Question

identify the function from the given graph.options: $y = (x + 2)^3$, $y = (x - 2)^3$, $y = x^3 + 2$, $y = x^3 - 2$

Explanation:

Step1: Analyze the parent function

The parent function of a cubic function is \( y = x^3 \), which passes through the origin \((0,0)\) and has a point of inflection there.

Step2: Check the y-intercept of the graph

From the graph, when \( x = 0 \), we can see that \( y = 2 \). Let's check the y-intercept (when \( x = 0 \)) for each option:

• For \( y=(x + 2)^3 \), when \( x = 0 \), \( y=(0 + 2)^3=8 \), which does not match the y-intercept of the graph.
• For \( y=(x - 2)^3 \), when \( x = 0 \), \( y=(0 - 2)^3=-8 \), which does not match the y-intercept of the graph.
• For \( y=x^3 + 2 \), when \( x = 0 \), \( y=0^3+2 = 2 \), which matches the y-intercept of the graph.
• For \( y=x^3 - 2 \), when \( x = 0 \), \( y=0^3-2=-2 \), which does not match the y-intercept of the graph.

Step3: Verify the shape and transformation

The graph of \( y=x^3+2 \) is a vertical shift of the parent function \( y = x^3 \) up by 2 units. The given graph has the same shape as the cubic function \( y = x^3 \) (since the coefficient of \( x^3 \) is 1 in all options, so the shape is the same) and is shifted up, which is consistent with \( y=x^3 + 2 \). Also, we can check a point. Let's take \( x=-1 \), for \( y=x^3 + 2 \), \( y=(-1)^3+2=-1 + 2 = 1 \). Looking at the graph, when \( x=-1 \), the y - value is around 1, which is consistent. For \( y=(x + 2)^3 \) at \( x=-1 \), \( y=(-1 + 2)^3=1 \), but we already saw the y-intercept is wrong. For \( y=(x - 2)^3 \) at \( x = 1 \), \( y=(1 - 2)^3=-1 \), which does not match the graph's behavior. For \( y=x^3-2 \) at \( x = 1 \), \( y=1 - 2=-1 \), which also does not match.

Answer:

\( y = x^3 + 2 \) (the third option: \( y = x^3 + 2 \))