Q. 65 - Precalculus Enhanced with Graphing Utilities

Question

Exploration Graph $y=x^3$. Then on the same screen graph $y={(x-1)}^3+2$. Could you have predicted the result? 

Step-by-Step Solution

Answer

The graph of the two functions on the same screen is 

Yes, we could have predicted the graph of $y={(x-1)}^3+2$ from the graph of $y=x^3$.

Step 1. Given information

The graph of the functions $y=x^3$ and $y={(x-1)}^3+2$ on the same coordinate plane is drawn as

Step 2. Prediction

Both the graphs have the same shape, so yes we could have predicted the result.

As $1$ is subtracted from the argument $x$ and $2$ is added to the output of the function $y=x^3$ to get the function $y={(x-1)}^3+2$.

So the graph of $y={(x-1)}^3+2$ is the graph of $y=x^3$ shifted horizontally right by one unit and shifted vertically up by two units.