2.3: Polynomial Functions and Their Graphs
A polynomial function is a function of the form
f(x) = anxn + an­1xn­1 + an­2xn­2 + ... + a1x + a0, where the coefficients are all real numbers and the exponents are all whole numbers.
The degree of the polynomial is n, the leading coefficient is an, and the constant term is a0.
Some familiar polynomials are those of degree 0, 1, 2, and 3.
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The graph of a polynomial function is smooth and continuous, which means that there are no sharp corners and no breaks in the graph. If the degree is 2 or higher, the graph will be curved.
The end behavior of the graph (behavior to the far left or far right) is determined by the leading term. (anxn). If the degree (n) is odd, the behavior on the ends will be different. If the degree is even, the behavior on the ends will be the same.
If the leading coefficient (an) is positive, the graph rises to the right.
If the leading coefficient (an) is negative, the graph falls to the right.
Determine the end behavior of the graph of each function below and make a rough sketch to illustrate the end behavior.
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Like quadratic functions, polynomial functions must have one y­intercept, but may have various numbers of x­intercepts, also called zeros of the function. If the function has an even degree, the graph does not have to have any x­
intercepts, but if it has an odd degree, the graph must have at least one x­
intercept.
Note that the maximum number of x­intercepts is the same as the degree of the polynomial, and the maximum number of turning points (local maximum or local minimum) is one less than the degree of the polynomial.
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To find the zeros of a polynomial function, set the function equal to 0, then factor and set each factor equal to 0. If a factor occurs twice (therefore giving a zero of multiplicity two), the graph will touch the x­axis at that point, but not cross it. This is precisely the behavior of the graph of a parabola which has its vertex on the x­axis.
Find the zeros of each polynomial function below and give the multiplicity for each zero. Tell whether the graph crosses the x­axis or is tangent to it at each zero. (Being tangent to it means touching it in one point, but not crossing it.)
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For each polynomial function, determine (a) the end behavior of the graph, (b) the x­intercepts and the behavior at each x­intercept, (c) the y­intercept, (d) whether the graph has y­axis symmetry, origin symmetry, or neither, and (e) the graph.
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The Intermediate Value Theorem tells us that if two points on a polynomial function lie on opposite sides of the x­axis, then the function must have a zero between them.
More formally stated:
Let f be a polynomial function with real coefficients. If f(a) and f(b) have opposite signs, then there is at least one value of c between a and b for which f(c)= 0. Equivalently, the equation f(x) = 0 has at least one real root between a and b.
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Given the graph of a polynomial function f(x), determine the zeros and their multiplicities, write an equation for the graph with a leading coefficient of 1 or ­1, and determine the y­intercept. Assume the scale on the x­axis is 1, but do not make this assumption for the y­axis.
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