Math

QuestionWhat transformation changes f(x)=2x9f(x)=2|x|-9 to g(x)=2x+9g(x)=-2|x|+9?

Studdy Solution

STEP 1

Assumptions1. The original function is f(x)=x9f(x)=|x|-9 . The transformed function is g(x)=x+9g(x)=-|x|+9
3. We are looking for the type of transformation that converts f(x)f(x) into g(x)g(x)

STEP 2

Let's first look at the general form of the absolute value function, which is y=axh+ky=a|x-h|+k. Here, aa affects the vertical stretch or shrink and the reflection across the x-axis, hh affects the horizontal shift, and kk affects the vertical shift.

STEP 3

Comparing f(x)=2x9f(x)=2|x|-9 with the general form, we can see that a=2a=2 and k=9k=-9 for f(x)f(x). This means the graph of f(x)f(x) is vertically stretched by a factor of2 and shifted down by9 units.

STEP 4

Similarly, comparing g(x)=2x+9g(x)=-2|x|+9 with the general form, we can see that a=2a=-2 and k=9k=9 for g(x)g(x). This means the graph of g(x)g(x) is vertically stretched by a factor of2, reflected across the x-axis, and shifted up by9 units.

STEP 5

Comparing the transformations of f(x)f(x) and g(x)g(x), we can see that the graph of f(x)f(x) is transformed into the graph of g(x)g(x) by reflecting it across the x-axis and shifting it up by18 units.
So, the transformation that converts the graph of f(x)f(x) into the graph of g(x)g(x) is a reflection across the x-axis and a vertical shift upwards by18 units.

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