Question

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

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

Short answer

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

Step-by-step solution

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=x3-3x+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=3satisfies the equation y=x3-3x+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 3,1check to see if x=3, y=1satisfies the equation y=x3-3x+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.