Using statistical methods, the financial department of a...

Using statistical methods, the financial department of a hospital arrived at the cost equation $$ C(x)=0.00048(x-500)^{3}+60,000 \quad 100 \leq x \leq 1,000 $$ where $C(x)$ is the cost in dollars for handling $x$ cases per month. (A) Describe how the graph of function $C$ can be obtained from the graph of one of the basic functions in Figure 1 on page $19 .$

Using statistical methods, the financial department of a hospital arrived at the cost equation
$$
C(x)=0.00048(x-500)^{3}+60,000 \quad 100 \leq x \leq 1,000
$$
where $C(x)$ is the cost in dollars for handling $x$ cases per month.
(A) Describe how the graph of function $C$ can be obtained from the graph of one of the basic functions in Figure 1 on page $19 .$

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Key Concepts

Vertical Translation

Vertical translation is achieved by adding a constant to the function, which moves the graph up or down without altering its shape. Adding 60,000 translates the entire graph upward by 60,000 units, adjusting its vertical position on the coordinate system.

Vertical Scaling

Vertical scaling involves multiplying the function by a constant factor, which stretches or compresses the graph along the y-axis. Here, the factor 0.00048 compresses the graph vertically, reducing the steepness of the original cubic curve.

Function Transformations

The process involves transforming a basic graph into a new function by applying operations such as shifts, stretches, compressions, and reflections. This approach allows one to modify the graph of a simple function—in this case, the cubic function—to match the characteristics of a more complex function.

Horizontal Shift

A horizontal shift occurs when the function argument is modified by adding or subtracting a constant. In this context, replacing x with (x - 500) shifts the basic graph 500 units to the right, repositioning the location of the key features of the graph along the x-axis.

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