MPF - Graphs of Polynomial Functions Lesson

Graphs of Polynomial Functions

What is a polynomial function? It is a function in the form $LaTeX: a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0=0$ $a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0=0$ text annotation indicator where equation image indicator $a_n$ is called the leading coefficient, $a_nx^n$ is called the leading term, and LaTeX: a_0 $a_0$ is called the constant. The coefficients are real numbers and the exponents are whole numbers. In order to be a polynomial function "n" must be a nonnegative integer. The degree of the polynomial function is "n" or the value of the term in the polynomial that has the highest exponent. Let's expand our table to identify the different types of polynomial functions.

Polynomial graphs have basic shapes, depending on the degree, as shown in the table.

Polynomial Function	Example	Degree	Leading Coefficient	Graph
Constant	$LaTeX: F\left(x\right)=2 \\ f\left(x\right)=2x^0$ $F\left(x\right)=2 \\ f\left(x\right)=2x^0$	0	2	Horizontal line
Linear	$F(x)=4x+3$	1	4	Line with slope
Quadratic	$F(x)=3x^2-x+5$	2	3	Parabola
Cubic	$LaTeX: F\left(x\right)=10x^3-7x^2+13$ $F\left(x\right)=10x^3-7x^2+13$	3	10	S-shape curve
Quartic	$LaTeX: F\left(x\right)=6x^4-3x^3+5x^2-1$ $F\left(x\right)=6x^4-3x^3+5x^2-1$	4	6	U-shaped curve with some potential "ups" and "downs" in the middle of the graph.

The graph of a polynomial function is continuous. It has no holes or breaks. It is also smooth, which means that it has no sharp corners. In general its domain is the set of all real numbers.

Notice you can draw it without ever lifting your pencil. In a previous class, you learned that a translation was a vertical and/or horizontal shift. We can also translate the graph of a polynomial function.

Get your graphing calculator (TI-83 or Ti-84) or use an online calculator and graph the following:

The following chart should summarize your discoveries:

To Graph	Draw by	Change in function
Vertical Shifts
Y = f(x) + k, k > 0	Shift graph "up" by k units	Add "k" to f(x)
Y = f(x) - k, k > 0	Shift graph "down" by k units	Subtract "k" from f(x)
Horizontal Shifts
Y = f(x + h), h > 0	Shift graph "left" by "h" units	Replace "x" with (x + h)
Y = f(x - h), h > 0	Shift graph "right" by "h" units	Replace "x" with (x - h)

Key Features of Polynomial Graphs

There are several key features of polynomial graphs (and all graphs). The first is the domain and the range. The domain of polynomial functions is always the set of all real numbers. The range is determined by the degree. If the degree is odd, the range is the set of all real numbers. If the degree is even, the range is from the lowest point up to ∞ or from -∞ to the highest point.

In the section of "rough sketches of graphs", we learned the importance of intercepts. Finding the zeros of a function will give you the x-intercepts and the constant term of the function is the y-intercept. Other key features are extrema, which are the maximums and minimums of a graph (typically these are the absolute minimums and maximums). For a parabola, this is the vertex.

The vertex of a quadratic polynomial is (h, k) when the equation is in the vertex form: $LaTeX: y=a\left(x-h\right)^2+k$ $y=a\left(x-h\right)^2+k$ .

If the quadratic polynomial is in standard form, which is LaTeX: y=ax^2+bx+c $y=ax^2+bx+c$ , the x - coordinate of the vertex is $LaTeX: -\frac{b}{2a}$ $-\frac{b}{2a}$ . The vertex would be $LaTeX: \left(-\frac{b}{2a},\:f\left(-\frac{b}{2a}\right)\right)$ $\left(-\frac{b}{2a},\:f\left(-\frac{b}{2a}\right)\right)$ . You substitute the x-coordinate into the function in order to find the y-coordinate. Also, recall from the rough sketch lesson that the x-coordinate is the number halfway between the x-intercepts. In polynomial functions, you may use the trace, zoom, maximum, minimum, or zeros features of a graphing calculator or graphing program to find the maximums and minimums. Extrema are often called turning points of a graph.

A polynomial function's absolute extrema occurs at the highest or lowest point for the function. The highest point on a function is called the absolute maximum, and the lowest point on a function is called the absolute minimum. The relative extrema occurs in a function when a point is higher or lower than any other points nearby in the graph. A relative maximum is the highest point for an interval of the graph, and a relative minimum is the lowest point for an interval of the graph.

Functions have a beginning and an end. There is a caveat to the previous statement. Functions with a domain of all real numbers do not have a beginning or an end. Functions with "even degrees" do have an absolute maximum or minimum, while the "opposite" end of these functions will go to positive OR negative infinity. While functions with "odd numbered degrees" will go to negative infinity AND positive infinity at opposite ends. Functions can be graphed and/or solved algebraically to determine the end behavior. The end behavior of a function can be determined graphically by looking at the graph and viewing how the graph begins and ends. The behavior of a function can be determined algebraically by using the leading coefficient of the equation and the degree.

If the leading coefficient is positive the graph ends in an upward direction (rises).
If the leading coefficient is negative, the graph ends in a downward direction (falls).

Degree of the Polynomial

Positive Leading Coefficient

Negative Leading Coefficient

Quadratic

graph with parabola that opens up

graph of cubic negative: parabola that opens down

Cubic

graph of cubic positive

graph of quartic positive

Quartic

image of quadratic negative leading coefficient

graph of quartic negative

Graphing by Hand and with a Calculator

When the characteristics of polynomial graphs are examined in detail we can find patterns that arise in their behavior.

Watch the following videos to see how to graph various polynomial functions "by hand" or "without the use of a calculator."

Watch the following videos to see how to graph quadratic equations. (Notice in the first video, that the x and y intercepts are in the table. Recall that you can get the x intercepts by factoring the equation and that the y intercept is the constant.)

Graphs of Polynomial Functions Self-Assessment

Write a Formula for a Polynomial Function given the Graph

If we know the zeros of polynomial functions, we can write the formulas based on the graphs. This is possible because a polynomial function written in factored form will have an x-intercept where each factor is equal to zero. Let's look at an example.

Given the following graph, we can determine that the zeros are x=-4, x=-2, x=0, and x=3.