Descartes Rule of Signs
What is the definition of Descartes Rule of Signs?
- max # of positive roots are the # of sign changes in f(x)
- max # of negative roots are the # of sign changes in f(-x)
How do you find vertical asymptotes algebraically?
non-canceling factors in denominator
What is another word for hole?
What are two different methods you could use to factor an equation?
Boxes, X-Games, Mental Math, Bottoms Up
If asked to find a horizontal asymptote, what part of the table/graph should you focus on?
The end behavior.
Possibilities of negative/positive roots decrease by what number?
2
How many different situations are there for determining a horizontal asymptote?
3
How do you determine the range?
All real values except for the y-value of the hole and the horizontal asymptote.
What is the parent function for a rational?
1/x
What is the shorthand notation for "from the left" and "from the right"?
A superscript of - or +.
How do you determine the number of zero roots? Give an example.
Take out the GCF. For example, if the GCF is x2 then the number of zero roots is two.
If the numerators highest degree is x2 and the denominators highest degree is x what can you say about the asymptotes of the graph?
There is no horizontal asymptote, but there is a slant asymptote.
How do you determine the y-intercept?
Yes
True or False: Approaching a point discontinuity from the left and from the right, you will approach the same number.
True
How do you determine the number of possible imaginary roots?
Look at the highest degree. Imaginary roots cannot exceed the highest degree, and must come in pairs.
What asymptotes affect the domain, and which affect the range?
Domain: vertical
Range: horizontal
What is the first step to determining any characteristic of a rational function?
Completely factor.
Suppose a polynomial function f(x) has roots x=4 with multiplicity 3 and x=4-i, what type of function is f(x)?
Quintic
Look for anywhere where y is undefined. Then from the left and from the right you would have to be approaching either positive or negative infinity.
List out all positive, negative, zero, and imaginary root possibilities for the following equation.
Positive:
Negative:
Zero:
Imaginary:
A vertical asymptote is considered what type of discontinuity?
Non-removable infinite discontinuity.
What is the process you take when determining a slant asymptote of a rational function with a divisor of 2x+8?
Take out a GCF of 2, then -4 would be the number on the outside of the synthetic division. Use the leading coefficients of the dividend. Your answer will need to be divided by the GCF of 2 that we took out in the beginning.
What is the definition of the Fundamental Theorem of Algebra?
A polynomial function whose degree is 'n' will have a total of 'n' roots.
How do you determine a point discontinuity from the table?
Looking at the table, identify and spots where y is undefined. From the left and from the right you must approach the same number, regardless of what that number may be.