Question

use the graph of y = f(x) shown to the right to graph the following function g. g(x) = -\frac{1}{3}f(x + 1) choose the correct graph below. \\(\bigcirc\\) a. \\(\bigcirc\\) b. \\(\bigcirc\\) c. \\(\bigcirc\\) d.

Explanation:

Step1: Horizontal shift

For the function \(g(x)=-\frac{1}{3}f(x + 1)\), compared to \(y = f(x)\), the transformation \(y=f(x)\to y = f(x+1)\) is a horizontal shift. According to the rule \(y = f(x + h)\) (where \(h>0\) shifts the graph of \(y = f(x)\) to the left by \(h\) units), here \(h = 1\), so the graph of \(y=f(x)\) is shifted to the left by 1 unit.

Step2: Vertical compression and reflection

The transformation \(y=f(x+1)\to y=-\frac{1}{3}f(x + 1)\). The factor \(\frac{1}{3}\) (since \(y=af(x)\) with \(|a|<1\)) compresses the graph of \(y = f(x+1)\) vertically by a factor of \(\frac{1}{3}\). The negative sign (\(y=-af(x)\)) reflects the graph of \(y=\frac{1}{3}f(x + 1)\) about the \(x\) - axis.

If we consider the key points of \(y = f(x)\):

• For the point \((0,3)\) on \(y = f(x)\), after \(x\to x+1\) (i.e., \(x'=x - 1\)), when \(x'=- 1\), and then \(y\to-\frac{1}{3}y\), the new \(y\) - value is \(- 1\). So the new point is \((-1,-1)\).
• For the point \((3,0)\) on \(y = f(x)\), after \(x\to x + 1\) ( \(x'=2\)), and \(y\to-\frac{1}{3}y\) ( \(y = 0\)), the new point is \((2,0)\).
• For the point \((6,-3)\) on \(y = f(x)\), after \(x\to x + 1\) ( \(x'=5\)), and \(y\to-\frac{1}{3}y\) ( \(y = 1\)), the new point is \((5,1)\).

By analyzing the transformations of key - points and the general shape (reflection and compression), we can match the graph.

Answer:

A.