Final answer:
The given equation y = (-5x)/(x^2+81) has a y-intercept at y = 0 and does not have any real x-intercepts. It is not symmetric about the y-axis or the x-axis but is symmetric about the origin.
Step-by-step explanation:
To list the intercepts and test for symmetry for the given equation y=(-5x)/(x2+81), we start by finding where the graph intersects the axes. For the y-intercept, we set x to 0 and solve for y. In this case, the y-intercept is y = 0, since plugging x = 0 into the equation gives us y = (-5*0)/(02+81) = 0.
To find the x-intercepts, we set y to 0 and solve for x. However, there are no real values of x that can satisfy this equation because the numerator would need to be 0, which is not possible with this equation as -5x can't be 0 unless x is 0, but then the denominator is positive yielding the fraction undefined at x = 0.
To test for symmetry, we check for three types: y-axis symmetry, origin symmetry, and x-axis symmetry. A graph is symmetric about the y-axis if replacing x with -x yields the original equation. For this function, replacing x with -x gives us y = (-5(-x))/((-x)2+81) which simplifies to y = (5x)/(x2+81), and so the graph is not symmetric about the y-axis. A graph is symmetric about the origin if replacing both x and y with -x and -y respectively gives the original equation. In this case, we get -y = (-5(-x))/((-x)2+81), which simplifies back to the original equation, hence the graph of the equation is symmetric about the origin. Lastly, a graph is symmetric about the x-axis if replacing y with -y results in the original equation. As seen, this equation does not satisfy that condition. Therefore the equation is only symmetric about the origin.