Question

question 2 (mandatory) (1 point) when compared to f(x) = √x, the function g(x) = 3√(x/2) + 6 is stretched vertically by a factor of 3, stretched horizontally by a factor of 2, and translated up 6 units. true false

Explanation:

Step1: Analyze the vertical stretch

For a function \(y = af(x)\), the vertical stretch factor is \(|a|\). Given \(g(x)=3\sqrt{\frac{x}{2}}+6=\frac{3}{\sqrt{2}}\sqrt{x}+6\). Comparing with \(f(x)=\sqrt{x}\), the vertical transformation part is \(y = \frac{3}{\sqrt{2}}\sqrt{x}\). The vertical stretch factor is \(\frac{3}{\sqrt{2}}
eq3\).

Step2: Analyze the horizontal stretch

For a function \(y = f(bx)\), the horizontal stretch factor is \(\frac{1}{|b|}\). Given \(g(x)=3\sqrt{\frac{x}{2}}+6 = 3\sqrt{\frac{1}{2}x}+6\). Here \(b=\frac{1}{2}\), so the horizontal stretch factor is \(2\).

Step3: Analyze the vertical translation

For a function \(y=f(x)+k\), the vertical translation is \(k\) units. Given \(g(x)=3\sqrt{\frac{x}{2}}+6\), the vertical translation is \(6\) units up.

Since the vertical stretch factor is not \(3\), the statement is False.

Answer:

False