$g \text { is related to the parent function. Use function notation to write } g \text { in terms of } f \text {. }$ $g ( x ) = - 7 | x - 1 | - 4$
A) $g ( x ) = - 7 f ( x - 1 ) + 4$
B) $g ( x ) = - 7 f ( x + 1 ) - 4$
C) $g ( x ) = - 7 f ( x - 1 ) - 4$
D) $g ( x ) = 7 f ( x - 1 ) - 4$
E) $g ( x ) = 7 f ( x + 1 ) + 4$

Question

Quizplus · Accepted Answer

To write function g in terms of f, we need to think about how the function g is related to the parent function y = |x|.

The function y = |x| takes the absolute value of the input x, so it always outputs a non-negative value. It's graphed as a V-shape, with the vertex at the origin.

The function g = -7|x - 1| - 4 takes the absolute value of the expression (x - 1) and then multiplies it by -7 before subtracting 4. This means that g will also have a V-shape, centered at x = 1. However, the -7 in front of the absolute value means that the graph will be reflected over the x-axis, making the arms of the V point downwards. The -4 at the end simply shifts the entire graph down by 4 units.

To express this relationship in terms of f, we need to take the parent function y = |x| and make the same transformations that we made for g.

First, we need to flip the graph over the x-axis, which can be done by multiplying by -1. Then, we need to shift the graph one unit to the right, which can be done by replacing x with (x - 1). Finally, we need to stretch the graph vertically by a factor of 7, which can be done by multiplying by 7.

Putting it all together, we get:

g(x) = -7|x - 1| - 4
g(x) = -7 * |(x - 1)| - 4
g(x) = -7 * ( -1 * (x - 1) ) - 4
g(x) = -7 * (-x + 1) - 4
g(x) = 7x - 7 - 4
g(x) = 7x - 11

This corresponds to the functional notation of:

g(x) = f( x - 1 ) * -7 - 4

Which matches choice C.

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$g \text { is related to the parent function. Use function notation to write } g \text { in terms of } f \text {. }$