Glossary
Complex zeros
Roots of a polynomial function that are complex numbers (involving the imaginary unit 'i'), which do not appear as x-intercepts on the real number line.
Example:
The polynomial has complex zeros at and , meaning its graph never crosses the x-axis.
Degree
The highest power of the variable in a polynomial function, which significantly influences its end behavior and number of possible roots.
Example:
In the polynomial , the degree is 5.
Domain
The set of all possible input values (x-values) for which a function is defined.
Example:
For the function representing the number of tickets sold for a concert, the domain would be non-negative integers, as you can't sell negative or fractional tickets.
End Behavior
Describes how the graph of a function behaves as the input (x) approaches positive infinity or negative infinity.
Example:
The end behavior of a polynomial like is that both ends of the graph point downwards.
Factor Theorem
A theorem stating that a polynomial $P(x)$ has a factor $(x-c)$ if and only if $P(c) = 0$ (i.e., 'c' is a zero of the polynomial).
Example:
According to the Factor Theorem, if for a polynomial , then must be a factor of .
Functions
A relation where each input (from the domain) has exactly one output (in the range). They are fundamental tools for modeling real-world scenarios.
Example:
The cost of a pizza is a function of the number of toppings, where each number of toppings corresponds to a single price.
Graphical analysis
The study of functions by examining their visual representation on a coordinate plane, revealing properties like intercepts, asymptotes, and end behavior.
Example:
By performing graphical analysis of a stock's price over time, an investor can identify trends and potential future movements.
Holes
A point of discontinuity in a rational function's graph where a factor in the numerator and denominator cancels out, resulting in a single missing point rather than an asymptote.
Example:
The function has a hole at , where the graph is defined everywhere else but has a single missing point.
Horizontal asymptotes
Horizontal lines that a function's graph approaches as the input (x) approaches positive or negative infinity, indicating the function's end behavior.
Example:
The function has a horizontal asymptote at , showing where the graph levels off at its far ends.
Inverses
A function that 'undoes' the action of another function; if f(a)=b, then the inverse function, f⁻¹(b)=a.
Example:
If a function converts Celsius to Fahrenheit, its inverse would convert Fahrenheit back to Celsius.
Leading coefficient
The coefficient of the term with the highest degree in a polynomial, which, along with the degree, determines the function's end behavior.
Example:
In , the leading coefficient is -3, indicating the graph will fall on both ends.
Linear Functions
Functions whose graphs are straight lines, characterized by a constant rate of change (slope).
Example:
The cost of a taxi ride, which includes a flat fee plus a per-mile charge, can be modeled by a linear function.
Long division
An algebraic method used to divide polynomials, often employed to find factors or simplify rational expressions.
Example:
To find the other factors of a polynomial after identifying one root, you can use long division to divide the polynomial by the known factor.
Modeling
The process of using mathematical functions to represent and analyze real-world phenomena or relationships.
Example:
We can use a quadratic function to model the trajectory of a basketball shot, predicting its height over time.
Numerical methods
Techniques that use numerical approximations to solve mathematical problems, often involving tables of values or iterative calculations.
Example:
Using a table of values to estimate the root of an equation is an example of a numerical method.
Polynomial function
A function consisting of a sum of terms, each of which is a constant multiplied by a variable raised to a non-negative integer power.
Example:
The function is a polynomial function.
Power functions
Functions of the form $f(x) = ax^n$, where 'a' is a real number and 'n' is a positive integer, forming the basic building blocks of polynomials.
Example:
The function is a power function.
Power rule
A fundamental concept in calculus for finding the derivative of power functions, indicating how the rate of change behaves for terms like $x^n$.
Example:
Using the power rule, we know that the rate of change of involves .
Quadratic Functions
Polynomial functions of degree 2, whose graphs are parabolas and whose rates of change are not constant but vary linearly.
Example:
The path of a projectile, like a thrown ball, often follows the shape of a quadratic function.
Range
The set of all possible output values (y-values) that a function can produce.
Example:
If a function models the height of a ball thrown into the air, its range would be all possible heights the ball reaches, from zero to its maximum height.
Rates of Change
A measure of how much a function's output changes in response to a change in its input, often calculated as the slope between two points.
Example:
The average speed of a car is a rate of change, calculated as the change in distance over the change in time.
Rational function
A function that can be expressed as the ratio of two polynomial functions, $f(x) = p(x)/q(x)$, where $q(x)$ is not the zero polynomial.
Example:
The function is a rational function.
Transformations
Changes applied to a function's graph, such as shifts (translations), stretches, compressions, or reflections, altering its position or shape.
Example:
Applying a vertical shift and a horizontal stretch are common transformations used to fit a basic sine wave to real-world data.
Vertical asymptotes
Vertical lines that a function's graph approaches but never touches, occurring at x-values where the denominator of a rational function is zero and the numerator is non-zero.
Example:
The function has a vertical asymptote at , meaning the graph shoots up or down infinitely as it gets closer to this line.