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.\
b. scale vertically by a factor of 6.\
c. shift 6 units to the right.\
d. shift 6 units to the left.\
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graph the function. choose the correct graph below.\
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a.\
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graph a: coordinate plane with -10 to 10 on x and y, curve starting near x=-6, y=0, increasing\
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b.\
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graph b: coordinate plane with -10 to 10 on x and y, steeper curve starting near x=-1, y=0\
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c.\
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graph c: coordinate plane with -10 to 10 on x and y, curve starting near x=-10, y=0, increasing\
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d.\
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graph d: coordinate plane with -10 to 10 on x and y, curve starting near x=10, y=0, increasing

Explanation:

Part a: Transformation Analysis

For the function \( g(x) = f(x + 6) \) where \( f(x)=\sqrt{x} \), the transformation rule for horizontal shifts is: if we have \( f(x + h) \), the graph of \( f(x) \) shifts \( h \) units to the left when \( h>0 \). Here, \( h = 6 \), so the graph of \( f(x)=\sqrt{x} \) (which has its starting point at \( (0,0) \)) will shift 6 units to the left. The correct option for the transformation is D. For the graph, the original \( f(x)=\sqrt{x} \) starts at \( (0,0) \) and increases slowly. After shifting 6 units left, the starting point becomes \( (-6,0) \), and the shape remains the same (square - root curve). Looking at the graphs, option A (with the curve starting near \( x=-6 \) on the left - hand side of the y - axis) matches this transformation.

Answer:

For the transformation: D. Shift 6 units to the left.
For the graph: A.