Glossary
Absolute Value Function
A function that includes the absolute value of an expression, resulting in a V-shaped graph.
Example:
The graph of f(x) = |x - 4| is an absolute value function that forms a 'V' shape with its vertex at (4,0).
Base (of an Exponential Function)
The constant 'b' in an exponential function f(x) = bˣ, which determines whether the function represents growth or decay.
Example:
In the function f(x) = 0.5ˣ, the base is 0.5, indicating exponential decay.
Degree of a Polynomial
The highest power of the variable in a polynomial function.
Example:
In the polynomial 4x⁵ + 2x² - 1, the degree of a polynomial is 5, indicating its highest exponent.
End Behavior
Describes the direction of the graph of a function as the input variable (x) approaches positive or negative infinity.
Example:
A quadratic function with a positive leading coefficient will have end behavior where both ends of the graph go up, like a parabola opening upwards.
Exponential Decay
A pattern of decrease in which the rate of decrease is proportional to the current quantity, occurring when the base 'b' in bˣ is between 0 and 1.
Example:
The reduction of a radioactive substance's mass over time is a process of exponential decay, halving at regular intervals.
Exponential Function
A function where the variable appears in the exponent, typically in the form f(x) = bˣ, where b is a positive constant not equal to 1.
Example:
The growth of a bacterial population often follows an exponential function, showing rapid increase over time.
Exponential Growth
A pattern of increase in which the rate of growth itself is proportional to the current quantity, occurring when the base 'b' in bˣ is greater than 1.
Example:
Compound interest on an investment is a classic example of exponential growth, where your money grows faster over time.
Factoring
The process of breaking down a polynomial into a product of simpler expressions (factors) to find its zeros or simplify it.
Example:
To find the roots of x² - 5x + 6 = 0, you can use factoring to rewrite it as (x-2)(x-3)=0, revealing the roots are 2 and 3.
Fundamental Theorem of Algebra
States that a polynomial of degree 'n' has exactly 'n' complex roots, counting multiplicity.
Example:
According to the Fundamental Theorem of Algebra, a polynomial like x³ - 8 will have exactly three complex roots, even if some are imaginary.
Graphical Analysis and Interpretation
The process of examining a function's graph to understand its behavior, key features, and implications in a given context.
Example:
Through graphical analysis and interpretation, you can quickly identify a function's zeros, turning points, and end behavior without needing to solve complex equations.
Horizontal Asymptote
A horizontal line that the graph of a function approaches but never touches as the input variable approaches positive or negative infinity.
Example:
The graph of an exponential function like y = 2ˣ has a horizontal asymptote at y=0, meaning it gets infinitely close to the x-axis but never reaches it.
Inverse of Exponential Functions
Logarithmic functions are the inverse of exponential functions, meaning they 'undo' each other; their graphs are reflections across the line y=x.
Example:
Since 2³ = 8, the inverse of exponential functions tells us that log₂(8) = 3.
Leading Coefficient
The coefficient of the term with the highest degree in a polynomial function.
Example:
For the polynomial -2x⁴ + 7x³ - x + 10, the leading coefficient is -2, which dictates the graph's end behavior.
Logarithmic Function
The inverse of an exponential function, expressed as f(x) = log_b(x), which answers the question 'to what power must 'b' be raised to get 'x'?'
Example:
The Richter scale, which measures earthquake intensity, uses a logarithmic function to represent large ranges of energy on a more manageable scale.
Maxima/Minima
The highest (maximum) or lowest (minimum) points on a function's graph, either locally within an interval or globally for the entire function.
Example:
In an optimization problem, finding the maxima/minima of a function can help determine the greatest possible area or the least possible cost.
Multiplicity of Zeros
The number of times a particular root appears as a solution to a polynomial equation, affecting how the graph interacts with the x-axis.
Example:
If a polynomial has a zero at x=3 with an even multiplicity of zeros, the graph will touch the x-axis at 3 but not cross it, bouncing off.
Polynomial Function
A function defined by a sum of terms, each consisting of a constant multiplied by a non-negative integer power of the variable, typically in the form f(x) = a₀ + a₁x + ... + aₙxⁿ.
Example:
The function f(x) = 5x³ - 2x + 7 is a polynomial function of degree 3.
Properties of Logarithms
Rules that govern how logarithms can be manipulated, including the Product Rule, Quotient Rule, and Power Rule.
Example:
Using the properties of logarithms, you can simplify log(2x) to log(2) + log(x), which is often useful in solving equations.
Rational Function
A function that can be expressed as the ratio of two polynomials, where the denominator is not zero.
Example:
The function f(x) = (x+1)/(x-3) is a rational function and will have a vertical asymptote where the denominator is zero.
Rational Root Theorem
A theorem that helps identify all possible rational roots of a polynomial equation with integer coefficients.
Example:
When trying to solve a complex polynomial, the Rational Root Theorem can narrow down the potential integer or fractional solutions to test.
Square Root Function
A function that includes the square root of a variable or expression, typically having a restricted domain.
Example:
The domain of the square root function f(x) = √(x+5) is x ≥ -5, as you cannot take the square root of a negative number.
Synthetic Division
A shorthand method for dividing a polynomial by a linear factor (x - k), often used to find roots or factor polynomials.
Example:
Using synthetic division with a known root can quickly reduce a cubic polynomial to a quadratic, making it easier to find the remaining roots.
Transformations (of Functions)
Changes applied to a function's graph, such as shifts (translations), reflections, stretches, or compressions.
Example:
Understanding transformations allows you to quickly sketch the graph of f(x) = (x-2)² + 3 by shifting the basic parabola y=x² two units right and three units up.
Turning Points
Points on a function's graph where it changes from increasing to decreasing (local maximum) or from decreasing to increasing (local minimum).
Example:
A cubic function can have up to two turning points, creating a 'hill' and a 'valley' on its graph.
Vertical Asymptote
A vertical line that the graph of a function approaches but never touches as the input variable approaches a certain value.
Example:
The graph of a logarithmic function like y = log(x) has a vertical asymptote at x=0, meaning it gets infinitely close to the y-axis but never crosses it.
Zeros (Roots)
The input values (x-values) for which a function's output (y-value) is zero; these are the x-intercepts of the graph.
Example:
Finding the zeros of a profit function helps a business determine its break-even points.