Q 89. - Precalculus Enhanced with Graphing Utilities

Question

Draw the graph of a function that has the following properties: domain: all real numbers; range: all real numbers; intercepts: $(0, -3)$ and $(3, 0)$; a local maximum value of $-2$ is at $-1$; a local minimum value of $-6$ is at $2$. Compare your graph with those of others. Comment on any differences.

Step-by-Step Solution

Answer

The required graph is shown below:

Step 1. Given information

The following properties are given.

• Domain: All real numbers; range: all real numbers;
• Intercepts: $(0, -3)$ and $(3, 0)$;
• A local maximum value of $-2$ is at $-1$; a local minimum value of $-6$ is at $2$.

Step 2. Draw the figure with the given properties.

First, plot the points $(0,-3), (3,0), (-1,-2),$ and $(2,-6)$ on a cartesian plane. Then, connect the points to make the graph.

It is given that the domain and range are all real numbers.

This implies that there is no asymptotes.

Since there is one local maxima and one local minima, the graph will be the graph of the cubic polynomial.

Thus, the required graph is shown below: