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When you analyze transformations, it helps to look at an important point (or points) on the parent function and see where that point moved.

The vertex of the absolute value parent function is [tex]$(0,0)$[/tex], but the transformed function has a vertex of [tex]$(-4,3)$[/tex]. This shows that the graph shifted left 4 units and up 3 units. Therefore, the transformed function is [tex]$g(x)=|x+4|+3$[/tex].

Check Your Understanding - Question 2 of 2
How is the graph of [tex]$h(x)=\left|\frac{1}{4} x\right|+6$[/tex] different from the graph of [tex]$f(x)=|x|$[/tex]?

A. The graph of [tex]$h(x)$[/tex] is stretched horizontally by a factor of 4 and shifted up 6 units.

B. The graph of [tex]$h(x)$[/tex] is compressed vertically by a factor of 4 and shifted right 6 units.

C. The graph of [tex]$h(x)$[/tex] is compressed horizontally by a factor of 4 and shifted up 6 units.

D. The graph of [tex]$h(x)$[/tex] is stretched vertically by a factor of 4 and shifted right 6 units.



Answer :

GinnyAnswer
To determine how the graph of [tex]\(h(x) = \left|\frac{1}{4}x\right| + 6\)[/tex] differs from the graph of [tex]\(f(x) = |x|\)[/tex], we need to analyze the transformations applied to the basic absolute value function [tex]\(f(x)\)[/tex].

1. Horizontal Stretch:
- In [tex]\(h(x) = \left|\frac{1}{4}x\right| + 6\)[/tex], the term [tex]\(\frac{1}{4}x\)[/tex] inside the absolute value indicates a horizontal transformation. Specifically, multiplying [tex]\(x\)[/tex] by [tex]\(\frac{1}{4}\)[/tex] stretches the graph horizontally by a factor of 4. This is because [tex]\(|x|\)[/tex] compressed horizontally would become [tex]\(|kx|\)[/tex] where [tex]\(0 < k < 1\)[/tex]; here [tex]\(k = \frac{1}{4}\)[/tex], and this represents a horizontal stretch by the reciprocal factor, which is 4.

2. Vertical Shift:
- The term [tex]\(+6\)[/tex] outside the absolute value indicates a vertical shift. Specifically, it shifts the graph up by 6 units.

Combining these transformations, we can summarize the effects on the graph of [tex]\(f(x) = |x|\)[/tex] to obtain the graph of [tex]\(h(x) = \left|\frac{1}{4}x\right| + 6\)[/tex] as follows:
- The graph is horizontally stretched by a factor of 4.
- The graph is shifted up by 6 units.

Let's match these transformations with the provided choices:
- Choice A: The graph of [tex]\(h(x)\)[/tex] is stretched horizontally by a factor of 4 and shifted up 6 units.
- Choice B: The graph of [tex]\(h(x)\)[/tex] is compressed vertically by a factor of 4 and shifted right 6 units. (Not correct regarding the vertical compression and horizontal shift.)
- Choice C: The graph of [tex]\(h(x)\)[/tex] is compressed horizontally by a factor of 4 and shifted up 6 units. (Not correct regarding the horizontal compression.)
- Choice D: The graph of [tex]\(h(x)\)[/tex] is stretched vertically by a factor of 4 and shifted right 6 units. (Not correct regarding vertical stretch and horizontal shift.)

The correct transformation matches Choice A: The graph of [tex]\(h(x)\)[/tex] is stretched horizontally by a factor of 4 and shifted up 6 units.

Therefore, the correct answer is A.

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