Graphs of polynomials: Challenge problems

Solve challenging problems that tackle the relationship between the features of a polynomial and its graph.

What you should be familiar with before taking this lesson

What you will do in this lesson

Now that we have learned about the features of the graphs of polynomial functions, let's put that knowledge to use!
In this set of problems, the equations of the polynomials are not completely given. This way, they force us to focus on a specific feature of the polynomial's graph.
Good luck!
1*) Which of the following could be the graph of y=ax3+bx2+cx+2y=ax^3+bx^2+cx+2, where aa, bb, and cc are real numbers?
Choose 1 answer:
Choose 1 answer:
In this case, we do not know the specific values of aa, bb, and cc, and so we will not be able to determine end behavior, xx-intercepts, or the intervals over which the polynomial is positive or negative.
We do, however, know that the constant term is 22. This will be the yy-intercept of the graph of y=ax3+bx2+cx+2y=ax^3+bx^2+cx+2, because when we input x=0x=0, we get y=2y=2.
So the yy-intercept of the graph of the function is (0,2)(0,2). The only graph that has this feature is graph CC.
2*) Which of the following could be the graph of y=2x5+p(x)y=-2x^5+p(x), where p(x)p(x) is a fourth degree polynomial?
Choose 1 answer:
Choose 1 answer:
In this case, we do not know the polynomial p(x)p(x), and so we will not be able to determine the xx- or yy-intercepts, or the intervals over which the polynomial is positive or negative.
We can, however, determine the function's end behavior. Since the degree of p(x)p(x) is less than the degree of 2x5-2x^5, we know that 2x5-2x^5 must be the leading term of the polynomial!
So the end behavior of y=2x5+p(x)y=-2x^5+p(x) will be the same as the end behavior of the monomial 2x5-2x^5.
Since the degree of 2x5\greenD{-2}x^\blueD5 is odd (5)(\blueD 5) and the leading coefficient is negative (2)(\greenD{-2}), the end behavior will be as follows:
As x+x\rightarrow +\infty, yy\rightarrow -\infty and as xx\rightarrow -\infty, y+y\rightarrow +\infty.
This feature is only seen in graph BB.
3*) Which of the following could be the graph of y=k(x2)m(x+1)ny=k(x-2)^m(x+1)^n, where kk is a real number, mm is an even integer, and nn is an odd integer?
Choose all answers that apply:
Choose all answers that apply:
We do not know the value of kk, and so we cannot make a claim regarding the yy-intercept, the end behavior, or the positive and negative intervals of the function's graph.
We do, however, know that 22 is a zero of even multiplicity. This means that the graph of the function must touch the xx-axis at (2,0)(2,0).
We also know that 1-1 is a zero of odd multiplicity. This means that the graph of the function must cross the xx-axis at (1,0)(-1,0).
The graphs that have these features are AA and DD.
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