The graph of the function f(x) = 4x^7 + 40x^6 + 100x^5 is a polynomial of degree 7, which means it can have up to 7 real roots and will behave in a certain way as x approaches positive and negative infinity.
Since the leading coefficient (4) is positive and the degree of the polynomial is odd, the graph will rise to infinity as x approaches positive infinity and fall to negative infinity as x approaches negative infinity.
Additionally, we can examine the behavior of the polynomial at its critical points, which are found by taking the derivative and setting it to zero. This analysis will reveal the turning points and help to sketch the curve accurately.
Overall, the polynomial will show a continuous curve with possible local maxima and minima due to its multiple terms, but the overall end behavior is dictated by the degree and leading coefficient.