Question

Find g(x), where g(x) is the translation 5 units left and 1 unit down of f(x)=x^2.
Write your answer in the form a(x–h)^2+k, where a, h, and k are integers.

Answer 1

To find the function \( g(x) \) that represents the translation of \( f(x) = x^2 \) by 5 units left and 1 unit down, we start by understanding how to apply translations to a function.

1. **Translation 5 units left**: To move a function left by \( h \) units, we replace \( x \) with \( x + h \). For a translation of 5 units left, we replace \( x \) with \( x + 5 \):
\[
f(x) = x^2 \quad \Rightarrow \quad f(x + 5) = (x + 5)^2
\]

2. **Translation 1 unit down**: To move a function down by \( k \) units, we subtract \( k \) from the function's output. Here, \( k = 1 \):
\[
g(x) = f(x + 5) - 1 = (x + 5)^2 - 1
\]

Now, we can simplify \( (x + 5)^2 - 1 \):
\[
(x + 5)^2 = x^2 + 10x + 25
\]
Thus,
\[
g(x) = x^2 + 10x + 25 - 1 = x^2 + 10x + 24
\]

Next, we want to express \( g(x) \) in the vertex form \( a(x-h)^2 + k \). We can complete the square for the quadratic \( x^2 + 10x + 24 \):

1. Take the coefficient of \( x \), which is \( 10 \), divide it by \( 2 \) to get \( 5 \), and square it to obtain \( 25 \).
2. Rewrite the expression:
\[
g(x) = x^2 + 10x + 25 - 1 = (x + 5)^2 - 1
\]

So, we have:
\[
g(x) = 1(x + 5)^2 - 1
\]

In the form \( a(x-h)^2 + k \), we can identify:
- \( a = 1 \)
- \( h = -5 \) (so, \( h \) is actually \( -5 \))
- \( k = -1 \)

Thus, the final form of \( g(x) \) is:
\[
g(x) = 1(x + 5)^2 - 1
\]

Therefore:
\[
\boxed{g(x) = (x + 5)^2 - 1}
\]