Question

the graph of an equation is given. (a) find the intercepts. (b) indicate whether the graph is symmetric with respect to the x - axis, the y - axis, the origin, or none of these. (a) the x - intercepts are the points on a graph at which the graph touches or crosses the x - axis (horizontal axis). at what point(s) do the x - intercept(s) occur? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the x - intercepts occur at points (type an ordered pair. use a comma to separate answers as needed. use integers or fractions for any numbers in the expression. type an exact answer, using π as needed.) b. there are no x - intercepts.

Explanation:

Step1: Identify x - intercepts

The x - intercepts are the points where the graph crosses the x - axis (y = 0). From the graph, the x - intercepts are at \(x=-\frac{\pi}{2},-\frac{\pi}{4},\frac{\pi}{4},\frac{\pi}{2}\). In ordered - pair form, they are \((-\frac{\pi}{2},0),(-\frac{\pi}{4},0),(\frac{\pi}{4},0),(\frac{\pi}{2},0)\).

Step2: Check for symmetry

1. X - axis symmetry: If \((x,y)\) is on the graph, then \((x, - y)\) should also be on the graph. The graph is not symmetric about the x - axis since for a point \((x,y)\) on the graph (e.g., \((0,1)\)), \((x, - y)=(0, - 1)\) is not on the graph.
2. Y - axis symmetry: If \((x,y)\) is on the graph, then \((-x,y)\) should be on the graph. The graph is symmetric about the y - axis because for any point \((x,y)\) on the graph, \((-x,y)\) is also on the graph.
3. Origin symmetry: If \((x,y)\) is on the graph, then \((-x,-y)\) should be on the graph. The graph is not symmetric about the origin since for the point \((0,1)\) on the graph, \((-0,-1)=(0, - 1)\) is not on the graph.

Answer:

(a) A. The x - intercepts occur at points \((-\frac{\pi}{2},0),(-\frac{\pi}{4},0),(\frac{\pi}{4},0),(\frac{\pi}{2},0)\)
(b) The graph is symmetric with respect to the y - axis.