Unit 56: Polynomial Graphs

In this section, we will build on knowledge we have learned in other units to analyze polynomial equations and their corresponding graphs.

To start, let’s go over some basic features of polynomial graphs.

These graphs may seem big and scary, but by understanding some of its key features, we can use process of elimination to get the correct answer choice.

Strategy:

  1. Look at the number of changes in direction of the graph to find the highest degree of the polynomial
    • The number of changes in direction will always equal the highest degree of the polynomial
  2. Look at where the graph crosses the x-axis
    • Each time a graph crosses the x-axis, it will have a solution at that point. We can find the equation by looking for the zeros of the graph.

Let’s apply this strategy to an example.

Which of the following could be the equation of the graph shown in the xy-plane?

A) (y=frac15(x-3){(x+2)}^2(x+5))
B) (y=frac15(x-3)(x+2)(x+5))
C) (y=frac15(x+3){(x-2)}^2(x-5))
D) (y=frac15(x+3)(x-2)(x-5))

First, we can see from the graph that there are four changes in direction for this function. This means that this is an (x^4) polynomial, and thus there must be four x‘s in our equation. This rules out options B and D because these equation only have 3 x‘s in them. Both options A and C have four x‘s, so these two equations could be correct.

Now let’s look at the zeros or x-intercepts of the graph. This graph has three zeros, which matches both A and D. However, when we solve for each x-intercept, we get the following factors: ((x+3)(x-2)(x-5)). This only matches with answer C.

Choice C is the correct answer.

Exponential Graphs

Most questions that deal with exponential graphs will ask you about the y-intercept of the graph. Remember that the basic form of an exponential function is (y=ab^x). To find the y-intercept from this equation, simply plug in zero for x. Remember that any number raised to the zeroth power is 1!

For example:

$$g(x)=13{(frac15)}^x +23$$

If the given function g is graphed in the xy-plane, where (y=g(x)), what is the (y)-intercept of the graph?

To solve this problem, we plug in (x=0):

(g(x)=y=13{(frac15)}^x+23)
(y=13(1))+23)
(y=36)

The y-intercept of the equation is 36.

Other Nonlinear Graphs

Another type of nonlinear graphs that you might encounter are rational functions.

These equations are written in the form: (f(x)=frac1{x+5})

A graph of the function is shown below.

For these graphs, it is important to remember that a function is undefined when dividing a number by 0. Thus, when the denominator of the equation is equal to 0, there will be a vertical asymptote at that point where the function is undefined. For the following graph, (x+5=0) when (x=-5), so there is a vertical asymptote where as the graph approaches (-5) the y-value of the graph is undefined.