Question

graph the equation shown below by transforming the given graph of the parent function.
$y = \sqrt3{\frac{1}{4}x}$

Explanation:

Step1: Identify the parent function

The parent function of a cube root function is \( y = \sqrt[3]{x} \). The given function is \( y=\sqrt[3]{\frac{1}{4}x} \).

Step2: Analyze the transformation

For a cube root function of the form \( y=\sqrt[3]{ax} \), when \( a=\frac{1}{4} \), this represents a horizontal stretch (since \( 0 < a< 1 \)) of the parent function \( y = \sqrt[3]{x} \) by a factor of \( \frac{1}{a}=4 \).

Step3: Determine key points transformation

• For the parent function \( y = \sqrt[3]{x} \), some key points are:
• When \( x = 0 \), \( y=0 \)
• When \( x = 1 \), \( y = 1 \)
• When \( x=-1 \), \( y=-1 \)
• When \( x = 8 \), \( y = 2 \)
• When \( x=-8 \), \( y=-2 \)
• For the transformed function \( y=\sqrt[3]{\frac{1}{4}x} \), we solve for \( x \) in terms of the parent function's \( x \)-values. Let \( \frac{1}{4}x = X \) (where \( X \) is the \( x \)-value of the parent function). Then \( x = 4X \).
• For \( X = 0 \) (parent's \( x = 0 \)), \( x=0 \), \( y = 0 \) (same as parent at this point)
• For \( X = 1 \) (parent's \( x = 1 \)), \( x=4\times1 = 4 \), \( y=\sqrt[3]{\frac{1}{4}\times4}=\sqrt[3]{1} = 1 \)
• For \( X=-1 \) (parent's \( x=-1 \)), \( x=4\times(-1)=-4 \), \( y=\sqrt[3]{\frac{1}{4}\times(-4)}=\sqrt[3]{-1}=-1 \)
• For \( X = 8 \) (parent's \( x = 8 \)), \( x=4\times8 = 32 \), \( y=\sqrt[3]{\frac{1}{4}\times32}=\sqrt[3]{8} = 2 \)
• For \( X=-8 \) (parent's \( x=-8 \)), \( x=4\times(-8)=-32 \), \( y=\sqrt[3]{\frac{1}{4}\times(-32)}=\sqrt[3]{-8}=-2 \)

Step4: Graph the transformed function

Using the transformed key points \((0,0)\), \((4,1)\), \((-4,-1)\), \((32,2)\), \((-32,-2)\), we can plot these points and draw the curve of \( y=\sqrt[3]{\frac{1}{4}x} \) which is a horizontal stretch of the parent cube root function by a factor of 4.

Answer:

To graph \( y=\sqrt[3]{\frac{1}{4}x} \), we horizontally stretch the graph of the parent function \( y = \sqrt[3]{x} \) by a factor of 4. Key points of the transformed function are \((0,0)\), \((4,1)\), \((-4,-1)\), \((32,2)\), \((-32,-2)\) (and other corresponding stretched points), and we draw the curve through these points.