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Chapter 7: Problem 30

Graph each equation in Exercises 21-32. Select integers for \(x\) from \(-3\) to 3 , inclusive. \(y=x^{3}-1\)

Short Answer

Expert verified
The graph of the equation \(y = x^{3} - 1\) for \(x\) values from -3 to 3 forms a cubic curve with the following points: (-3,-28), (-2,-9), (-1,-2), (0,-1), (1,0), (2,7), (3,26).

Step by step solution

01

Understand the Equation

The given equation is \(y = x^{3} - 1\). This is a cubic function where for each value of \(x\), we cube it and then subtract 1 to get its corresponding \(y\)-value.
02

Identify the Range of \(x\)-values

The exercise provides the range for the \(x\)-values, which are integers from -3 to 3. So, the \(x\)-values we will be substituting into our equation will be -3, -2, -1, 0, 1, 2, 3.
03

Calculate Corresponding \(y\)-values

Substitute each \(x\)-value into the equation to find the corresponding \(y\)-value. Here are the calculated values: When \(x = -3\), \(y = (-3)^{3} - 1 = -28\),When \(x = -2\), \(y = (-2)^{3} - 1 = -9\),When \(x = -1\), \(y = (-1)^{3} - 1 = -2\),When \(x = 0\), \(y = (0)^{3} - 1 = -1\),When \(x = 1\), \(y = (1)^{3} - 1 = 0\),When \(x = 2\), \(y = (2)^{3} - 1 = 7\),When \(x = 3\), \(y = (3)^{3} - 1 = 26\
04

Plot the Graph

Now we need to sketch a graph of this equation using these points. Create a graph with the \(x\)-axis and \(y\)-axis. Label the \(x\)-axis from -3 to 3, and the \(y\)-axis according to the calculated \(y\)-values. Now plot the points (-3,-28), (-2,-9), (-1,-2), (0,-1), (1,0), (2,7), (3,26) and join them to form a smooth curve which will represent our equation.

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