To solve this problem, you should first find the x-intercepts of the function, then the y-intercept, and finally determine the end behavior of the function.
Step 1. Find the x-intercepts:
The x-intercepts of a function can be found by setting the function equal to zero and solving for x. So, for the function h(x) = x(x-1)(x+3)(x+5), set it equal to zero:
x(x-1)(x+3)(x+5) = 0
We find that the x-intercepts of the function are at the points are -5, -3, 0, and 1 because if we substitute these into x(x-1)(x+3)(x+5), the product will be zero. Therefore, the function has a value of zero at these four points along the x-axis.
Step 2. Find the y-intercept:
The y-intercept of a function can be found by setting x = 0 in the function and solving for the function. So, for h(x) = x(x-1)(x+3)(x+5), set x equal to zero:
h(x) = 0(0-1)(0+3)(0+5) = 0
Therefore, the y-intercept of the function is 0, meaning the function touch the y-axis at the origin (0,0).
Step 3. Determine the end behavior:
The end behavior of a polynomail function can be determined by inspecting the highest degree term of the function which is in the form of x^n. In this case, the function can be expanded as a fourth degree polynomial:
h(x) = x^4 -3x^2 -15x
Since the highest power of x (which is 4, even number) has a positive coefficient, the function's ends should both go towards positive infinity. Thus, as x goes to -∞, h(x) goes to ∞, and as x goes to ∞, h(x) also goes to ∞.
So to summarize, the function h(x)=x(x-1)(x+3)(x+5) has x-intercepts at -5, -3, 0, and 1, a y-intercept at 0, and its end behavior is that as x goes to -∞ or ∞, h(x) goes to ∞.