Question

the polynomial function $f(x)$ is graphed below. select all the features of the graph.
there are 4 different real zeros
there are 5 different real zeros
there are 2 relative maximums
the leading coefficient is positive
there is 1 relative maximum
the degree of $f(x)$ is even.
the degree of $f(x)$ is odd.
the leading coefficient is negative

Explanation:

1. Real Zeros: The graph intersects the x - axis at 4 distinct points (we count the number of times the graph crosses or touches the x - axis. Here, it crosses the x - axis 4 times, so there are 4 different real zeros.
2. Relative Maximums: A relative maximum is a point where the function changes from increasing to decreasing. Looking at the graph, we can see that there are 2 such points (the "peaks" in the graph), so there are 2 relative maximums.
3. Degree and Leading Coefficient: The end - behavior of a polynomial is determined by the degree (even or odd) and the leading coefficient (positive or negative). For the given graph, as \(x

ightarrow+\infty\), \(f(x)
ightarrow-\infty\) and as \(x
ightarrow-\infty\), \(f(x)
ightarrow+\infty\). The end - behavior of a polynomial \(y = a_nx^n+\cdots+a_0\) is: if \(n\) is odd, the ends of the graph go in opposite directions. If \(a_n>0\), as \(x
ightarrow+\infty\), \(y
ightarrow+\infty\) and as \(x
ightarrow-\infty\), \(y
ightarrow-\infty\); if \(a_n < 0\), as \(x
ightarrow+\infty\), \(y
ightarrow-\infty\) and as \(x
ightarrow-\infty\), \(y
ightarrow+\infty\). Since our graph has opposite end - behaviors (one end going up and one going down) and as \(x
ightarrow+\infty\), \(f(x)\) is negative and as \(x
ightarrow-\infty\), \(f(x)\) is positive, the degree is odd and the leading coefficient is negative.

Answer:

• There are 4 different real zeros
• There are 2 relative maximums
• The degree of \(f(x)\) is odd.
• The leading coefficient is negative