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What kind of transformation converts the graph of $f(x) = 8|x - 5| + 5$ into the graph of $g(x) = 8|x - 5| + 10$ ? $\newline$ Choices: $\newline$ (A) translation $5$ units up $\newline$ (B) translation $5$ units left $\newline$ (C) translation $5$ units down $\newline$ (D) translation $5$ units right

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Describe function transformations

Full solution

Q. What kind of transformation converts the graph of

f(x) = 8|x - 5| + 5

into the graph of

g(x) = 8|x - 5| + 10

?

\newline

Choices:

\newline

(A) translation

5

units up

\newline

(B) translation

5

units left

\newline

(C) translation

5

units down

\newline

(D) translation

5

units right

Identify Vertex: Identify the vertex of the function $f(x) = 8|x - 5| + 5$ . $\newline$ The vertex of the absolute value function $f(x) = 8|x - 5| + 5$ is at the point where the expression inside the absolute value is zero, which is at $x = 5$ . The y-coordinate of the vertex is the constant term outside the absolute value, which is $+5$ . Therefore, the vertex of $f(x)$ is $(5, 5)$ .
Identify Vertex: Identify the vertex of the function $g(x) = 8|x - 5| + 10$ . Similarly, the vertex of the absolute value function $g(x) = 8|x - 5| + 10$ is at the point where the expression inside the absolute value is zero, which is at $x = 5$ . The $y$ -coordinate of the vertex is the constant term outside the absolute value, which is $+10$ . Therefore, the vertex of $g(x)$ is $(5, 10)$ .
Determine Transformation: Determine the transformation from $f(x)$ to $g(x)$ . Comparing the vertices of $f(x)$ and $g(x)$ , we see that the $x$ -coordinate remains the same ( $5$ ), but the $y$ -coordinate has increased from $5$ to $10$ . This indicates a vertical shift. The change in the $y$ -coordinate is from $5$ to $10$ , which is an increase of $5$ units.
Identify Transformation Type: Identify the type of transformation based on the change in the $y$ -coordinate. $\newline$ Since the $y$ -coordinate of the vertex has increased by $5$ units, the transformation is a translation $5$ units up.

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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{]}

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

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

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\newline

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