Q 48. - Precalculus Enhanced with Graphing Utilities

Question

Is there a function whose graph is symmetric with respect to the x-axis? Explain.

Step-by-Step Solution

Answer

Yes, there is a function whose graph is symmetric with respect to the x-axis.

Step 1. Given information.

Is there a function whose graph is symmetric with respect to the x-axis.

Step 2. Concept used.

A symmetric function is a function in several variable which remains unchanged for any permutation of the variables. 

$f\left(x\right)=-f\left(x\right)$

Step 3. Explanation.

If a graph does not change when reflected over a line or rotated around a point, the graph is symmetric with respect to that line or point. 

The above graph is symmetric with respect to the x-axis (y = 0). 

Let us consider some points from the graph (0, 2), (1, 3), (3, 4)... are the points on the graph.

Also the points (0, -2), (1, -3), (3, -4)... are also on the graph.

When we reflect the green graph about the x-axis x values are never change.

Note that if (x, y) is a point on the graph, then (x, - y) is also a point on the graph. 

Step 4. Conclusion.

Yes, there is a function whose graph is symmetric with respect to the x-axis.