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Math 150 Lecture Notes
Polynomial Functions
A polynomial function of degree n is a function of the form P(x) = anxn^ + an-1 xn-1^ + · · · + a 1 x + a 0 Where n is a nonnegative integer and an ≠ 0.
The numbers a 0 , a 1 , a 2 , ⋅ ⋅ ⋅ , an are called the coefficients of the polynomial.
The number a 0 is the constant coefficient or constant term.
The number an, the coefficient of the highest power, is the leading coefficient, and the term anxn is the leading term.
Graphs of polynomial functions are smooth curves with no breaks or corners.
The end behavior of a polynomial is a description of what happens as x becomes large in the positive or negative direction.
Notation: x → ∞ means “x becomes large in the positive direction” x → - ∞ means “x becomes large in the negative direction”
For any polynomial, the end behavior is determined by the term that contains the highest power of x.
If P is a polynomial function, then c is called a zero of P if P(c) = 0. In other words, the zeros of P are the solutions or roots of the polynomial equation P(x) = 0.
If P is a polynomial and c is a real number, then the following are equivalent.
- c is a zero of P.
- x = c is a solution of the equation P(x) = 0.
- x – c is a factor of P(x).
- x = c is an x-intercept of the graph of P.
Intermediate Value Theorem for Polynomials If P is a polynomial function and P(a) and P(b) have opposite signs, then there exists at least one value c between a and b for which P(c) = 0.
Graphing Polynomials Functions
- Find all the real zeros or x-intercepts. Break it down into linear factors by factoring methods and/or quadratic formula.
- Plot the x-intercepts and determine the shape near the intercepts according to the multiplicity of the factor.
- Determine the end behavior of the polynomial.
- Make a table of values including test points to determine where the graph is above or below the x-axis and y-intercept.
- Plot the test points and y-intercept and sketch a smooth curve passing through the points and having the required end behavior.
If P(x) = anxn^ + an-1 xn-1^ + · · · + a 1 x + a 0 is a polynomials of degree n, then the graph of P has at most n – 1 local extrema.
Example 1: Sketch the graph of the function by transforming the graph of the parent function. Indicate all x- and y-intercepts on the graph. f(x) = -½ (x – 2)^5 + 16
