Answer :
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.
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.
