Glossary
Axis of Symmetry
A vertical line that passes through the vertex of a parabola, dividing it into two mirror-image halves.
Example:
If a parabola's vertex is at (3, 5), its axis of symmetry is the line x = 3.
Base (b) / Growth Factor
In an exponential function f(x) = a ⋅ bˣ, 'b' determines the rate of change. If b > 1, it's growth; if 0 < b < 1, it's decay.
Example:
In the function P(t) = 500(1.07)ᵗ, the growth factor of 1.07 indicates a 7% increase per time period.
Completing the Square
A method used to solve quadratic equations or to convert a quadratic function from standard form to vertex form by creating a perfect square trinomial.
Example:
Using completing the square can transform y = x² + 6x + 2 into y = (x+3)² - 7, revealing the vertex.
Complex Zeros
Zeros of a polynomial function that involve imaginary numbers (i = √-1), always occurring in conjugate pairs (a + bi and a - bi).
Example:
A quadratic equation with a negative discriminant will have two complex zeros.
Continuity (Polynomial)
A property of polynomial functions indicating that their graphs are smooth and unbroken, without any jumps, holes, or asymptotes.
Example:
Because polynomial functions exhibit continuity, you can draw their entire graph without lifting your pencil.
Continuously Compounded Interest
Interest that is calculated and added to the principal infinitely many times over a given period, modeled by the formula A(t) = P ⋅ eʳᵗ.
Example:
An investment account that offers continuously compounded interest will grow slightly faster than one compounded daily or monthly.
Degree (Polynomial)
The highest power of the variable in a polynomial function, which significantly influences its shape and end behavior.
Example:
The degree of the polynomial f(x) = 5x⁴ - 2x² + 7 is 4.
Discriminant (Δ)
The part of the quadratic formula under the square root, Δ = b² - 4ac, which determines the number and type of solutions a quadratic equation has.
Example:
If the discriminant is negative, there are no real solutions, meaning the parabola doesn't cross the x-axis.
Domain (Rational Function)
The set of all possible input values (x-values) for which a rational function is defined, excluding any values that make the denominator zero.
Example:
For f(x) = 1/(x-5), the domain includes all real numbers except x = 5.
Doubling Time/Half-Life
Doubling time is the period required for a quantity to double in size, while half-life is the time required for a quantity to reduce to half its initial value, both common in exponential models.
Example:
The half-life of Carbon-14 is about 5,730 years, meaning that much time passes for half of a sample to decay.
End Behavior (Polynomial)
Describes how the graph of a polynomial function behaves as x approaches positive infinity (x → ∞) or negative infinity (x → -∞), determined by its degree and leading coefficient.
Example:
For an odd-degree polynomial with a positive leading coefficient, the end behavior is that the graph falls to the left and rises to the right.
Exponential Decay
Occurs in an exponential function when the base (b) is between 0 and 1, causing the function's value to decrease and approach zero asymptotically.
Example:
The amount of a radioactive substance decreasing over time is an example of exponential decay.
Exponential Function
A function of the form f(x) = a ⋅ bˣ, where 'a' is the initial value and 'b' is the base or growth factor. Its graph never crosses the x-axis.
Example:
The population of a town growing by 5% each year can be modeled by an exponential function.
Exponential Growth
Occurs in an exponential function when the base (b) is greater than 1, causing the function's value to increase rapidly over time.
Example:
A rapidly spreading virus might show exponential growth in the number of infected individuals.
Extraneous Roots/Solutions
Solutions obtained during the algebraic process of solving an equation (especially rational or radical equations) that do not satisfy the original equation when substituted back in, often because they lead to division by zero or undefined terms.
Example:
When solving a rational equation, you must always check for extraneous roots that might make the original denominator zero.
Factoring (Quadratic)
A method of solving quadratic equations by expressing the quadratic expression as a product of two linear factors.
Example:
To solve x² - 5x + 6 = 0, you can use factoring to get (x-2)(x-3) = 0, so x=2 or x=3.
Growth/Decay Rate (r)
The percentage by which a quantity increases or decreases per unit of time, often expressed as a decimal in formulas like A(t) = A₀(1 + r)ᵗ.
Example:
If a car depreciates by 15% annually, its decay rate is 0.15.
Holes (Rational Function)
A point discontinuity in the graph of a rational function that occurs when a common factor in the numerator and denominator cancels out, making the function undefined at that specific point.
Example:
The function f(x) = (x²-4)/(x-2) has a hole at x = 2, because (x-2) is a common factor that cancels out.
Horizontal Asymptote
A horizontal line (y = c) that the graph of a rational function approaches as x tends towards positive or negative infinity, determined by comparing the degrees of the numerator and denominator.
Example:
The function f(x) = (2x+1)/(x-3) has a horizontal asymptote at y = 2.
Initial Value (a)
In an exponential function f(x) = a ⋅ bˣ, 'a' represents the starting amount or the y-intercept of the graph when x = 0.
Example:
If you invest 1000 is the initial value of your investment.
Leading Coefficient (Polynomial)
The coefficient of the term with the highest degree in a polynomial function, which helps determine the end behavior of the graph.
Example:
In the polynomial f(x) = -3x⁵ + 2x² - 1, the leading coefficient is -3.
Multiplicity (of Zeros)
The number of times a particular zero appears as a root of a polynomial equation, which affects how the graph behaves at the x-axis (crossing for odd multiplicity, touching for even).
Example:
If a zero has an even multiplicity, like (x-2)², the graph will touch the x-axis at x=2 and turn around, rather than crossing it.
Oblique (Slant) Asymptote
A diagonal line that the graph of a rational function approaches when the degree of the numerator is exactly one greater than the degree of the denominator.
Example:
The function f(x) = (x² + 1)/x has an oblique asymptote because the numerator's degree is one higher than the denominator's.
Parabola
The U-shaped graphical representation of a quadratic function, which opens either upward (if a > 0) or downward (if a < 0).
Example:
The iconic Gateway Arch in St. Louis is shaped like an inverted parabola.
Polynomial Function
A function consisting of sums of terms, each of which is a constant multiplied by a variable raised to a non-negative integer power, like f(x) = a₀ + a₁x + ... + aₙxⁿ.
Example:
A cubic function like f(x) = x³ - 2x + 1 is a type of polynomial function.
Principal Amount (P)
In financial applications, 'P' refers to the initial amount of money invested or borrowed, before any interest is added.
Example:
If you take out a 20,000.
Quadratic Formula
A formula used to find the solutions (roots) of any quadratic equation in the form ax² + bx + c = 0: x = (-b ± √(b² - 4ac)) / 2a.
Example:
When factoring doesn't work easily, the quadratic formula is a reliable way to solve for x.
Quadratic Function
A polynomial function of degree 2, typically written in the general form f(x) = ax² + bx + c, where a ≠ 0.
Example:
The path of a thrown ball can often be modeled by a quadratic function.
Rational Function
A function that can be expressed as the ratio of two polynomial functions, P(x)/Q(x), where Q(x) is not equal to zero.
Example:
The function f(x) = (x+1)/(x-3) is a rational function.
Turning Points
Points on the graph of a polynomial function where the graph changes from increasing to decreasing or vice versa, corresponding to local maxima or minima.
Example:
A cubic polynomial can have up to two turning points, creating a 'hill' and a 'valley' in its graph.
Vertex
The highest or lowest point on a parabola, representing the maximum or minimum value of the quadratic function.
Example:
For a parabola opening upwards, the vertex is the lowest point, indicating the minimum value of the function.
Vertical Asymptote
A vertical line (x = c) that the graph of a rational function approaches but never touches, occurring at x-values where the denominator is zero and the numerator is not.
Example:
The function f(x) = 1/x has a vertical asymptote at x = 0.
X-intercepts (Roots/Zeros - Quadratic)
The points where the parabola crosses the x-axis, also known as the roots or zeros of the quadratic equation, where f(x) = 0.
Example:
Finding the x-intercepts of a quadratic equation helps determine when a projectile hits the ground.
Y-intercept (Quadratic)
The point where the parabola crosses the y-axis, which for f(x) = ax² + bx + c is always (0, c).
Example:
In the equation y = x² + 2x + 5, the y-intercept is (0, 5).
Zeros (Roots - Polynomial)
The x-values for which a polynomial function equals zero, corresponding to the x-intercepts of its graph.
Example:
If a polynomial has zeros at x = -1, x = 2, and x = 4, its graph crosses the x-axis at these points.