Question

graph the equation shown below by transforming the given graph of the parent function.
$y = \frac{1}{2}x^3$

Explanation:

Step1: Identify the parent function

The parent function for \( y = \frac{1}{2}x^3 \) is \( y = x^3 \), which is a cubic function. The graph of \( y = x^3 \) passes through points like \((-2, -8)\), \((-1, -1)\), \((0, 0)\), \((1, 1)\), \((2, 8)\).

Step2: Analyze the transformation

The given function is \( y = \frac{1}{2}x^3 \), which is a vertical compression of the parent function \( y = x^3 \) by a factor of \( \frac{1}{2} \). For a vertical compression by a factor of \( a \) (where \( 0 < a < 1 \)), we multiply the \( y \)-values of the parent function's points by \( a \).

Step3: Transform key points

• For \( x = -2 \): Parent \( y = (-2)^3 = -8 \), transformed \( y = \frac{1}{2}(-8) = -4 \)
• For \( x = -1 \): Parent \( y = (-1)^3 = -1 \), transformed \( y = \frac{1}{2}(-1) = -0.5 \)
• For \( x = 0 \): Parent \( y = 0^3 = 0 \), transformed \( y = \frac{1}{2}(0) = 0 \)
• For \( x = 1 \): Parent \( y = 1^3 = 1 \), transformed \( y = \frac{1}{2}(1) = 0.5 \)
• For \( x = 2 \): Parent \( y = 2^3 = 8 \), transformed \( y = \frac{1}{2}(8) = 4 \)

Step4: Graph the transformed function

Plot the transformed points \((-2, -4)\), \((-1, -0.5)\), \((0, 0)\), \((1, 0.5)\), \((2, 4)\) and draw a smooth curve through them, which will be a vertically compressed version of the parent cubic function's graph.

Answer:

To graph \( y = \frac{1}{2}x^3 \), vertically compress the parent cubic function \( y = x^3 \) by a factor of \( \frac{1}{2} \). Plot transformed points (e.g., \((-2, -4)\), \((-1, -0.5)\), \((0, 0)\), \((1, 0.5)\), \((2, 4)\)) and draw a smooth curve.