Question

activity 3
mathia connection
vertical dilations and vertical
translations of functions
let’s consider more transformations of the
basic function f(x) = x.
1 describe the transformations performed on f(x) to produce g(x). then, graph g(x).
write the function equation in general form.
① g(x) = 2·f(x) + 7
② g(x) = 3·f(x) − 1
③ ( g(x) = \frac{1}{3}·f(x) + 2 )
④ ( g(x) = \frac{1}{2}·f(x) − 3 )

Explanation:

Step1: Recall function transformation rules

For a function \(y = a\cdot f(x)+k\), \(a\) represents vertical dilation (if \(|a|> 1\), vertical stretch; if \(0 < |a|<1\), vertical compression) and \(k\) represents vertical translation (if \(k>0\), shift up; if \(k < 0\), shift down).

Step2: Analyze \(g(x)=2\cdot f(x)+7\)

Since \(a = 2>1\), there is a vertical stretch by a factor of \(2\). Since \(k = 7>0\), there is a vertical translation \(7\) units up. The general form is \(g(x)=2x + 7\).

Step3: Analyze \(g(x)=3\cdot f(x)-1\)

Since \(a = 3>1\), there is a vertical stretch by a factor of \(3\). Since \(k=-1<0\), there is a vertical translation \(1\) unit down. The general form is \(g(x)=3x-1\).

Step4: Analyze \(g(x)=\frac{1}{3}\cdot f(x)+2\)

Since \(0<\frac{1}{3}<1\), there is a vertical compression by a factor of \(\frac{1}{3}\). Since \(k = 2>0\), there is a vertical translation \(2\) units up. The general form is \(g(x)=\frac{1}{3}x + 2\).

Step5: Analyze \(g(x)=\frac{1}{2}\cdot f(x)-3\)

Since \(0<\frac{1}{2}<1\), there is a vertical compression by a factor of \(\frac{1}{2}\). Since \(k=-3<0\), there is a vertical translation \(3\) units down. The general form is \(g(x)=\frac{1}{2}x-3\).

Answer:

a. Vertical stretch by a factor of \(2\) and vertical translation \(7\) units up; \(g(x)=2x + 7\)
b. Vertical stretch by a factor of \(3\) and vertical translation \(1\) unit down; \(g(x)=3x-1\)
c. Vertical compression by a factor of \(\frac{1}{3}\) and vertical translation \(2\) units up; \(g(x)=\frac{1}{3}x + 2\)
d. Vertical compression by a factor of \(\frac{1}{2}\) and vertical translation \(3\) units down; \(g(x)=\frac{1}{2}x-3\)