Question

graph the equation shown below by transforming the given graph of the parent function.
$y = \frac{1}{3} \vert x \vert$

Explanation:

Step1: Identify the parent function

The parent function here is \( y = |x| \), which has a V - shape with vertex at the origin \((0,0)\), and the slope of the right - hand side (for \(x\geq0\)) is \(1\) and the slope of the left - hand side (for \(x < 0\)) is \(- 1\).

Step2: Analyze the transformation

The given function is \( y=\frac{1}{3}|x| \). This is a vertical scaling of the parent function \(y = |x|\). The general form of a vertical scaling of a function \(y = f(x)\) is \(y = a\cdot f(x)\), where if \(0 < a<1\), the graph is vertically compressed by a factor of \(a\). Here, \(a=\frac{1}{3}\), so we need to vertically compress the graph of \(y = |x|\) by a factor of \(\frac{1}{3}\).

For the parent function \(y = |x|\), when \(x = 3\), \(y=|3| = 3\). For the function \(y=\frac{1}{3}|x|\), when \(x = 3\), \(y=\frac{1}{3}\times|3| = 1\). When \(x = 6\), for the parent function \(y = |6|=6\), and for \(y=\frac{1}{3}|x|\), \(y=\frac{1}{3}\times6 = 2\). Similarly, for \(x=- 3\), the parent function has \(y = 3\) and the transformed function has \(y = 1\), and for \(x=-6\), the parent function has \(y = 6\) and the transformed function has \(y = 2\).

To graph \(y=\frac{1}{3}|x|\):

• The vertex remains at \((0,0)\) because there is no horizontal or vertical shift (the form \(y=\frac{1}{3}|x|\) has no \(h\) (horizontal shift) or \(k\) (vertical shift) in the vertex form \(y=a|x - h|+k\), here \(h = 0\) and \(k = 0\)).
• For the right - hand side (\(x\geq0\)), the slope of \(y = |x|\) is \(1\), and for \(y=\frac{1}{3}|x|\), the slope is \(\frac{1}{3}\). So we can plot points by taking \(x\) values, calculating \(y=\frac{1}{3}|x|\), and then drawing a line with slope \(\frac{1}{3}\) from the origin for \(x\geq0\) and a line with slope \(-\frac{1}{3}\) from the origin for \(x < 0\).

For example, let's take some key points:

• When \(x = 0\), \(y=\frac{1}{3}\times|0|=0\) (vertex at \((0,0)\)).
• When \(x = 3\), \(y = 1\) (point \((3,1)\)).
• When \(x=-3\), \(y = 1\) (point \((- 3,1)\)).
• When \(x = 6\), \(y = 2\) (point \((6,2)\)).
• When \(x=-6\), \(y = 2\) (point \((-6,2)\)).

We then connect these points to form the V - shaped graph of \(y=\frac{1}{3}|x|\), which is a vertically compressed version of the parent function \(y = |x|\) by a factor of \(\frac{1}{3}\).

Answer:

To graph \(y=\frac{1}{3}|x|\), vertically compress the graph of \(y = |x|\) by a factor of \(\frac{1}{3}\). The vertex is at \((0,0)\), and the graph has a slope of \(\frac{1}{3}\) for \(x\geq0\) and \(-\frac{1}{3}\) for \(x < 0\). Key points include \((0,0)\), \((3,1)\), \((-3,1)\), \((6,2)\), \((-6,2)\) etc., and the graph is a V - shape narrower (vertically compressed) than \(y = |x|\).