Q. 2 - Precalculus Enhanced with Graphing Utilities

Question

Which of the following points are on the graph of $y=x^3-3x+1$?

$$\begin{gathered} \left(a\right) \left(-2,-1\right) \\ \left(b\right) \left(2,3\right) \\ \left(c\right) \left(3,1\right) \end{gathered}$$

Step-by-Step Solution

Answer

The point $\left(a\right) \left(-2,-1\right) \mathrm{and}\left(b\right) \left(2,3\right)$ are on the graph.

Step 1. Given information.

Which of the following points are on the graph of $y=x^3-3x+1$.

$$\begin{gathered} \left(a\right) \left(-2,-1\right) \\ \left(b\right) \left(2,3\right) \\ \left(c\right) \left(3,1\right) \end{gathered}$$

Part (a) Step 1. For the point - 2 , - 1 , check to see if x = - 2 ,   y = - 1 satisfies the equation y = x 3 - 3 x + 1 .

$$\begin{gathered} y=x^3-3x+1 \\ -1={\left(-2\right)}^3-3\left(-2\right)+1 \\ -1=-8+6+1 \\ -1=-1 \end{gathered}$$

The equation is satisfied, so the point $(-2,-1)$ is on the graph. 

Part (b) Step 1. For the point 2 , 3 check to see if x = 2 ,   y = 3 satisfies the equation y = x 3 - 3 x + 1 .

$$\begin{gathered} y=x^3-3x+1 \\ 3=2^3-3\left(2\right)+1 \\ 3=8-6+1 \\ 3=3 \end{gathered}$$

The equation is satisfied, so the point $\left(3,2\right)$is on the graph. 

Part (c) Step 1. For the point 3 , 1 check to see if x = 3 ,   y = 1 satisfies the equation y = x 3 - 3 x + 1 .

$$\begin{gathered} y=x^3-3x+1 \\ 1=3^3-3\left(3\right)+1 \\ 1=27-9+1 \\ 1\ne 19 \end{gathered}$$

The equation is not satisfied, so the point $\left(3,1\right)$ is not on the graph.