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Understanding the roots of a polynomial is fundamental in math as they represent the solutions of the equation when set to zero. The process of finding these roots can often be simplified by factoring the polynomial—a mathematical version of simplifying by breaking a number into its component multipliers.

For the polynomial function given by h(x)=-x^3-2x^2+15x, the roots are found by setting h(x) to 0 and solving for x. Factoring out the common factor of x simplifies the equation, which can then be factored further until we land upon the zeros of the function: x=0, x=3, and x=-5. These points are where the graph of our polynomial will intersect the x-axis. It's useful to note that polynomials of higher degrees may require more complex methods to find their roots, such as synthetic division or the Rational Root Theorem.

Once the roots of the polynomial are identified, graphing the polynomial becomes more intuitive. Knowing where the graph intersects the x-axis (at the roots) helps in creating an accurate visual representation, which in terms of learning, aids in better understanding the function's behavior.

Considering our function h(x), after plotting the roots on an xy-coordinate grid, we can add the y-intercept, which is also a root, as the point (0,0). It acts as a pivotal point in the graph. A rough sketch can then be created by drawing a curve through these points, making sure that the graph's shape is influenced by the end behavior and reflects the polynomial’s degree and leading coefficient. It’s crucial to remember that every polynomial graph is continuous, meaning it can be drawn without lifting the pencil from the paper. Visual learners especially benefit from this sketching step, as it provides a clear picture of how the function behaves overall.

The end behavior of a polynomial function refers to the direction the graph heads as it moves towards infinity or negative infinity. This is pivotal in sketching correct graphs because it gives us a boundary condition, setting conceptual 'bookends' on our graph.

The highest degree term in our function h(x) dictates the overall shape and direction of the graph for large values of x. Since we have a degree 3 polynomial with a negative coefficient, this means as x approaches negative infinity, h(x) approaches infinity. Conversely, as x approaches positive infinity, h(x) heads towards negative infinity, creating an 's' shaped curve. It's a bit like predicting the future of the graph beyond the confines of our plotted points; these predicted trends are a cornerstone in understanding polynomial functions, especially for higher-degree polynomials where direct calculation of all roots might not be feasible.