# Graphs of polynomials: Challenge problems

CCSS Math: HSF.IF.C.7c
Solve challenging problems that tackle the relationship between the features of a polynomial and its graph.

#### 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 $y=ax^3+bx^2+cx+2$, where $a$, $b$, and $c$ are real numbers?
In this case, we do not know the specific values of $a$, $b$, and $c$, and so we will not be able to determine end behavior, $x$-intercepts, or the intervals over which the polynomial is positive or negative.
We do, however, know that the constant term is $2$. This will be the $y$-intercept of the graph of $y=ax^3+bx^2+cx+2$, because when we input $x=0$, we get $y=2$.
So the $y$-intercept of the graph of the function is $(0,2)$. The only graph that has this feature is graph $C$.
2*) Which of the following could be the graph of $y=-2x^5+p(x)$, where $p(x)$ is a fourth degree polynomial?
In this case, we do not know the polynomial $p(x)$, and so we will not be able to determine the $x$- or $y$-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 $p(x)$ is less than the degree of $-2x^5$, we know that $-2x^5$ must be the leading term of the polynomial!
So the end behavior of $y=-2x^5+p(x)$ will be the same as the end behavior of the monomial $-2x^5$.
Since the degree of $\greenD{-2}x^\blueD5$ is odd $(\blueD 5)$ and the leading coefficient is negative $(\greenD{-2})$, the end behavior will be as follows:
As $x\rightarrow +\infty$, $y\rightarrow -\infty$ and as $x\rightarrow -\infty$, $y\rightarrow +\infty$.
This feature is only seen in graph $B$.
3*) Which of the following could be the graph of $y=k(x-2)^m(x+1)^n$, where $k$ is a real number, $m$ is an even integer, and $n$ is an odd integer?
We do not know the value of $k$, and so we cannot make a claim regarding the $y$-intercept, the end behavior, or the positive and negative intervals of the function's graph.
We do, however, know that $2$ is a zero of even multiplicity. This means that the graph of the function must touch the $x$-axis at $(2,0)$.
We also know that $-1$ is a zero of odd multiplicity. This means that the graph of the function must cross the $x$-axis at $(-1,0)$.
The graphs that have these features are $A$ and $D$.
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