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

Question

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:

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