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### Question 1:
We are given a linear function \( f(x) = 4x - 8 \) and asked to find the transformed function after translating it by 3 units to the right and 2 units up.

The general translation rule:
- Moving \( a \) units to the right: replace \( x \) with \( (x - a) \), so \( f(x - 3) \).
- Moving \( b \) units up: add \( b \) to the whole function, so \( f(x - 3) + 2 \).

Apply this to \( f(x) = 4x - 8 \):
1. \( f(x - 3) = 4(x - 3) - 8 = 4x - 12 - 8 = 4x - 20 \)
2. Adding 2 gives: \( g(x) = 4x - 20 + 2 = 4x - 18 \)

**Answer: D**

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### Question 2:
We are given the absolute value function \( f(x) = |2x + 4| \) and asked for the equation after the translation by the vector \( T = \left(\begin{array}{c} -2 \\ 1 \end{array}\right) \), which means shifting 2 units to the left and 1 unit up.

For translation:
1. Shift left by 2: Replace \( x \) with \( x + 2 \), so \( f(x + 2) = |2(x + 2) + 4| = |2x + 4 + 4| = |2x + 8| \).
2. Shift up by 1: Add 1, so \( g(x) = |2x + 8| + 1 \).

**Answer: E**

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### Question 3:
We are given the quadratic function \( f(x) = x^2 - 6x + 5 \) and asked to find the transformed function after shifting it 3 units to the left and 2 units up.

Steps:
1. Shift 3 units left: Replace \( x \) with \( x + 3 \), so \( f(x +

Given various functions (linear, absolute value, quadratic) and translations, solve for the transformed function using specific translation rules for shifting functions.

Learn how to apply translations to different types of functions, including linear, quadratic, and absolute value. This exercise involves shifting functions horizontally and vertically based on given translations, with a step-by-step solution for each type. Suitable for Grades 10-12.

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