Question

The graph of the functionf (x)=x^2-8 will be shifted to the right 4 units and shifted up 14 units. What is the equation of the translated
function?

Answer 1

To find the equation of the translated function, we start with the original function:

\[
f(x) = x^2 - 8
\]

1. **Shifting Right**: To shift the graph to the right by 4 units, we replace \( x \) with \( x - 4 \):

\[
f(x - 4) = (x - 4)^2 - 8
\]

2. **Shifting Up**: To shift the graph up by 14 units, we add 14 to the function:

\[
f(x - 4) + 14 = (x - 4)^2 - 8 + 14
\]

This can be simplified:

\[
f(x - 4) + 14 = (x - 4)^2 + 6
\]

Now, we can expand \( (x - 4)^2 \):

\[
(x - 4)^2 = x^2 - 8x + 16
\]

So,

\[
f(x - 4) + 14 = x^2 - 8x + 16 + 6 = x^2 - 8x + 22
\]

Thus, the equation of the translated function is:

\[
f_{\text{translated}}(x) = x^2 - 8x + 22
\]