College Algebra, Chapter 4, 4.2, Section 4.2, Problem 72

Graph the family of polynomials $P(x) = x^3 + cx; c = 2, 0 , -2 , -4$ in the same viewing rectangle. Explain what are the effects of changing the value of $c$.


Based from the graph, when the value of $c$ is positive, its graph will never cross the $x$-axis other than the origin. On the other hand, when the value of $c$ is negative, its graph crosses the $x$-axis that makes the functions local extrema defined. All the functions have the same end behaviours and their graphs are all symmetric to the origin. As the value of $c$ increases positively, the graph of $P(x) = x^3$ where $c$ is 0 became more compressed. On the other hand, when the value of $c$ increases negatively its graph expands.