Question

graph the equation shown below by transforming the given graph of the parent function.
$y = \frac{1}{4}x^3$

Explanation:

Step1: Identify the parent function

The parent function for \( y = \frac{1}{4}x^3 \) is \( y = x^3 \), which is a cubic function. The graph of \( y = x^3 \) passes through the origin \((0,0)\), and for \( x = 1 \), \( y = 1 \); for \( x = -1 \), \( y = -1 \); for \( x = 2 \), \( y = 8 \); for \( x = -2 \), \( y = -8 \), etc.

Step2: Analyze the transformation

The given function is \( y=\frac{1}{4}x^3 \), which is a vertical compression of the parent function \( y = x^3 \) by a factor of \( \frac{1}{4} \). A vertical compression by a factor of \( a \) (where \( 0 < a < 1 \)) means that for each \( x \)-value, the corresponding \( y \)-value of the transformed function is \( a \) times the \( y \)-value of the parent function.

Step3: Transform key points

• For the parent function \( y = x^3 \), when \( x = 1 \), \( y = 1^3=1 \). For the transformed function \( y=\frac{1}{4}x^3 \), when \( x = 1 \), \( y=\frac{1}{4}(1)^3=\frac{1}{4} \). But looking at the given graph, the blue dot at \( x = 1 \) in the parent graph (original) is at \( y = 1 \), so we need to adjust it to \( y=\frac{1}{4} \)? Wait, no, maybe the given graph is the parent function \( y = x^3 \). Let's check the points:
• At \( x = 2 \), parent function \( y = 2^3 = 8 \). For \( y=\frac{1}{4}x^3 \), when \( x = 2 \), \( y=\frac{1}{4}(8)=2 \).
• At \( x=-2 \), parent function \( y = (-2)^3=-8 \). For \( y=\frac{1}{4}x^3 \), when \( x = -2 \), \( y=\frac{1}{4}(-8)=-2 \).
• At \( x = 1 \), \( y=\frac{1}{4}(1)=\frac{1}{4} \)
• At \( x=-1 \), \( y=\frac{1}{4}(-1)=-\frac{1}{4} \)

So to graph \( y=\frac{1}{4}x^3 \), we take the key points of \( y = x^3 \) (like \((-2,-8)\), \((-1,-1)\), \((0,0)\), \((1,1)\), \((2,8)\)) and multiply their \( y \)-coordinates by \( \frac{1}{4} \) to get the new points \((-2,-2)\), \((-1,-\frac{1}{4})\), \((0,0)\), \((1,\frac{1}{4})\), \((2,2)\) and then plot these points and draw the curve.

Answer:

To graph \( y = \frac{1}{4}x^3 \), perform a vertical compression of the parent function \( y = x^3 \) by a factor of \( \frac{1}{4} \). Transform key points of \( y = x^3 \) (e.g., \((-2, -8)\to(-2, -2)\), \((-1, -1)\to(-1, -\frac{1}{4})\), \((0, 0)\to(0, 0)\), \((1, 1)\to(1, \frac{1}{4})\), \((2, 8)\to(2, 2)\)) and draw the cubic curve through these transformed points.