Questions: A manufacturer's cost is given by C=400 n^(1/3)+64 where C is the cost and n is the number of parts produced. Describe the transformations needed to obtain the graph of this function from the graph of y=n^(1/2).

A manufacturer's cost is given by C=400 n^(1/3)+64 where C is the cost and n is the number of parts produced. Describe the transformations needed to obtain the graph of this function from the graph of y=n^(1/2).

Transcript text: HW 4.1B Question 10, 4.1-32 HW Score: $90 \%, 9$ of 10 points Points: 0 of 1 Save A manufacturer's cost is given by $C=400 \sqrt[3]{n}+64$ where $C$ is the cost and $n$ is the number of parts produced. Describe the transformations needed to obtain the graph of this function from the graph of $y=\sqrt{n}$.

Solution

Solution Steps

To transform the graph of $ y = \sqrt{n} $ to the graph of $ C = 400 \sqrt[3]{n} + 64 $, we need to consider the following transformations:

Root Change: Change the square root to a cube root, which alters the shape of the graph.
Vertical Stretch: Multiply the cube root by 400, which stretches the graph vertically.
Vertical Shift: Add 64 to the entire function, which shifts the graph upward by 64 units.

To solve the problem, we need to describe the transformations needed to obtain the graph of the function $ C = 400 \sqrt[3]{n} + 64 $ from the graph of $ y = \sqrt{n} $.

Step 1: Understand the Given Function

The given function is $ C = 400 \sqrt[3]{n} + 64 $. This function is a transformation of the cube root function $ y = \sqrt[3]{n} $.

Step 2: Identify the Basic Function

The basic function we are starting with is $ y = \sqrt[3]{n} $, which is the cube root function.

Step 3: Apply Vertical Stretch

The function $ C = 400 \sqrt[3]{n} + 64 $ includes a coefficient of 400 in front of the cube root. This represents a vertical stretch by a factor of 400. The transformation can be described as: