Question

Given the parent function of
f
(
x
)
=
1
5
x
f(x)=15
x
, write the equation that would represent a vertical shift down 6 units and a horizontal shift left 8 units.



g
(
x
)
=
g(x)=

Answer 1

To shift the function \( f(x) = \frac{1}{5}x \) vertically down 6 units and horizontally left 8 units, we need to apply transformations accordingly.

1. **Vertical Shift Down 6 Units**: To shift down by 6 units, we subtract 6 from the function:
\[
f(x) - 6
\]

2. **Horizontal Shift Left 8 Units**: To shift left by 8 units, we replace \( x \) with \( x + 8 \):
\[
f(x + 8)
\]

Combining both transformations, we start with \( f(x) \), make the horizontal shift, and then apply the vertical shift:

Starting from:
\[
f(x) = \frac{1}{5}x
\]
1. Replace \( x \) with \( x + 8 \):
\[
f(x + 8) = \frac{1}{5}(x + 8)
\]
2. Now, subtract 6 from the entire function:
\[
g(x) = f(x + 8) - 6 = \frac{1}{5}(x + 8) - 6
\]

So, overall, the equation representing both transformations is:
\[
g(x) = \frac{1}{5}(x + 8) - 6
\]

Simplifying this gives:
\[
g(x) = \frac{1}{5}x + \frac{8}{5} - 6
\]

To combine the constants:
\(-6\) can be written as \(-30/5\), so:
\[
g(x) = \frac{1}{5}x + \frac{8}{5} - \frac{30}{5} = \frac{1}{5}x - \frac{22}{5}
\]

Thus, the final transformed function is:
\[
g(x) = \frac{1}{5}x - \frac{22}{5}
\]