Glossary
Bottom Heavy (HA Rule)
A rule for rational functions where the degree of the numerator is less than the degree of the denominator, resulting in a horizontal asymptote at y = 0.
Example:
For f(x) = (x + 1) / (x^2 + 5), the denominator's degree is higher, making it bottom heavy, so the horizontal asymptote is y = 0.
Equal Degree (HA Rule)
A rule for rational functions where the degree of the numerator equals the degree of the denominator, resulting in a horizontal asymptote at the ratio of the leading coefficients.
Example:
Given f(x) = (7x^2 - 3x) / (2x^2 + 9), the degrees are equal, so the equal degree rule tells us the horizontal asymptote is y = 7/2.
Exponential Functions
Functions where the variable appears as an exponent, characterized by extremely rapid growth or decay.
Example:
The spread of a virus or the growth of an investment with compound interest can often be modeled by exponential functions due to their fast growth.
Growth Rates of Functions
A hierarchy that describes how quickly different types of functions increase or decrease as their input approaches infinity.
Example:
Understanding the growth rates of functions helps predict that an exponential model will eventually surpass any polynomial model in magnitude.
Horizontal Asymptote (HA)
A horizontal line (y = c) that a function's graph approaches as x tends towards positive or negative infinity, indicating the function's end behavior.
Example:
If a function models the concentration of a drug in the bloodstream over time, a horizontal asymptote might represent the maximum concentration the drug can reach.
Limit
The value that a function's y-value approaches as its x-value approaches a certain input, or as x approaches infinity or negative infinity.
Example:
As x gets infinitely large, the function f(x) = 1/x approaches a limit of 0.
Limits at Infinity
The behavior of a function's y-value as the x-value approaches positive or negative infinity, describing the function's end behavior.
Example:
To understand the long-term behavior of a population model, we might evaluate its limits at infinity to see if it stabilizes.
Oscillating Functions
Functions that repeatedly vary between two or more values without approaching a single limit as the input approaches infinity.
Example:
The function f(x) = cos(x) is an oscillating function because its values continuously cycle between -1 and 1, never settling on a single limit as x approaches infinity.
Rational Functions
Functions expressed as a ratio of two polynomials, typically in the form p(x)/q(x).
Example:
The function f(x) = (3x^2 + 5) / (x^2 - 4) is a rational function often analyzed for its horizontal asymptotes.
Squeeze Theorem
A theorem used to find the limit of a function by comparing it to two other functions whose limits are known and equal.
Example:
To prove that lim (x->ā) sin(x)/x = 0, you can use the Squeeze Theorem by bounding sin(x)/x between -1/x and 1/x.
Top Heavy (HA Rule)
A rule for rational functions where the degree of the numerator is greater than the degree of the denominator, indicating no horizontal asymptote.
Example:
The function f(x) = (x^3 - 2x) / (x^2 + 1) is top heavy, meaning its graph will not approach a specific y-value as x goes to infinity.
Vertical Asymptotes
Vertical lines that a function's graph approaches as the x-value approaches a certain constant, typically where the denominator of a rational function is zero and the numerator is non-zero.
Example:
The function f(x) = 1/(x-5) has a vertical asymptote at x = 5, as the function's value approaches infinity or negative infinity near this point.