Rational Functions and End Behavior
A function that can be expressed as the ratio of two polynomials.
What is a rational function?
A line x=a where the function approaches infinity or negative infinity as x approaches a.
What is a vertical asymptote?
A method used to divide polynomials similar to the long division of numbers.
What is polynomial long division?
Shifts the graph of the function horizontally and/or vertically without changing its shape.
What is a translation of a function?
Defined by different expressions for different intervals of the domain.
What is a piecewise function?
Determined by the degrees of the numerator and denominator polynomials.
What is the end behavior of a rational function?
Set the denominator equal to zero and solve for x to find these in a rational function.
What are vertical asymptotes?
Used to divide a polynomial by another polynomial of a lower or equal degree.
When is polynomial long division used?
Replace x with x−h for a right shift and x+h for a left shift to achieve this.
What is a horizontal translation?
Graph each expression on its corresponding interval and ensure the pieces fit together according to the function's definition.
How do you graph a piecewise function?
As x approaches infinity, the graph approaches this if the degrees of the numerator and denominator are the same or the denominator's degree is greater.
What is a horizontal asymptote?
Occurs where both the numerator and denominator are zero at the same x value, indicating a removable discontinuity.
What is a hole in a rational function?
Provides a formula for expanding the power of a binomial.
What is the Binomial Theorem?
Add k to the function for an upward shift and f(x)−k for a downward shift to achieve this.
What is a vertical translation?
A mathematical representation of a real-world situation using a function.
What is a function model?
To determine this, compare the degrees of the numerator and denominator polynomials and, if they are equal, take the ratio of their leading coefficients.
What is the horizontal asymptote of a rational function?
A single point where the function is undefined due to canceling factors, as opposed to where the function grows unbounded.
What is the difference between a hole and a vertical asymptote?
The coefficients in the expansion of (a+b)n.
What are binomial coefficients?
Stretches or compresses the graph of a function vertically or horizontally.
What is a dilation of a function?
Analyze the situation, determine the type of relationship, choose an appropriate function, and fit it to the data.
What is the process of selecting a function model?
If the numerator's degree is greater than the denominator's degree, the function's end behavior approaches this.
What is infinity or negative infinity, indicating no horizontal asymptote?
Factor the numerator and denominator and cancel common factors to identify these in the graph of a rational function.
What are holes in the graph of a rational function?
Divide the numerator by the denominator, and the quotient without the remainder gives this.
What is a slant asymptote?
Multiply the function by a constant a. If |a|>1, the graph stretches; if |a|<1, it compresses.
What is a vertical dilation?
Can model situations where a rule or relationship changes at different points, such as tax brackets or speed limits.
How can you use piecewise functions in modeling real-world situations?