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Question

# Suppose the graph of f is given. Describe how the graph of each function can be obtained from the graph of f. $\frac{1}{4} f(x-3)+5$

Solution

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Graph of $\dfrac{1}{4}f(x-3)+5$ can be obtained from $f(x)$ by three combined translations.

$f(x) \rightarrow f(x-3)$ is shifting 3 units to the right

$f(x-3)\rightarrow \dfrac{1}{4}f(x-3)$ is vertical shrink by a factor of $\dfrac{1}{4}$

$\dfrac{1}{4}f(x-3)\rightarrow \dfrac{1}{4}f(x-3)+5$ is shifting 5 units upward

So, graph of $\dfrac{1}{4}f(x-3)+5$ can be obtained from $f(x)$ by shifting 3 units to the right, followed by vertical shrink by a factor of $\dfrac{1}{4}$ and shifting 5 units up.

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