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Chapter 2: Problem 63

Explain how to use the graph of \(y=x^{3}\) to produce the graph of \(P(x)=(x-2)^{3}+1\).

Short Answer

Expert verified
To produce the graph of \(P(x) = (x-2)^{3}+1\) from \(y=x^{3}\), shift the graph 2 units to the right and 1 unit upwards.

Step by step solution

01

Understanding the Base Function

The original function is \(y=x^{3}\). This function forms a cubic curve. For \(x<0\), \(y<0\); for \(x>0\), \(y>0\); and at \(x=0\), \(y\)=0. The curve increases as \(x\) increases.
02

Shifting the Graph Horizontally

The transformation \((x-2)^{3}\) shifts the graph to the right by 2 units. That is, every point \((x,y)\) on the original graph \(y=x^{3}\) goes to the point \((x+2, y)\) on the new graph. Therefore, the whole shape of the graph shifts to the right by 2 units while maintaining its form and orientation.
03

Shifting the Graph Vertically

The transformation \((x-2)^{3}+1\) shifts the graph upward by 1 unit. That is, every point \((x,y)\) on the graph obtained in Step 2 goes to the point \((x, y+1)\) on the new graph. Therefore, the whole shape of the graph shifts upward by 1 unit while maintaining its form and orientation.

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