Question

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

Explanation:

Step1: Identify Parent Function

The parent function is \( y = |x| \), which has a V - shape with vertex at \((0,0)\), slope \(1\) for \(x\geq0\) and slope \(- 1\) for \(x < 0\).

Step2: Analyze Transformation

The given function is \( y=3|x| \). For a function of the form \( y = a|x| \), when \(|a|>1\), it is a vertical stretch of the parent function \( y = |x| \) by a factor of \(|a|\). Here, \(a = 3>1\), so we vertically stretch the graph of \( y = |x| \) by a factor of \(3\).

Step3: Apply Transformation to Points

Take a point on \( y = |x| \), say \((1,1)\). After vertical stretch by factor \(3\), the \(y\) - coordinate is multiplied by \(3\), so the new point is \((1,3\times1)=(1,3)\). Similarly, for \((2,2)\), the new point is \((2,3\times2)=(2,6)\), for \(( - 1,1)\) the new point is \((-1,3\times1)=(-1,3)\), for \(( - 2,2)\) the new point is \((-2,3\times2)=(-2,6)\) and the vertex \((0,0)\) remains \((0,0)\) since \(3\times0 = 0\).

Step4: Graph the Transformed Function

Plot the new points \((-2,6)\), \((-1,3)\), \((0,0)\), \((1,3)\), \((2,6)\) etc. and draw the two lines (for \(x\geq0\) with slope \(3\) and for \(x < 0\) with slope \(- 3\)) passing through these points to get the graph of \( y = 3|x| \).

Answer:

To graph \( y = 3|x| \), vertically stretch the graph of \( y=|x| \) by a factor of 3. The vertex remains at \((0,0)\), and points on \( y = |x| \) like \((x,y)\) become \((x,3y)\). Plot points such as \((-2,6)\), \((-1,3)\), \((0,0)\), \((1,3)\), \((2,6)\) and draw the V - shaped graph with steeper slopes (slope \(3\) for \(x\geq0\), slope \(-3\) for \(x < 0\)) than the parent function \( y = |x| \).