Question

use shifts and scalings to transform the graph of $f(x)=\sqrt{x}$ into the graph of $g$. use a graphing utility to check your work.\
a. $g(x)=f(x + 6)$ b. $g(x)=2f(3x - 4)$ c. $g(x)=\sqrt{x - 2}$ d. $g(x)=3\sqrt{x - 4}-6$\
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a. which transformation is used to transform the graph of $f$ into the graph of $g$?\
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a. shift 6 units up.\
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b. scale vertically by a factor of 6.\
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c. shift 6 units to the right.\
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d. shift 6 units to the left.

Explanation:

To determine the transformation from \( f(x)=\sqrt{x} \) to \( g(x) = f(x + 6)\), we use the rules of function transformations. For a function \( y = f(x + h)\), if \( h>0 \), the graph of \( f(x) \) is shifted \( h \) units to the left. Here, \( h = 6\) (since \( g(x)=f(x+6) \) is in the form \( f(x+h) \) with \( h = 6 \)), so the graph of \( f(x) \) is shifted 6 units to the left to get the graph of \( g(x) \). Option A is incorrect as there is no vertical shift (the transformation is horizontal). Option B is incorrect as there is no vertical scaling (the coefficient of \( f \) is 1). Option C is incorrect because a shift to the right would be of the form \( f(x - h) \), not \( f(x + h) \).

Answer:

D. Shift 6 units to the left.