Question

answer attempt 1 out of 2
the degree of f(x) is even and the leading coefficient is negative. there are 2 different real zeros and
relative extremes.

Explanation:

Step1: Analyze the graph's end - behavior

For a polynomial function, the end - behavior is determined by the degree (even or odd) and the leading coefficient (positive or negative). The graph of the function has both ends going down (as \(x
ightarrow+\infty\) and \(x
ightarrow-\infty\), \(f(x)
ightarrow-\infty\)). For a polynomial, if the degree is even, the ends of the graph have the same behavior. If the leading coefficient is negative and the degree is even, both ends go down. So the degree is even and the leading coefficient is negative.

Step2: Count the real zeros

A real zero of a function \(y = f(x)\) is a value of \(x\) where \(f(x)=0\), that is, where the graph intersects the \(x\) - axis. From the graph, we can see that the graph intersects the \(x\) - axis at 2 distinct points. So there are 2 different real zeros.

Step3: Count the relative extrema

Relative extrema (relative maxima and relative minima) occur at the "peaks" and "valleys" of the graph. Looking at the graph, we can see that there are 3 turning points (a valley and two peaks or vice - versa). The number of relative extrema of a polynomial function is related to its degree. For a polynomial of degree \(n\), the number of relative extrema is at most \(n - 1\). Each turning point corresponds to a relative extremum. By visually inspecting the graph, we can see that there are 3 relative extrema (2 relative maxima and 1 relative minimum or 2 relative minima and 1 relative maximum).

Answer:

The degree of \(f(x)\) is even and the leading coefficient is negative. There are 2 different real zeros and 3 relative extrema.