Question

determine features of polynomial graph
question
the polynomial function ( f(x) ) is graphed below. fill in the form below regarding the features of this graph.
(graph of a polynomial function with y-axis symmetry, crossing x-axis at two points, having a relative minimum at the origin and two relative maxima)
answer
attempt 1 out of 2
the degree of ( f(x) ) is (\boxed{quad}) and
the leading coefficient is (\boxed{quad}).
there are (\boxed{quad}) different real zeros and
(\boxed{quad}) relative minimums.

Explanation:

Step1: Determine the degree

The end - behavior of a polynomial: If the ends of the graph of a polynomial function go in opposite directions (one up and one down or vice - versa), the degree is odd. If the ends go in the same direction (both down or both up), the degree is even. Here, both ends of the graph of \(f(x)\) go down (as \(x\to+\infty\) and \(x\to-\infty\), \(y\to-\infty\)), so the degree of \(f(x)\) is even. Also, the number of turning points (relative maxima and minima) of a polynomial of degree \(n\) is at most \(n - 1\). Looking at the graph, we can see that there are 3 turning points (2 relative maxima and 1 relative minimum). So if the number of turning points \(T=n - 1\), and \(T = 3\), then \(n=4\) (since \(n-1 = 3\Rightarrow n = 4\)). So the degree of \(f(x)\) is 4 (even).

Step2: Determine the leading coefficient

For a polynomial \(y=a_nx^n+\cdots+a_0\), if the degree \(n\) is even:

• If \(a_n>0\), the graph opens up (both ends go up).
• If \(a_n<0\), the graph opens down (both ends go down). Since the graph of \(f(x)\) opens down (both ends go to \(-\infty\)), the leading coefficient is negative.

Step3: Determine the number of real zeros

The real zeros of a polynomial are the \(x\) - intercepts (where the graph crosses or touches the \(x\) - axis). Looking at the graph, we can see that the graph crosses the \(x\) - axis at 2 distinct points (on the left and on the right of the \(y\) - axis). So there are 2 different real zeros.

Step4: Determine the number of relative minima

A relative minimum is a point where the function changes from decreasing to increasing. Looking at the graph, we can see that there is 1 relative minimum (the lowest point between the two relative maxima).

Answer:

The degree of \(f(x)\) is \(\boldsymbol{4}\) (even) and the leading coefficient is \(\boldsymbol{\text{negative}}\). There are \(\boldsymbol{2}\) different real zeros and \(\boldsymbol{1}\) relative minimums.