Question

graph the equation shown below by transforming the given graph of the parent function.
$y = 3\sqrt3{x}$

Explanation:

Step1: Identify the parent function

The parent function for \( y = 3\sqrt[3]{x} \) is \( y=\sqrt[3]{x} \). The graph of the parent function \( y = \sqrt[3]{x} \) passes through points like \((-8, -2)\), \((-1, -1)\), \((0, 0)\), \((1, 1)\), \((8, 2)\) (these are typical points for the cube - root function \( y=\sqrt[3]{x} \)).

Step2: Analyze the transformation

The given function is \( y = 3\sqrt[3]{x} \). This is a vertical stretch of the parent function \( y=\sqrt[3]{x} \) by a factor of 3. For a vertical stretch by a factor of \( a>1 \) of a function \( y = f(x) \), the new function is \( y=a\cdot f(x) \). In this case, \( f(x)=\sqrt[3]{x} \) and \( a = 3 \).

To find the new points, we take the \( y \) - coordinates of the points on the parent function and multiply them by 3.

• For the point \((-8,-2)\) on \( y = \sqrt[3]{x} \), when we apply the transformation \( y = 3\sqrt[3]{x} \), the new \( y \) - coordinate is \( 3\times(-2)=-6 \), so the new point is \((-8, - 6)\).
• For the point \((-1,-1)\) on \( y=\sqrt[3]{x} \), the new \( y \) - coordinate is \( 3\times(-1)=-3 \), so the new point is \((-1, - 3)\).
• For the point \((0,0)\) on \( y=\sqrt[3]{x} \), the new \( y \) - coordinate is \( 3\times0 = 0 \), so the new point is \((0,0)\).
• For the point \((1,1)\) on \( y=\sqrt[3]{x} \), the new \( y \) - coordinate is \( 3\times1=3 \), so the new point is \((1,3)\).
• For the point \((8,2)\) on \( y=\sqrt[3]{x} \), the new \( y \) - coordinate is \( 3\times2 = 6 \), so the new point is \((8,6)\).

Step3: Graph the transformed function

Plot the new points \((-8, - 6)\), \((-1, - 3)\), \((0,0)\), \((1,3)\), \((8,6)\) and draw a smooth curve through them. The shape of the graph will be similar to the parent cube - root function, but it will be stretched vertically. The part of the graph for \( x\geq0 \) will be steeper (since the \( y \) - values are multiplied by 3, making the rise over run larger) and the part for \( x < 0 \) will also be steeper in the negative \( y \) - direction.

Answer:

To graph \( y = 3\sqrt[3]{x} \), vertically stretch the graph of the parent function \( y=\sqrt[3]{x} \) by a factor of 3. Plot points like \((-8, - 6)\), \((-1, - 3)\), \((0,0)\), \((1,3)\), \((8,6)\) and draw a smooth curve through them. The graph will have the same general shape as the cube - root function but will be stretched vertically, passing through the calculated points.