Which table shows that y= -f(x) results in a reflection of the function graphed over the. -axis

On graph f(x)= |x+1| points (-1,0),(0,1),(-2,1),(1,2),(-3,2),(2,3),(-4,3)

Tables
x -f(x)
-1 -2
1 0
-2 3

x -f(x)
1 -2
0 -1
-3 -2

x -f(x)
1 2
-2 1
-3 2

x -f(x)
4 -3
2 -1
3 -2

1 answer

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:

  1. First Table:

    • \( -1 \rightarrow -2 \)
    • \( 1 \rightarrow 0 \)
    • \( -2 \rightarrow 3 \)

    Does not match.

  2. Second Table:

    • \( 1 \rightarrow -2 \)
    • \( 0 \rightarrow -1 \)
    • \( -3 \rightarrow -2 \)

    Matches!

    • \((1, -2)\)
    • \((0, -1)\)
    • \((-3, -2)\)
  3. Third Table:

    • \( 1 \rightarrow 2 \)
    • \( -2 \rightarrow 1 \)
    • \( -3 \rightarrow 2 \)

    Does not match.

  4. 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.