Chapter 2.4, Problem 65AYU

In Problems 64-71, use a graphing utility.

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

To determine

To graph: Graph the equations $y = x^3$ and $y = {\left(x - 1\right)}^3 + 2$. Then conclude that the result can be predicted or not.

Answer to Problem 65AYU

Explanation of Solution

To sketch:

Graph the following equation with the graphing utility:

1. $y = x^3$

2. $y = {\left(x - 1\right)}^3 + 2$

Interpretation:

The equation $y = x^3$ is cubic equation passes through the vertex point $\left(0,0\right)$.

The equation $y = {\left(x - 1\right)}^3 + 2$ is modified version of $y = x^3$ with subtracting 1 from the variable $x$ and adding 2 with the equation.

As the value 1 subtracted from the cubic equation, its vertex point shifted to $\left(1, 0\right)$ in the positive $x\text{-axis}$.

As the value 2 added to the entire equation, the given curve moves upwards $\text{(along}y\text{-axis)}$ with 2 units.

Therefore the new vertex point of the modified equation $y = {\left(x - 1\right)}^3 + 2$ must be $\left(1,2\right)$.

The result of the graph can be predicted easily.