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) select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the intercept(s) of the graph are. (type an ordered pair. use a comma to separate answers as needed. use integers or fractions for any numbers in the expression. type each answer only once. type an exact answer, using π as needed.) b. there are no intercepts.

Explanation:

Step1: Find x - intercepts

Set \(y = 0\). The graph intersects the \(x\)-axis at \(x=-\frac{\pi}{2},-\frac{\pi}{4},\frac{\pi}{4},\frac{\pi}{2}\). So the \(x\)-intercepts are \((-\frac{\pi}{2},0),(-\frac{\pi}{4},0),(\frac{\pi}{4},0),(\frac{\pi}{2},0)\).

Step2: Find y - intercept

Set \(x = 0\). The graph intersects the \(y\)-axis at \(y = 1\), so the \(y\)-intercept is \((0,1)\).

for part (b):

Step1: Test for x - axis symmetry

Replace \(y\) with \(-y\). If the resulting equation is the same as the original, it is symmetric about the \(x\)-axis. The graph is not symmetric about the \(x\)-axis since replacing \(y\) with \(-y\) changes the graph.

Step2: Test for y - axis symmetry

Replace \(x\) with \(-x\). The graph is symmetric about the \(y\)-axis because if we replace \(x\) with \(-x\), the shape of the graph remains the same.

Step3: Test for origin symmetry

Replace \(x\) with \(-x\) and \(y\) with \(-y\). The graph is not symmetric about the origin since doing these replacements changes the graph.

Answer:

A. The intercept(s) of the graph are \((-\frac{\pi}{2},0),(-\frac{\pi}{4},0),(0,1),(\frac{\pi}{4},0),(\frac{\pi}{2},0)\)