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Let's break down the given problem step by step to solve for \( g(-4) \):

### Function Definition
You are given a transformation of a function \( f(x) \), such that:

\[
g(x) = a \cdot f(bx) + c
\]

Three transformations are applied to \( f(x) \):
1. **Horizontal dilation** by a factor of 2: This implies \( b = \frac{1}{2} \), since the dilation factor is applied inversely in transformations.
2. **Vertical dilation** by a factor of 3: This means \( a = 3 \).
3. **Vertical translation** by 5 units: This means \( c = 5 \).

Thus, the transformation becomes:

\[
g(x) = 3 \cdot f\left(\frac{x}{2}\right) + 5
\]

### Find \( g(-4) \)

Now, we need to evaluate \( g(-4) \):

\[
g(-4) = 3 \cdot f\left(\frac{-4}{2}\right) + 5 = 3 \cdot f(-2) + 5
\]

From the table, we know that:

\[
f(-2) = 5
\]

Substituting this value into the equation for \( g(-4) \):

\[
g(-4) = 3 \cdot 5 + 5 = 15 + 5 = 20
\]

Thus, the value of \( g(-4) \) is **20**, which corresponds to answer choice **D**.

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Would you like further explanation or have additional questions?

Here are 5 related questions to expand your understanding:
1. How do horizontal and vertical dilations affect the graph of a function?
2. What is the impact of a vertical translation on the range of a function?
3. How can you generalize the transformation formula for different dilation and translation factors?
4. How would the result change if the vertical dilation factor were 4 instead of 3?
5. What is the significance of the horizontal dilation being applied inversely in the transformation?

**Tip:** When dealing with transformations, always adjust horizontal changes with inverse factors (e.g., dividing by the dilation factor instead of multiplying).

The table gives values for a polynomial function f at selected values of x. Let g(x) = a f(bx) + c, where a, b, and c are positive constants. In the xy-plane, the graph of g is constructed by applying three transformations to the graph of f: a horizontal dilation by a factor of 2, a vertical dilation by a factor of 3, and a vertical translation by 5 units. What is the value of g(-4)?

Learn how to solve a function transformation problem where g(x) = a f(bx) + c with specific values for a, b, and c. This example uses a horizontal dilation by a factor of 2, vertical dilation by a factor of 3, and vertical translation by 5 units to find the value of g(-4). Suitable for Grades 10-12.

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