QuestionFind the transformation of the functions $f(x) = |x|$ and $g(x) = 2f(x-3)$ .

Studdy Solution

STEP 1

Assumptions1. The function $f(x) = |x|$ is the absolute value function. . The function $g(x) =f(x-3)$ is a transformation of the function $f(x)$ .
3. The transformation involves a horizontal shift, a vertical stretch, and possibly a reflection.

STEP 2

First, let's identify the transformation of the function $f(x)$ to $g(x)$ .The general form of a transformation of a function is $g(x) = af(b(x - h)) + k$ , where- $a$ is the vertical stretch or compression factor. If $a <0$ , there is also a reflection about the x-axis. - $b$ is the horizontal stretch or compression factor. If $b <0$ , there is also a reflection about the y-axis. - $h$ is the horizontal shift. - $k$ is the vertical shift.

STEP 3

Now, let's compare the function $g(x) =2f(x-3)$ with the general form of a transformation.
Here, we can see that- $a =2$ , so there is a vertical stretch by a factor of2. - $b =1$ , so there is no horizontal stretch or compression. - $h =3$ , so there is a horizontal shift of3 units to the right. - $k =0$ , so there is no vertical shift.

STEP 4

So, the transformation of the function $f(x) = |x|$ to $g(x) =2f(x-3)$ involves- A vertical stretch by a factor of2. - A horizontal shift of3 units to the right.

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