Look at the following absolute valui $$ g(x)=\left|-\frac{1}{4} x\right|+8 $$ Vertical Translation? Rofizontal Translation? $$ \square $$ Dilation? $$ \square $$

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We start with the function

g(x) = | –(1/4)x | + 8

Notice that the absolute value makes the negative sign irrelevant, so we can rewrite the function as

g(x) = (1/4)|x| + 8

Now, comparing g(x) to the basic absolute value function f(x) = |x|, we can see the following transformations:

1. Vertical Translation: The “+8” at the end means that the graph is shifted upward by 8 units.

2. Horizontal Translation: There is no term of the form |x – h| inside the absolute value; therefore, there is no horizontal translation.

3. Dilation: The factor (1/4) multiplying the absolute value affects the “width” of the graph. Since (1/4) is inside the absolute value function (applied to x), it represents a horizontal dilation. More precisely, an expression of the form |ax| corresponds to a horizontal scaling by a factor of 1/|a|. Here, a = 1/4, so the horizontal dilation factor is 1/(1/4) = 4. This means the graph is stretched horizontally by a factor of 4 (it is wider than f(x)=|x|).

Summary:
• Vertical Translation? Yes, 8 units up.
• Horizontal Translation? No.
• Dilation? Yes, a horizontal stretching (by a factor of 4).
- **Vertical Translation**: 8 units up - **Horizontal Translation**: None - **Dilation**: Horizontally stretched by a factor of 4

Look at the following absolute valui \[ g(x)=\left|-\frac{1}{4} x\right|+8 \] Vertical Translation? Rofizontal Translation? \( \square \) Dilation? \( \square \)

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