How to Draw the graph.

Question

The graph of f(x) only has zeros at x = -5, x = -3, x = 1, and x = 7. The graph has a relative maximum at x = 5.

Answer 1

For the polynomial described, we are told where it crosses the x-axis (at -5, -3, 1 and 7) so it's of degree 4.

 

Since a 'relative maximum' occurs at 5, therefore the graph will be always negative on the interval (7,infinity).

 

The positivity/negativity ALTERNATES as you examine the graph from left to right, traveling through the various subintervals established by the zeros.  The 'end behavior' is y values become more and more negative as x values increase beyond 7, or as x values decrease (leftward) beyond -5