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. 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. (b) choose the correct answer below. select all that apply. a. the graph is symmetric with respect to the x - axis. b. the graph is symmetric with respect to the origin. c. the graph is symmetric with respect to the y - axis. d. the graph is not symmetric.

Explanation:

Step1: Find x - intercepts

The x - intercepts are the points where the graph crosses the x - axis (y = 0). From the graph, the x - intercepts are the x - values of the points where the curve intersects the x - axis. Let's assume the x - intercepts are $x_1$ and $x_2$. By observing the graph, we can see that the x - intercepts are $x=- \frac{\pi}{2},x = \frac{\pi}{2}$. So the x - intercepts as ordered pairs are $(-\frac{\pi}{2},0)$ and $(\frac{\pi}{2},0)$.

Step2: Find y - intercepts

The y - intercept is the point where the graph crosses the y - axis (x = 0). By observing the graph, when x = 0, y = 0. So the y - intercept is the ordered pair (0,0). The set of all intercepts as ordered pairs is $(-\frac{\pi}{2},0),(0,0),(\frac{\pi}{2},0)$.

Step3: Check for symmetry

• X - axis symmetry: If $(x,y)$ is on the graph, then $(x, - y)$ must also be on the graph. By observing the graph, if we take a point $(x,y)$ on the graph, the point $(x,-y)$ is not on the graph. So the graph is not symmetric about the x - axis.
• Y - axis symmetry: If $(x,y)$ is on the graph, then $(-x,y)$ must be on the graph. By observing the graph, if we take a point $(x,y)$ on the graph, the point $(-x,y)$ is not on the graph. So the graph is not symmetric about the y - axis.
• Origin symmetry: If $(x,y)$ is on the graph, then $(-x,-y)$ must be on the graph. By observing the graph, if we take a point $(x,y)$ on the graph, the point $(-x,-y)$ is not on the graph. So the graph is not symmetric about the origin.

Answer:

(a) The intercepts of the graph are $(-\frac{\pi}{2},0),(0,0),(\frac{\pi}{2},0)$
(b) D. The graph is not symmetric.