When f(x) = x3

Question

is reflected vertically, the new function is f(x) =

Complete the table below for the vertical reflection of the function f(x) =x3

x −3 −2 −1 0 1 2 3
-f(x)

Answer 1

To find the function that reflects \( f(x) = x^3 \) vertically, we need to take the negative of the function. Thus, the new function will be:

\[ f(x) = -x^3 \]

Now, let’s complete the table by calculating \( -f(x) \) for the given values of \( x \):

1. For \( x = -3 \): \[ f(-3) = -(-3^3) = -(-27) = 27 \]

2. For \( x = -2 \): \[ f(-2) = -(-2^3) = -(-8) = 8 \]

3. For \( x = -1 \): \[ f(-1) = -(-1^3) = -(-1) = 1 \]

4. For \( x = 0 \): \[ f(0) = -(0^3) = -0 = 0 \]

5. For \( x = 1 \): \[ f(1) = -(1^3) = -1 \]

6. For \( x = 2 \): \[ f(2) = -(2^3) = -8 \]

7. For \( x = 3 \): \[ f(3) = -(3^3) = -27 \]

Now, we can fill in the table:

\[ \begin{array}{|c|c|} \hline x & -f(x) \ \hline -3 & 27 \ -2 & 8 \ -1 & 1 \ 0 & 0 \ 1 & -1 \ 2 & -8 \ 3 & -27 \ \hline \end{array} \]

So, the completed table is shown above.