Polynomial Functions Help (page 2)
Introduction to Polynomial Functions
A polynomial function is a function in the form f(x) = a n x n + a n −1 x n −1 + ... + a 1 x + a 0 , where each a i is a real number and the powers on x are whole numbers. There is no x under a root sign and no x in a denominator. The number a i is called a coefficient . For example, in the polynomial function f(x) = −2 x 3 + 5 x 2 − 4 x + 8, the coefficients are −2, 5, −4, and 8. The constant term (the term with no variable) is 8. The powers on x are 3, 2, and 1. The degree of the polynomial (and polynomial function) is the highest power on x . In this example, the degree is 3. Quadratic functions are degree 2. Linear functions of the form f(x) = mx + b (if m ≠ 0) are degree 1. Constant functions of the form f(x) = b are degree 0 (this is because x 0 = 1, making f(x) = bx 0 ).
Leading Term and Leading Coefficient - Determining the Behavior of a Graph
The leading term of a polynomial (and polynomial function) is the term having x to the highest power. Usually, but not always, the leading term is written first. The leading coefficient is the coefficient on the leading term. In our example, the leading term is −2 x 3 , and the leading coefficient is −2. By looking at the leading term only, we can tell roughly what the graph looks like. The graph of any polynomial will either go up on both ends, go down on both ends, or go up on one end and down on the other. This is called the end behavior of the graph. The figures below illustrate the end behavior of polynomial functions. The shape of the dashed part of the graph depends on the individual function.
This graph goes up on both ends.
This graph goes down on both ends.
This graph goes down on the left and up on the right.
This graph goes up on the left and down on the right.
If the degree of the polynomial is an even number, the graph will look like the graph in Figure 7.1 or in Figure 7.2. If the leading coefficient is a positive number, the graph will look like the graph in Figure 7.1. If the leading coefficient is a negative number, the graph will look like the graph in Figure 7.2. If the degree of the polynomial is an odd number, the graph will look like the one in Figure 7.3 or in Figure 7.4. If the leading coefficient is a positive number, the graph will look like the graph in Figure 7.3. If the leading coefficient is a negative number, the graph will look like the graph in Figure 7.4.
How can one term in a polynomial function give us this information? For polynomial functions, the leading term dominates all of the other terms. For x -values large enough (both large positive numbers and large negative numbers), the other terms don’t contribute much to the size of the y -values.
Match the graph of the given function with one of the graphs in Figures 7.1-7.4.
- f(x) = 4 x 5 + 6 x 3 −2 x 2 + 8 x + 11
We only need to look at the leading term, 4 x 5 . The degree, 5, is odd, and the leading coefficient, 4, is positive. The graph of this function looks like the one in Figure 7.3.
- P(x ) = 5 + 2 x −6 x 2
The leading term is −6 x 2 . The degree, 2, is even, and the leading coefficient, −6, is negative. The graph of this function looks like the one in Figure 7.2.
- h(x) = −2 x 3 + 4 x 2 −7 x + 9
The leading term is −2 x 3 . The degree, 3, is odd, and the leading coefficient, −2, is negative. The graph of this function looks like the one in Figure 7.4.
- g(x) = x 4 + 4 x 3 − 8 x 2 + 3 x − 5
The leading term is x 4 . The degree, 4, is even, and the leading coefficient, 1, is positive. The graph of this function looks like the one in Figure 7.1 .
Finding the x-Intercepts of a Polynomial Function
Finding the x -intercepts (if any) for the graph of a polynomial function is very important. The x -intercept of any graph is where the graph touches the x -axis. This happens when the y -coordinate of the point is 0. We found the x -intercepts for some quadratic functions by factoring and setting each factor equal to zero. This is how we will find the x -intercepts for polynomial functions. It is not always easy to do. In fact, some polynomials are so hard to factor that the best we can do is approximate the x -intercepts (using graphing calculators or calculus). This will not be the case for the polynomials in this book, however. Every polynomial here will be factorable using techniques covered later.
Zeros of a Polynomial
Because an x -intercept for f(x) = a n x n + a n −1 x n −1 + ... + a 1 x + a 0 is a solution to the equation 0 = a n x n + a n −1 x n −1 + ... + a 1 x + a 0 , x -intercepts are also called zeros of the polynomial. All of the following statements have the same meaning for a polynomial. Let c be a real number, and let P(x) be a polynomial function.
- c is an x -intercept of the graph of P (x).
- c is a zero for P(x) .
- x − c is a factor of P(x) .
- x − 1 is a factor means that 1 is an x -intercept and a zero.
- x + 5 is a factor means that −5 is an x -intercept and a zero.
- x is a factor means that 0 is an x -intercept and a zero.
- 3 is a zero means that x − 3 is a factor and 3 is an x -intercept.
We can find the zeros of a function (or at least the approximate zeros) by looking at its graph.
The x -intercepts of the graph in Figure 7.5 are 2 and −2. Now we know that x − 2 and x + 2 (which is x − (−2)) are factors of the polynomial.
The graph of the polynomial function in Figure 7.6 has x -intercepts of −1, 1, and 2. This means that x − 1, x − 2, and x + 1 (as x − (−1)) are factors of the polynomial.
The x -intercepts for the graph in Figure 7.7 are −3, 0, and 2, making x + 3, x (as x − 0), and x − 2 factors of the polynomial.
Matching a Graph to its Polynomial Function Examples
Now that we know about the end behavior of the graphs of polynomial functions and the relationship between x -intercepts and factors, we can look at a polynomial and have a pretty good idea of what its graph looks like. In the next set of examples, we will match the graphs from the previous section with their polynomial functions.
Match the functions with the graphs in Figures 7.5-7.7.
Because f(x) is a polynomial whose degree is even and whose leading coefficient is positive, we will look for a graph that goes up on the left and up on the right. Because the factors are x 2 , x + 3, and x − 2, we will also look for a graph with x -intercepts 0, −3, and 2. The graph in Figure 7.7 satisfies these conditions.
Because g(x) is a polynomial whose degree is odd and whose leading coefficient is negative, we will look for a graph that goes up on the left and down on the right. The factors are x − 1, x − 2, and x + 1, we will also look for a graph with 1, 2, and −1 as x -intercepts. The graph in Figure 7.6 satisfies these conditions.
Because P(x) is a polynomial whose degree is odd and whose leading term is positive, we will look for a graph that goes down on the left and up on the right. The x -intercepts are 2 and −2. The graph in Figure 7.5 satisfies these conditions.
Practice problems for this concept can be found at: Polynomial Functions Practice Test.
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