Glossary
Arithmetic Sequence
A sequence of numbers where the difference between consecutive terms is constant.
Example:
The sequence 2, 5, 8, 11, ... is an arithmetic sequence because each term increases by 3.
Base (of an exponential function)
The constant factor by which the output of an exponential function is repeatedly multiplied for each unit increase in the input.
Example:
In the function , the base is 1.05, indicating a 5% growth per unit.
Common Difference
The constant value added to each term of an arithmetic sequence to get the next term.
Example:
In the sequence 10, 8, 6, 4, the common difference is -2.
Common Ratio
The constant factor by which each term of a geometric sequence is multiplied to get the next term.
Example:
In the sequence 100, 50, 25, 12.5, the common ratio is 0.5.
Constant Proportion
A consistent multiplicative factor by which output values change for each unit increase in the input, characteristic of exponential relationships.
Example:
If an investment grows by 7% each year, it has a constant proportion of 1.07.
Constant Rate of Change
A consistent increase or decrease in the output value for each unit increase in the input value, characteristic of linear relationships.
Example:
If a car travels at a steady 60 miles per hour, its distance covered has a constant rate of change of 60 mph.
Continuous Function
A function whose graph can be drawn without lifting the pen, meaning there are no breaks, jumps, or holes.
Example:
The temperature outside over a day is a continuous function of time.
Discrete Sequence
A sequence whose terms are distinct and separate, typically defined only for integer inputs.
Example:
The number of students in a classroom is a discrete sequence of values, as you can't have half a student.
Domain
The set of all possible input values (x-values) for which a function or sequence is defined.
Example:
For the function , the domain is all non-negative real numbers, .
Exponential Change (Multiplication)
The characteristic of exponential functions and geometric sequences where output values change by multiplying by a constant factor.
Example:
If a population doubles every year, that's an exponential change (multiplication).
Exponential Function
A function where the independent variable appears as an exponent, characterized by a constant multiplicative rate of change.
Example:
The growth of a bacterial colony that doubles every hour can be modeled by an exponential function.
Geometric Sequence
A sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
Example:
The sequence 3, 6, 12, 24, ... is a geometric sequence because each term is multiplied by 2.
Initial Value
The starting quantity or output value of a function or sequence when the input is zero or the first term.
Example:
If a plant is 5 cm tall when you start measuring, its initial value is 5 cm.
Linear Change (Addition)
The characteristic of linear functions and arithmetic sequences where output values change by adding a constant amount.
Example:
If your savings increase by $50 each month, that's a linear change (addition).
Linear Function
A function whose graph is a straight line, characterized by a constant rate of change.
Example:
The cost of a taxi ride might be modeled by a linear function where there's a flat fee plus a constant charge per mile.
Point-Slope Form
A linear equation form, $y - y_1 = m(x - x_1)$, used to express the equation of a line given a point $(x_1, y_1)$ and the slope $m$.
Example:
A line passing through (2, 3) with a slope of 4 can be written in point-slope form as .
Slope
The measure of the steepness and direction of a line, calculated as the ratio of the vertical change to the horizontal change.
Example:
A road with a slope of 0.05 rises 5 feet for every 100 feet of horizontal distance.
y-intercept
The point where a graph crosses the y-axis, representing the output value when the input is zero.
Example:
In the equation , the y-intercept is 5, meaning the line crosses the y-axis at (0, 5).