Answer:
The graph of [tex]y=\sqrt{4x+16}+5[/tex] is the graph of [tex]y=\sqrt{x}[/tex] translated 4 units left, stretched horizontally by a factor of 1/4, and translated 5 units up.
Step-by-step explanation:
Transformations
[tex]\textsf{For }a > 0[/tex]
[tex]f(x+a) \implies f(x) \: \textsf{translated}\:a\:\textsf{units left}[/tex]
[tex]f(x)+a \implies f(x) \: \textsf{translated}\:a\:\textsf{units up}[/tex]
[tex]y=f(ax) \implies f(x) \: \textsf{stretched parallel to the x-axis (horizontally) by a factor of} \: \dfrac{1}{a}[/tex]
Given function
[tex]y=\sqrt{4x+16}+5[/tex]
Parent function
Parent functions are the simplest form of a given family of functions.
[tex]y=\sqrt{x}[/tex]
The graph of the parent function is related to the graph of the given function by a series of transformations. To determine the series of transformations, work out the steps of how to go from the parent function to the given function.
Factor the expression under the square root sign:
[tex]y=\sqrt{4(x+4)}+5[/tex]
Transformations
Parent function:
[tex]f(x)=\sqrt{x}[/tex]
Translated 4 units left:
[tex]f(x+4)=\sqrt{x+4}[/tex]
Horizontally stretched by a factor of 1/4 (compressed by a factor of 4):
[tex]\begin{aligned}f(4(x+4)) & =\sqrt{4(x+4)}\\ & = \sqrt{4x+16} \end{aligned}[/tex]
Translated 5 units up:
[tex]f(4x+16)+5=\sqrt{4x+16}+5[/tex]
Therefore, the graph of [tex]y=\sqrt{4x+16}+5[/tex] is the graph of [tex]y=\sqrt{x}[/tex] translated 4 units left, stretched horizontally by a factor of 1/4, and translated 5 units up.
Learn more about graph transformations here;
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