Math

QuestionExplain how g(x)=2x3g(x) = 2|x-3| is derived from f(x)=xf(x) = |x| and how to graph both functions.

Studdy Solution

STEP 1

Assumptions1. The function f(x)=xf(x) = |x| is the absolute value function. . The function g(x)=f(x3)g(x) =f(x-3) is a transformation of the function f(x)f(x).
3. The transformation involves a horizontal shift and a vertical stretch.

STEP 2

First, let's understand the function f(x)=xf(x) = |x|. This is the absolute value function, which always returns the non-negative value of xx. The graph of this function is a V shape with the vertex at the origin (0,0).

STEP 3

Now, let's understand the function g(x)=2f(x3)g(x) =2f(x-3). This function is a transformation of f(x)f(x). The value inside the function f(x3)f(x-3) indicates a horizontal shift, and the coefficient2 outside the function indicates a vertical stretch.

STEP 4

The horizontal shift is determined by the value inside the function f(x3)f(x-3). The graph of f(x)f(x) is shifted to the right by3 units because of the 3-3 inside the function.

STEP 5

The vertical stretch is determined by the coefficient2 outside the function. The graph of f(x)f(x) is stretched vertically by a factor of2 because of this coefficient.

STEP 6

To graph the function g(x)g(x), we start by graphing the function f(x)f(x), then apply the transformations.

STEP 7

First, graph the function f(x)=xf(x) = |x|. This is a V shape with the vertex at the origin (0,0).

STEP 8

Next, apply the horizontal shift. Move the graph of f(x)f(x) to the right by3 units.

STEP 9

Finally, apply the vertical stretch. Stretch the graph of f(x)f(x) vertically by a factor of2.

STEP 10

The graph of g(x)=2f(x3)g(x) =2f(x-3) is now a V shape with the vertex at the point (3,0) and is twice as steep as the graph of f(x)=xf(x) = |x|.

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