Question

When f(x) = (x-1)^2 is reflected over the x-axis the new function is f(x) = __
Complete the table
f(x) = (x-1) ^2
x 0 2 4 6 8
-f(x) __ __ __ __ __

Answer 1

To reflect the function \( f(x) = (x - 1)^2 \) over the x-axis, we need to take the negative of the function. Therefore, the new function will be \( f(x) = -(x - 1)^2 \).

Now let's calculate the values of \( -f(x) \) for the given values of \( x \):

1. **When \( x = 0 \)**:
\[
f(0) = (0 - 1)^2 = 1
\]
\[
-f(0) = -1
\]

2. **When \( x = 2 \)**:
\[
f(2) = (2 - 1)^2 = 1
\]
\[
-f(2) = -1
\]

3. **When \( x = 4 \)**:
\[
f(4) = (4 - 1)^2 = 9
\]
\[
-f(4) = -9
\]

4. **When \( x = 6 \)**:
\[
f(6) = (6 - 1)^2 = 25
\]
\[
-f(6) = -25
\]

5. **When \( x = 8 \)**:
\[
f(8) = (8 - 1)^2 = 49
\]
\[
-f(8) = -49
\]

Now, we can fill in the table:

\[
\begin{array}{|c|c|c|}
\hline
x & f(x) & -f(x) \\
\hline
0 & 1 & -1 \\
2 & 1 & -1 \\
4 & 9 & -9 \\
6 & 25 & -25 \\
8 & 49 & -49 \\
\hline
\end{array}
\]

So, the completed table is:

- When \( x = 0 \), \( -f(x) = -1 \)
- When \( x = 2 \), \( -f(x) = -1 \)
- When \( x = 4 \), \( -f(x) = -9 \)
- When \( x = 6 \), \( -f(x) = -25 \)
- When \( x = 8 \), \( -f(x) = -49 \)