Glossary
Constant Difference
A characteristic of linear growth observed in tables, where consecutive y-values change by the same fixed amount when x-values change by a constant amount.
Example:
If a table shows temperatures increasing by 2 degrees every hour, there's a constant difference of 2.
Constant Percent Rate of Change
The consistent percentage by which the dependent variable changes for every unit increase in the independent variable in an exponential relationship.
Example:
An investment earning 5% interest annually has a constant percent rate of change of 5%.
Constant Rate of Change
The consistent amount by which the dependent variable changes for every unit increase in the independent variable in a linear relationship.
Example:
A car traveling at a steady 60 mph has a constant rate of change in distance over time.
Constant Ratio
A characteristic of exponential growth observed in tables, where the ratio between consecutive y-values is the same when x-values change by a constant amount.
Example:
If a bacterial population doubles every hour, the constant ratio between consecutive hourly counts is 2.
Curved Line
The graphical representation of an exponential function, showing a rapid increase or decrease rather than a steady, straight path.
Example:
The graph of compound interest over many years will form a curved line that gets steeper over time.
Exponential Growth
A type of growth where a quantity increases or decreases by a constant percentage over equal intervals of time or input, leading to rapid change.
Example:
The spread of a viral video on social media often demonstrates exponential growth.
Growth Factor
In an exponential function $y = ab^x$, 'b' is the growth factor, calculated as (1 + r), where 'r' is the percent rate of change as a decimal.
Example:
If a population grows by 10% each year, the growth factor is 1.10.
Horizontal Asymptote
A horizontal line that an exponential graph approaches but never actually touches as the independent variable (x) gets very large or very small.
Example:
In a model of drug concentration in the bloodstream, the horizontal asymptote might represent the minimum effective dose that the concentration never drops below.
Linear Growth
A type of growth where a quantity increases or decreases by the same fixed amount over equal intervals of time or input.
Example:
If you earn an extra $50 every week, your savings account balance shows linear growth.
Percent Rate of Change
A general term referring to how one quantity changes in relation to another, specifically a constant percentage in exponential functions.
Example:
The annual inflation rate is an example of a percent rate of change.
Rate of Change
A general term referring to how one quantity changes in relation to another, specifically a constant numerical amount in linear functions.
Example:
The rate of change of water in a leaky bucket might be 0.5 liters per hour.
Slope
In a linear function, the slope represents the constant rate of change, indicating the steepness and direction of the line.
Example:
A positive slope on a graph showing sales over time means the sales are consistently increasing.
Slope-Intercept Form
The equation $y = mx + b$, which is used to represent linear functions, where 'm' is the slope and 'b' is the y-intercept.
Example:
To model the cost of a taxi ride with a 2 per mile, you'd use the slope-intercept form: y = 2x + 3.
Straight Line
The graphical representation of a linear function, indicating a consistent rate of change.
Example:
Plotting the distance traveled by a car at a steady speed will always result in a straight line on a graph.
Y-Intercept
The point where a graph crosses the y-axis, representing the initial value or starting amount when the independent variable (x) is zero.
Example:
In a savings account model, the y-intercept would be the initial deposit you made.