Analyzing Graphs of Polynomial Functions

Instructor: Melanie Olczak

Melanie has taught high school Mathematics courses for the past ten years and has a master's degree in Mathematics Education.

This lesson will explain the graph of a polynomial function by identifying properties including end behavior, real and non-real zeros, odd and even degree, and relative maxima or minima. We will then sketch a graph using this information.

Graphs of Polynomial Functions

Have you ever been on a roller coaster? Did you know that the path of a roller coaster can be represented by a polynomial function? A polynomial function is an equation with multiple terms that has variables and exponents. The graphs of polynomial functions contain a great deal of information. We can find the information by looking at the graph and equation and we can graph the polynomial if we are given the information or equation.


polynomial function1


By looking at the graph we can determine the end behavior, real and non-real zeros, if the graph is odd or even, and the relative extrema.

The end behavior of a graph is what is happening to the y-values as the x-values get bigger and smaller, meaning as x approaches positive and negative infinity.

We look to the left and the right of the y-axis and determine what is happening on our graph. To the left of the x-axis, x is getting smaller or approaching negative infinity. The graph is going up, so y is getting bigger or approaching positive infinity.


end behavior


On the right of the y-axis, the x-values are getting bigger or approaching positive infinity. The graph is going down, so y is getting smaller or approaching negative infinity.


end behavior


The zeros of a function refer to the points where the function f(x) is equal to zero. These are also called the x-intercepts, the points where the graph crosses the x-axis.


zeros


Since two of the x-intercepts do not cross at integer points, we are approximating those zeros.

In this example, there are three points where the graph crosses the x-axis. Because there are three x-intercepts, there are three real zeros. A real zero is an x-intercept of the graph. This graph is a cubic function because the highest exponent is three. This means that there are three zeros. If there were not three x-intercepts, we would have some non-real zeros. A non-real zero is a solution to the equation that is an imaginary number. The number of total zeros, both real and non-real, is the same as the highest exponent of the function.

By looking at the graph, we can also determine if the function is even or odd. An even function is a function that is symmetrical with respect to the y-axis. That means if we fold our graph along the y-axis it matches up perfectly.


even


The red function is our original function, while the green is the reflected image. As you can see the red and green functions do not overlap. Because they do not match up, this function is not even.

An odd function is a function that is symmetrical with respect to the origin. Symmetrical with respect to the origin means that if we reflect the graph in the origin, by changing the signs of all our points, the graphs should match up perfectly.


odd


Again, the red function is our original function and the green is the reflected image. Do these images match up perfectly? The answer is no, as the green one is shifted to the left. This function is neither even nor odd.

We can find points on the graph called extrema. There are two types of extrema, minima and maxima. Minima are low points of the graph. The minimum can either be absolute or relative. An absolute minimum is the lowest point of the entire graph. A relative minimum is a low point on the graph, but not the lowest of all points on the graph. The same is true for maxima. Absolute maximum refers to the highest point on the graph, while a relative maximum is a high point on the graph, but not the highest of all points on the graph.

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