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Chapter 1: Problem 20

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$f(x)=(x-1)^{3}+2$$

Short Answer

Expert verified
The graph of the function \(f(x)=(x-1)^{3}+2\), obtained using a graphing utility, is a cubic curve shifted 1 unit to the right and 2 units upward from the position of the base cubic curve \(f(x)=x^{3}\). The viewing window from -10 to 10 for both x-axis and y-axis is suitable to present all significant features of this graph.

Step by step solution

01

Identifying the Parent Function and Its Transformation

The given function \(f(x)=(x-1)^{3}+2\) is a basic cube function \(f(x)=x^{3}\), but it's translated 1 unit to the right and 2 units up. That's because in the function \(f(x)=(x-h)^{3}+k\), the value of 'h' determines the horizontal shift and the value of 'k' determines the vertical shift. So, here 'h' is 1 (right shift) and 'k' is 2 (upward shift).
02

Determining the Viewing Window

The range of the cubic function is all real numbers, and there is no break or asymptote in its graph. Therefore, the suitable viewing window would be from -10 to 10 for both x and y-axis to view all significant aspects of the graph. However, this can vary as per the precise requirement or the nature of the graphing utility being used.
03

Plotting the Function Using a Graphing Utility

Now, you enter the translated function \(f(x)=(x-1)^{3}+2\) into the graphing utility and set the viewing window as decided in the previous step. Upon graphing, observe that the produced graph is a cubic curve translated from its original position according to the shift values determined in Step 1.

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Most popular questions from this chapter

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$h(x)=\sqrt{x+2}+3$$

A company produces a product for which the variable cost is 12.30 dollars per unit and the fixed costs are 98,000 dollars. The product sells for 17.98 dollars. Let \(x\) be the number of units produced and sold. (a) The total cost for a business is the sum of the variable cost and the fixed costs. Write the total cost \(C\) as a function of the number of units produced. (b) Write the revenue \(R\) as a function of the number of units sold. (c) Write the profit \(P\) as a function of the number of units sold. (Note: \(P=R-C\) ).

The table shows the numbers of tax returns (in millions) made through e-file from 2003 through \(2010 .\) Let \(f(t)\) represent the number of tax returns made through e-file in the year \(t .\) (Source: Internal Revenue Service) $$\begin{array}{|c|c|}\hline \text { Year } & \text { Number of Tax Returns Made Through E-File } \\\\\hline 2003 & 52.9 \\\2004 & 61.5 \\\2005 & 68.5 \\\2006 & 73.3 \\\2007 & 80.0 \\\2008 & 89.9 \\\2009 & 95.0 \\\2010 & 98.7 \\\\\hline\end{array}$$ (a) Find \(\frac{f(2010)-f(2003)}{2010-2003}\) and interpret the result in the context of the problem. (b) Make a scatter plot of the data. (c) Find a linear model for the data algebraically. Let \(N\) represent the number of tax returns made through e-file and let \(t=3\) correspond to 2003 (d) Use the model found in part (c) to complete the table. $$\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|}\hline t & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\hline N & & & & & & & & \\ \hline\end{array}$$ (e) Compare your results from part (d) with the actual data. (f) Use a graphing utility to find a linear model for the data. Let \(x=3\) correspond to \(2003 .\) How does the model you found in part (c) compare with the model given by the graphing utility?

Evaluate the function for the indicated values. \(k(x)=\left[\frac{1}{2} x+6\right]\) (a) \(k(5)\) (b) \(k(-6.1)\) (c) \(k(0.1)\) (d) \(k(15)\)

Write the area \(A\) of a circle as a function of its circumference \(C\).
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