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. (type an ordered pair. use a comma to separate answers as needed.) b. there are no intercepts.

Explanation:

Step1: Find x - intercepts

Set \(y = 0\). From the graph, the graph does not cross the \(x\) - axis. So there are no \(x\) - intercepts.

Step2: Find y - intercepts

Set \(x = 0\). The graph intersects the \(y\) - axis at \((0,4)\) and \((0, - 4)\).

Step3: Check for symmetry about x - axis

Replace \(y\) with \(-y\). If the equation of the graph remains the same, it is symmetric about the \(x\) - axis. For the given graph, if we have a point \((x,y)\) on the graph, the point \((x, - y)\) is also on the graph, so it is symmetric about the \(x\) - axis.

Step4: Check for symmetry about y - axis

Replace \(x\) with \(-x\). If the equation of the graph remains the same, it is symmetric about the \(y\) - axis. For the given graph, if we have a point \((x,y)\) on the graph, the point \((-x,y)\) is also on the graph, so it is symmetric about the \(y\) - axis.

Step5: Check for symmetry about origin

Replace \(x\) with \(-x\) and \(y\) with \(-y\). If the equation of the graph remains the same, it is symmetric about the origin. Since the graph is symmetric about both \(x\) - axis and \(y\) - axis, it is symmetric about the origin.

Answer:

(a) A. The intercept(s) of the graph are \((0,4),(0, - 4)\)
(b) The graph is symmetric with respect to the \(x\) - axis, the \(y\) - axis, and the origin.