Question

use transformations of ( f(x) = sqrt{x} ) to graph the following function. ( g(x) = 3sqrt{x + 6} - 5 ) choose the correct graph below. \\( \bigcirc a. \\) \\( \bigcirc b. \\) \\( \bigcirc c. \\) \\( \bigcirc d. \\) (images of graphs for each option are shown)

Explanation:

Step1: Identify base function

Base function: $f(x)=\sqrt{x}$, domain $x\geq0$, passes through $(0,0), (1,1), (4,2)$

Step2: Horizontal shift left 6 units

Transform to $f(x+6)=\sqrt{x+6}$, domain $x\geq-6$, passes through $(-6,0), (-5,1), (-2,2)$

Step3: Vertical stretch by factor 3

Transform to $3\sqrt{x+6}$, points become $(-6,0), (-5,3), (-2,6)$

Step4: Vertical shift down 5 units

Transform to $g(x)=3\sqrt{x+6}-5$, points become $(-6,-5), (-5,-2), (-2,1)$

Step5: Match to graph

Check which graph has domain $x\geq-6$, passes through $(-6,-5), (-5,-2), (-2,1)$ (this corresponds to Option C)

Answer:

C. <The graph with domain $x\geq-6$, passing through $(-6,-5), (-5,-2), (-2,1)$>