Let us analyze the problem step by step.

We are tasked to compare the y-intercepts of the functions $f(x)$ and $h(x)$:

Information Provided:

1. $f(x) = 3(x^2 + 2)$:

   • Expand $f(x)$:
     $f(x) = 3x^2 + 6.$
   • The y-intercept of $f(x)$ occurs when $x = 0$:
     Substituting $x = 0$ into $f(x)$:
     $f(0) = 3(0^2 + 2) = 6.$ Hence, the y-intercept of $f(x)$ is 6.

2. The table for $h(x)$:

   • For $h(x)$, the y-intercept is given by the value of $h(x)$ when $x = 0$.
     From the table, $h(0) = -6$.
     Thus, the y-intercept of $h(x)$ is -6.

Comparing y-intercepts:

• $f(x)$: $6$
• $h(x)$: $-6$
  The difference between the y-intercepts is:
  $6 - (-6) = 12.$

Thus, the y-intercept of $f(x)$ is 12 units above the y-intercept of $h(x)$.

Correct Answer:

D. The y-intercept of $f(x)$ is 12 units above the y-intercept of $h(x)$.

Would you like further clarification or details on any step?

Related Questions:

1. How do you find the y-intercept of any function from its equation?
2. How does shifting a quadratic function vertically affect its y-intercept?
3. How do you determine a function's intercepts from a table?
4. What role does the coefficient of $x^2$ play in a quadratic function's graph?
5. How do transformations like $f(x) + c$ affect a function's intercepts?

Tip:

When comparing functions, always check intercepts at $x = 0$ and analyze how transformations like shifts and scaling influence their values!