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?

Short answer

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

Step-by-step solution

Given information

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

Plot the points

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

The graph is shown below:

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:

Determine the symmetry with respect to x-axis.

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

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.

Write the conclusion

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