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. type each answer only once.) b. there are no intercepts.

Explanation:

Step1: Find x - intercepts

Set \(y = 0\) and find the x - values where the graph crosses the x - axis. From the graph, the x - intercepts are the points where the curve intersects the x - axis. The x - intercepts are \((- 2,0)\) and \((2,0)\).

Step2: Find y - intercepts

Set \(x = 0\) and find the y - value where the graph crosses the y - axis. From the graph, the y - intercept is \((0,0)\).

Step3: Check for symmetry about x - axis

Replace \(y\) with \(-y\) in the equation of the graph. If the resulting equation is the same as the original, the graph is symmetric about the x - axis. For the given graph, if we replace \(y\) with \(-y\), the graph does not look the same, so it is not symmetric about the x - axis.

Step4: Check for symmetry about y - axis

Replace \(x\) with \(-x\) in the equation of the graph. If the resulting equation is the same as the original, the graph is symmetric about the y - axis. For the given graph, if we replace \(x\) with \(-x\), the graph does not look the same, so it is not symmetric about the y - axis.

Step5: Check for symmetry about the origin

Replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation of the graph. If the resulting equation is the same as the original, the graph is symmetric about the origin. For the given graph, if we replace \(x\) with \(-x\) and \(y\) with \(-y\), the graph looks the same, so it is symmetric about the origin.

Answer:

(a) The intercepts of the graph are \((-2,0),(0,0),(2,0)\)
(b) The graph is symmetric with respect to the origin.