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What kind of transformation converts the graph of $f(x) = -4|x - 7| + 3$ into the graph of $g(x) = -4|x - 7| - 1$ ? $\newline$ Choices: $\newline$ (A) translation $4$ units left $\newline$ (B) translation $4$ units up $\newline$ (C) translation $4$ units right $\newline$ (D) translation $4$ units down

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Describe function transformations

Full solution

Q. What kind of transformation converts the graph of

f(x) = -4|x - 7| + 3

into the graph of

g(x) = -4|x - 7| - 1

?

\newline

Choices:

\newline

(A) translation

4

units left

\newline

(B) translation

4

units up

\newline

(C) translation

4

units right

\newline

(D) translation

4

units down

Identify Vertex of $f(x)$ : Identify the vertex of the function $f(x) = -4|x - 7| + 3$ . The vertex of the absolute value function $f(x) = -4|x - 7| + 3$ is at the point where $x - 7 = 0$ , which is $x = 7$ . The $y$ -coordinate of the vertex is the constant term, which is $+3$ . Therefore, the vertex of $f(x)$ is at $(7, 3)$ .
Identify Vertex of $g(x)$ : Identify the vertex of the function $g(x) = -4|x - 7| - 1$ . Similarly, the vertex of the absolute value function $g(x) = -4|x - 7| - 1$ is at the point where $x - 7 = 0$ , which is $x = 7$ . The $y$ -coordinate of the vertex is the constant term, which is $-1$ . Therefore, the vertex of $g(x)$ is at $(7, -1)$ .
Determine Transformation: Determine the transformation from $f(x)$ to $g(x)$ . The transformation from $f(x)$ to $g(x)$ involves a change in the $y$ -coordinate of the vertex from $+3$ to $-1$ . This is a vertical shift. To find the amount of the shift, calculate the difference between the $y$ -coordinates of the vertices of $f(x)$ and $g(x)$ : $g(x)$ $0$ . Since the $y$ -coordinate decreased from $g(x)$ $2$ to $-1$ , this is a translation $g(x)$ $4$ units down.

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\newline

\newline

\text{}(A)g(x) = 9|x+2| - 4\text{]}

\newline

\text{}(B)g(x) = 9|x-2| - 4\text{]}

\newline

\text{}(C)g(x)=-9|x-2| +4\text{]}

\newline

\text{(D)}g(x) = -9|x + 2| + 4\text{]}

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\newline

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