Exercise 2 Page 355

We are given a few statements that describe a quartic function and one that does not. Let's try to provide an example for each option given about a quartic function.

The End Behavior of the Function is Up and Up

Let's graph a quartic function that has and up and up end behavior.

As we can see from the graph, the end behavior can be up and up.

The Function Has $4$ Zeros

The real zeros of a function are the $x\text{-}$intercepts of its graph. We will graph a quartic function that has $4$ zeros.

Looking at the graph, we can conclude that the function can have four zeros.

The Function Has $4$ Turning Points

Let's try to graph a quartic function with $4$ turning points.

The graph of a polynomial function of degree $4$ can have at most $3$ turning points. Therefore, the statement is never true.

The Function Has Complex Roots

A quartic function has always four roots. We will try to graph a quartic function that has two real roots. This would mean that its other two roots are complex.

This statement can also be true.

Conclusion

We found that a quartic function can never have four turning points. This corresponds to statement H.