Question

the polynomial function $f(x)$ is graphed below. fill in the form below regarding the features of this graph.

the degree of $f(x)$ is dropdown with odd/even and the leading coefficient is dropdown. there are blank different real zeros and blank relative minimums.

Explanation:

Step1: Determine Degree Parity

The ends of the polynomial graph both go up (as \( x \to \pm\infty \), \( f(x) \to +\infty \)). For a polynomial, if the leading coefficient is positive and the degree is even, both ends rise. So the degree is even.

Step2: Leading Coefficient Sign

Since both ends of the graph rise (as \( x \to \infty \) and \( x \to -\infty \), \( f(x) \to \infty \)), the leading coefficient is positive (because for even degree, positive leading coefficient gives upward - ending ends).

Step3: Count Real Zeros

A real zero of a polynomial is a point where the graph intersects the \( x \) - axis. By looking at the graph, we can see that the graph crosses the \( x \) - axis 4 times. So there are 4 different real zeros.

Step4: Count Relative Minimums

A relative minimum is a point where the function changes from decreasing to increasing. Looking at the graph, we can identify the "valleys" of the graph. By carefully examining the graph, we can see that there are 3 relative minimums.

Answer:

The degree of \( f(x) \) is even and the leading coefficient is positive. There are 4 different real zeros and 3 relative minimums.