Math Problem Statement

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)?

Solution

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

Function Definition

You are given a transformation of a function f(x)f(x), such that:

g(x)=af(bx)+cg(x) = a \cdot f(bx) + c

Three transformations are applied to f(x)f(x):

  1. Horizontal dilation by a factor of 2: This implies b=12b = \frac{1}{2}, since the dilation factor is applied inversely in transformations.
  2. Vertical dilation by a factor of 3: This means a=3a = 3.
  3. Vertical translation by 5 units: This means c=5c = 5.

Thus, the transformation becomes:

g(x)=3f(x2)+5g(x) = 3 \cdot f\left(\frac{x}{2}\right) + 5

Find g(4)g(-4)

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

g(4)=3f(42)+5=3f(2)+5g(-4) = 3 \cdot f\left(\frac{-4}{2}\right) + 5 = 3 \cdot f(-2) + 5

From the table, we know that:

f(2)=5f(-2) = 5

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

g(4)=35+5=15+5=20g(-4) = 3 \cdot 5 + 5 = 15 + 5 = 20

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


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).

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Math Problem Analysis

Mathematical Concepts

Function transformations
Horizontal and vertical dilations
Vertical translations

Formulas

g(x) = a f(bx) + c
Horizontal dilation: f(bx) where b = 1/2
Vertical dilation: a * f(x) where a = 3
Vertical translation: f(x) + c where c = 5

Theorems

Function Transformation Theorem

Suitable Grade Level

Grades 10-12