Q. 87 - Precalculus Enhanced with Graphing Utilities

Question

Draw a graph that contains the points $(-2,5),(-1,3)$, and $(0,2)$ that is symmetric with respect to the $y$-axis. Compare your graph with those of other students; comment on any similarities. Can a graph contain these points and be symmetric with respect to the $x$-axis? The origin? Why or why not?

Step-by-Step Solution

Answer

The graph contain the given points and symmetric with respect to $x$-axis, $y$-axis and the origin.

Step 1: Given information

The given points are $(-2,5),(-1,3)$ and $(0,2)$.

Step 2: Plot the points

Start by plotting the points $(-2,5),(-1,3)$, and $(0,2)$.

The graph is shown below:

Step 3: Determine the symmetry with respect to y -axis..

By reflecting these points over either the $x$ or $y$ axes, it is clear that a graph can be drawn through these points that are either $x$-axis or $y$-axis symmetric.

Observe the green points are the reflections of the given points over the $y$-axis. Drawing any curve through these points, being sure to make the left and right sides of the curve symmetric, will yield a $y$-axis symmetric curve.

The graph is shown below:

Step 4: Determine the symmetry with respect to x -axis.

Similarly, an $x$-axis symmetric curve can be drawn through these points.

Step 5: Determine the symmetric with respect to origin.

For origin symmetry, rotate the points halfway around the origin. An origin symmetric curve can be drawn through these points.

Step 6: Write the conclusion

The graph is symmetric about the $x$-axis, $y$-axis and origin.