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What kind of transformation converts the graph of $f(x) = 4|x + 9| + 3$ into the graph of $g(x) = 4|x + 7| + 3$ ? $\newline$ Choices: $\newline$ (A) translation $2$ units right $\newline$ (B) translation $2$ units up $\newline$ (C) translation $2$ units left $\newline$ (D) translation $2$ units down

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Describe function transformations

Full solution

Q. What kind of transformation converts the graph of

f(x) = 4|x + 9| + 3

into the graph of

g(x) = 4|x + 7| + 3

?

\newline

Choices:

\newline

(A) translation

2

units right

\newline

(B) translation

2

units up

\newline

(C) translation

2

units left

\newline

(D) translation

2

units down

Compare Expressions: To determine the type of transformation, we need to compare the expressions inside the absolute value function of $f(x)$ and $g(x)$ . In $f(x)$ , the expression is $|x + 9|$ , and in $g(x)$ , it is $|x + 7|$ . We can see that the number inside the absolute value has decreased by $2$ , going from $+9$ to $+7$ .
Horizontal Shift: This decrease by $2$ inside the absolute value function indicates a horizontal shift. Since the value inside the absolute value has decreased, the graph has moved to the right. A decrease inside the absolute value corresponds to a rightward shift of the graph.
Vertical Component: The vertical component of the function, which is outside the absolute value, has not changed. It remains $+3$ in both $f(x)$ and $g(x)$ . This means there is no vertical shift.
Overall Transformation: Therefore, the transformation that converts the graph of $f(x) = 4|x + 9| + 3$ into the graph of $g(x) = 4|x + 7| + 3$ is a translation $2$ units to the right.

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\newline

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\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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