Question

draw a complete graph so that it has symmetry with respect to the origin.

Explanation:

Step1: Recall origin - symmetry property

A graph is symmetric about the origin if for every point \((x,y)\) on the graph, the point \((-x,-y)\) is also on the graph.

Step2: Identify given points

We have the points \((0,0)\), \((\frac{\pi}{4},1)\) and \((\frac{\pi}{2},0)\).

Step3: Find symmetric points

For the point \((\frac{\pi}{4},1)\), its symmetric point about the origin is \((-\frac{\pi}{4}, - 1)\). For the point \((\frac{\pi}{2},0)\), its symmetric point about the origin is \((-\frac{\pi}{2},0)\).

Step4: Sketch the graph

Plot the original points and their symmetric - about - the - origin points and draw a smooth curve passing through these points to complete the graph with origin symmetry.

Answer:

Sketch the graph by plotting \((0,0)\), \((\frac{\pi}{4},1)\), \((\frac{\pi}{2},0)\), \((-\frac{\pi}{4}, - 1)\) and \((-\frac{\pi}{2},0)\) and drawing a smooth curve through them.