Precalculus Chapter 2: Polynomial and Rational Functions - LIONS
A polynomial with degree "n".
All exponents are nonnegative (positive) integers
Domain of a Polynomial function
all real numbers
polynomial function
-A function of the form f(x)=anx^n+an-1x^n-1+...a1x+a0 -an, an-1, ..., a1, a0 are real numbers. -n is a nonnegative (positive) integer. -No x's under a radical -No x's on the bottom of a fraction
Rational Functions
-A function of the form r(x)=f(x)/g(x) -f and g are the polynomial functions -g is NOT the zero polynomial
Horizontal Asymptotes
-Describe end behavior -The line y=c is a horizontal asymptote -A graph may intersect its horizontal asymptote 3 cases... BOB0 BOT1 same y= a/b -If r(x) is proper, then there is a HA at y=0 -If f(x) and g(x) have equal degrees, then there is a HA at y=an/bn (ratio)
Graphing a polynomial function
-Graphed smooth and continuously -Domain is all real numbers -No breaks, holes, or gaps a) Find the x- and y-intercepts of the graph. b) Determine whether the graph crosses or touches the x-axis at each x-intercept. c) Determine the end behavior of the graph. d) Sketch the curve.
Multiplicity for Repeated Zeros
-If a zero is of even multiplicity, then the graph bounces off the x-axis at that point. Ex: f(x)=(x-3)^4 -If a zero is of odd multiplicity, then the graph crosses the x-axis at that point. Ex: f(x)=(x+1)^5
Odd degree w/ end behavior
-Left side points in the opposite direction as right side -Odd=opposite
Turning points
-Points where a graph changes from increasing to decreasing or vice versa. -A polynomial function of degree "n" has no more than n-1 turning points.
Finding restrictions of rational functions
-Restrictions only come from the DENOMINATOR -Factor denominator and solve for x
Finding the domain of a rational function
-The domain consists of all real numbers EXCEPT those for which the denominator equals 0.
Real zeros
-The number of real zeros is less than or equal to the degree. -The number of real zeros is less than or equal to n (exponent).
Oblique Asymptotes
-aka slant asymptotes -If r(x) is improper AND the degrees differ by 1, then there is an OA at y=mx+b.
Conjugate Pairs Theorem
-f(x) is a complex polynomial w/ real coefficients -If a+bi is a zero of f, then the complex conjugate a-bi is also a zero of f. Ex: i= square root -1
Finding the degree of a polynomial function
1. Combine like terms. ... 2. Drop all of the constants and coefficients. ... 3. Put the terms in decreasing order of their exponents. ... 4. Find the power of the largest term. ... 5. Identify this number as the degree of the polynomial. ... 6. Know that the degree of a constant is zero.
1 root
Crosses x-axis once
0 roots
Does not touch or cross x-axis
The Remainder Theorem
Given: f(x) / (x-k) Then: f(k)=r (the remainder) (k,r) is a point on the graph of f(x) (After using synthetic division, if their is a remainder then the factor is NOT a root)
Even Degree w/ end behavior
Left side points in the same direction as right side
If a factor has ODD MULTIPLICITY...
The graph CROSSES the x-axis
If a factor has EVEN MULTIPLICITY
The graph bounces off of the x-axis
End Behavior
The way a graph points to - and + infinity
Synthetic Division
Used to find the zeros of a polynomial function.
2 roots
crosses x-axis twice
A polynomial function degree determines...
its number of roots (all zeros)
Vertical Asymptotes
x=c "x" ALWAYS EQUALS A CONSTANT -For any rational, if a vertical asymptote exists values of "y" will either increase or decrease. 1. Write r(x) in simplest form. 2. Set the denominator equal to 0. 3. Solve.
