Question

if the formula $y = x^{3}$ is changed by adding seven as shown in red below. which of the following best describes the resulting change for each of the functions?
function
$f(x)=x^{3}+7$
$f(x)=(x + 7)^{3}$
transformation
a. the +7 would directly affect the x - values, so the graph would shift horizontally.
b. the +7 would have no effect.
c. the +7 would directly affect the y - values, so the graph would shift vertically.

Explanation:

Step1: Analyze $f(x)=x^{3}+7$

For a function $y = f(x)+k$, when $k = 7$ (in this case), for each $x$-value, the $y$-value of the original function $y=x^{3}$ is increased by 7. This is a vertical - shift transformation.

Step2: Analyze $f(x)=(x + 7)^{3}$

For a function $y=f(x + h)$, when $h = 7$ (in this case), the graph of the original function $y=x^{3}$ is shifted horizontally. The $x$-values are affected. For the original function, to get the same $y$-value in the new function, we need to use an $x$ that is 7 units less than before.

For $f(x)=x^{3}+7$, the answer is c. The +7 would directly affect the y - values, so the graph would shift vertically.
For $f(x)=(x + 7)^{3}$, the answer is a. The +7 would directly affect the x - values, so the graph would shift horizontally.

Answer:

For $f(x)=x^{3}+7$: c. The +7 would directly affect the y - values, so the graph would shift vertically.
For $f(x)=(x + 7)^{3}$: a. The +7 would directly affect the x - values, so the graph would shift horizontally.