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Chapter 1: Problem 19

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$f(x)=x^{3}-1$$

Short Answer

Expert verified
To properly graph the function \(f(x) = x^{3}-1\), it's important to understand it is a downward shifted version of the cubic parent function \(f(x) = x^{3}\). Subsequently, locate a few points by substituting x-values into the function, such as (-2, -7), (0, -1), (2, 7). The suitable viewing window might be -3 to 3 on the x-axis, and -8 to 8 on the y-axis to capture the key features of the function.

Step by step solution

01

- Understand the Function

Before we begin graphing, understanding the function structure is crucial. Here we have a cubic function \(f(x) = x^{3}-1\). It is a modified version of the parent function \(f(x) = x^{3}\), where the entire function is shifted down by 1 unit due to the \(-1\).
02

- Determine the Shape

A typical cubic function has a shape of 'S'. However, depending on the coefficients, the function can also be flattened, stretched, or reflect along either the x-axis or y-axis. In the case of \(f(x) = x^{3}-1\), there's no coefficient of x that could cause such transformations, hence, the S-shape will be maintained, only shifted 1 unit downwards.
03

- Test Some Values

To ensure our understanding of the function is sound, let's plug a few x-values and calculate the corresponding y-values. When x is -2, 0, and 2, the function yields y-values of -7, -1 and 7 respectively. These points lie in the 'S' shape curve we're expecting.
04

- Determine the Viewing Window in the Graphing Utility

An appropriate viewing window will make meaningful features of the graph visible. Given, the calculations in Step 3, a good viewing window might be -3 to 3 on the x-axis, and -8 to 8 on the y-axis. This should include the points calculated and other important aspects of the function.
05

- Graph the Function

With the viewing window defined, now input the function in the graphing utility to obtain the plot. If all was done correctly, a downward shifted 'S' shape should be visible, representing the function \(f(x) = x^{3}-1\).

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