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 and, if necessary, fill in the answer box to complete your choice. a. the intercept(s) of the graph are (0,4),(0, - 4). (type an ordered pair. use a comma to separate answers as needed.) b. there are no intercepts. (b) select all that apply. a. the graph is symmetric with respect to the y - axis. b. the graph is symmetric with respect to the x - axis. c. the graph is symmetric with respect to the origin. d. the graph is not symmetric.

Explanation:

Step1: Identify x - intercepts and y - intercepts

The x - intercepts are the points where the graph crosses the x - axis (y = 0), and the y - intercepts are the points where the graph crosses the y - axis (x = 0). From the graph, the y - intercepts are (0, 4) and (0, - 4), and there are no x - intercepts.

Step2: Check for symmetry

• Y - axis symmetry: Replace x with - x in the equation of the graph. If the equation remains the same, the graph is symmetric about the y - axis. Visually, the left - hand side of the graph is a mirror image of the right - hand side. So the graph is symmetric about the y - axis.
• X - axis symmetry: Replace y with - y in the equation of the graph. If the equation remains the same, the graph is symmetric about the x - axis. But from the graph, if we reflect the upper part about the x - axis, it does not match the lower part, so it is not symmetric about the x - axis.
• Origin symmetry: Replace x with - x and y with - y in the equation of the graph. If the equation remains the same, the graph is symmetric about the origin. Since it is not symmetric about the x - axis, it is not symmetric about the origin either.

Answer:

(a) A. (0, 4),(0, - 4)
(b) A. The graph is symmetric with respect to the y - axis.