of the function graphed over the x-axis, we need to understand what \( -f(x) \) means.
The function \( f(x) = |x + 1| \) takes positive values (or zero) for all inputs, and thus \( -f(x) \) will take those same values and invert their signs. More specifically, for each point \((x, f(x))\), the corresponding point for the transformed function \( (x, -f(x)) \) will be at \((x, -y)\).
Let’s evaluate the provided points:
The original points for \( f(x) = |x + 1| \) are:
- \( (-1, 0) \)
- \( (0, 1) \)
- \( (-2, 1) \)
- \( (1, 2) \)
- \( (-3, 2) \)
- \( (2, 3) \)
- \( (-4, 3) \)
Now, let's derive the corresponding points for \( -f(x) \):
- For \( x = -1, f(-1) = 0 \) ⇒ \( -f(-1) = 0 \) → point is \((-1, 0)\)
- For \( x = 0, f(0) = 1 \) ⇒ \( -f(0) = -1 \) → point is \((0, -1)\)
- For \( x = -2, f(-2) = 1 \) ⇒ \( -f(-2) = -1 \) → point is \((-2, -1)\)
- For \( x = 1, f(1) = 2 \) ⇒ \( -f(1) = -2 \) → point is \((1, -2)\)
- For \( x = -3, f(-3) = 2 \) ⇒ \( -f(-3) = -2 \) → point is \((-3, -2)\)
- For \( x = 2, f(2) = 3 \) ⇒ \( -f(2) = -3 \) → point is \((2, -3)\)
- For \( x = -4, f(-4) = 3 \) ⇒ \( -f(-4) = -3 \) → point is \((-4, -3)\)
Now, let's check the tables provided to see which one matches these results:
-
First Table:
- \( -1 \rightarrow -2 \)
- \( 1 \rightarrow 0 \)
- \( -2 \rightarrow 3 \)
Does not match.
-
Second Table:
- \( 1 \rightarrow -2 \)
- \( 0 \rightarrow -1 \)
- \( -3 \rightarrow -2 \)
Matches!
- \((1, -2)\)
- \((0, -1)\)
- \((-3, -2)\)
-
Third Table:
- \( 1 \rightarrow 2 \)
- \( -2 \rightarrow 1 \)
- \( -3 \rightarrow 2 \)
Does not match.
-
Fourth Table:
- \( 4 \rightarrow -3 \)
- \( 2 \rightarrow -1 \)
- \( 3 \rightarrow -2 \)
Does not match.
The second table is the one that reflects the results of \( y = -f(x) \), confirming that it shows a reflection of the function graphed over the x-axis.