Q. 85 - Precalculus Enhanced with Graphing Utilities

Question

Draw a graph that contains the points $(0,1),(1,3)$ , and $(3,5)$ and is symmetric with respect to the $x$-axis. Compare your graph with those of other students. Are most of the graphs almost straight lines? How many are "curved"? Discuss the various ways that these points might be connected.

Step-by-Step Solution

Answer

These points cannot be connected using a straight line. If a student has a graph that looks like a straight line, then it is possible that the graph was not drawn accurately. Drawing a graph that connects all the points given requires a slight curve.

The graph is shown below:

Step 1: Given information

The given points are $(0,1),(1,3)$, and $(3,5)$.

Step 2: Make the graph.

Plot the points $(0,1),(1,3)$ and $(3,5)$.

Reflect the point $(0,1)$ about the $x$-axis to get into the point $(0,-1)$.

Using the same procedure, the other two reflected points are: $(1,-3),(3,-5)$

Draw a smooth curve that passes through the plotted points.

Step 3: Draw the line segment curve.

In other case, the plotted points can be connected by segments but still respect the $x$-axis as its axis of symmetry as shown below.

Step 4: Write the conclusion

The 3 given points and the other 3 reflected points can be connected by a single smooth curve.

In other case, the plotted points can be connected by appropriate segments still respecting the $x$-axis as its axis of symmetry.

The graph is shown below: