Question

the function $f(x) = x^3$ is shown in red in the graph below. it is translated both vertically and horizontally. what is the equation of the blue function, $g(x)$?

graph with red curve $f(x)=x^3$ and blue curve $g(x)$

options:

• $g(x)=(x + 5)^3 + 2$
• $g(x)=(x - 5)^3 + 2$
• $g(x)=(x - 5)^3 - 2$
• $g(x)=(x + 5)^3 - 2$

Explanation:

Step1: Identify parent function vertex

The parent function $f(x)=x^3$ has a point (inflection point) at $(0,0)$.

Step2: Locate translated vertex

The blue function's inflection point is at $(-3,-2)$? No, correction: Observe the shift: red $f(x)=x^3$ has inflection at $(0,0)$; blue graph's inflection is shifted **left 3? No, wait, look at the x-axis: red crosses (0,0), blue crosses x at x=-3? No, wait the options have 5. Wait, no: the red graph's inflection is (0,0), blue graph's inflection is at $(-5,-2)$? Wait no, let's check the shift rules:
For horizontal shift: $f(x-h)$ shifts right h, $f(x+h)$ shifts left h.
Vertical shift: $f(x)+k$ shifts up k, $f(x)-k$ shifts down k.
Looking at the graph: the red $x^3$ (inflection at (0,0)) is shifted left 5 units (so $x$ becomes $x+5$) and down 2 units (subtract 2).

Step3: Match to formula

Apply shifts to $f(x)=x^3$:
$g(x) = (x+5)^3 - 2$

Answer:

$\boldsymbol{g(x)=(x+5)^3 - 2}$ (the fourth option)