Essential Concepts

• Linear growth involves adding a constant amount to the previous value in a sequence.
• In linear growth, the next value in a sequence [latex](P_n)[/latex] is found by adding a constant difference to the previous value [latex](P_{n-1})[/latex]. This is known as the recursive formula for linear growth: [latex]P_n =P_{n−1}+d[/latex]
• The explicit formula for linear growth is [latex]P_n=P_0+dn[/latex], where [latex]P_0[/latex] is the initial value, [latex]d[/latex] is the constant difference, and [latex]n[/latex] is the number of time periods.
• Use the explicit formula to directly calculate future values in a linear growth scenario.
• Exponential growth occurs when a quantity increases by a fixed percentage of the whole in each time period.
• The recursive formula for exponential growth is [latex]P_n = (1+r)P_{n-1}[/latex], and the explicit formula is [latex]P_n = (1+r)^{n}P_0[/latex] or equivalently, [latex]P_n= P_0(1+r)^{n}[/latex], where [latex]P_0[/latex] is the initial amount, [latex]r[/latex] is the growth rate, and [latex]n[/latex] is the number of time periods.
• Using the explicit formula to quickly calculate future values in an exponential growth scenario.
• The term [latex](1+r)[/latex] is the growth multiplier, where [latex]r[/latex] is the growth rate.
• Mathematical models are tools that represent real-world situations using mathematical language and symbols to simplify, explain, and predict phenomena. These models can be physical, conceptual, or mathematical, each serving a unique purpose in interpreting and understanding real-world scenarios.
• The process of modeling involves identifying variables, collecting data, and formulating a model that captures the relationship between variables.
• Models are not perfect representations and should be used with an understanding of their limitations and assumptions.
• The independent variable, representing the input quantity is graphed on the horizontal axis.
• The dependent variable, representing the output quantity is graphed on the vertical axis.
• Scatterplots are graphical representations of data points in two dimensions, typically used to observe relationships between variables.
• Creating a scatterplot involves plotting data points on a graph, where each point represents a pair of values from the dataset.
• Spreadsheet software like Microsoft Excel can be used to create scatterplots and analyze data efficiently.
• Scatterplots help in visually identifying trends, patterns, or correlations in the data, which are crucial for developing mathematical models.
• Trendlines in scatterplots represent the best fit for the given data, often used to identify the underlying trend in the dataset.
• Adjusting data inputs, like changing years to the number of years since a starting point, can significantly affect the model’s accuracy and relevance.
• Linear growth occurs when the output quantity changes by a constant amount per unit of input, resulting in a straight-line pattern in a graph.
• A linear model can be created to describe real-world situations of linear growth or decay, using the slope-intercept form of a linear equation, [latex]y=mx+b[/latex]. 
• The slope ([latex]m[/latex]) in the linear equation represents the rate of change, and the y-intercept ([latex]b[/latex]) represents the starting value.
• Spreadsheet software can be used to build models for making predictions in scenarios of linear growth, enhancing efficiency and accuracy.
• Linear regression involves finding the best-fitting line for a set of data, often using technology to ensure accuracy.
• The process of least squares regression is used to determine this line by minimizing the distances between the data points and the line itself.
• The correlation coefficient, denoted as [latex]r[/latex], is a numerical measure between [latex]-1[/latex] and [latex]1[/latex] that indicates the strength and direction of the relationship between variables. A positive correlation coefficient, [latex]r \gt 0[/latex], would indicate a positive slope while a negative correlation coefficient [latex]r \lt 0[/latex], would indicate a negative slope. 
• The coefficient of determination, known as [latex]R^2[/latex], quantifies how well the regression line represents the data.
• [latex]R^2[/latex] is equivalent to the square of the correlation coefficient [latex]r[/latex] and will always be a positive number between [latex]0\%[/latex] and [latex]100\%[/latex].
• [latex]R^2[/latex] should be interpreted and written as a percentage.
• In scatterplots, a strong linear relationship is shown by data points lying close to a line, while a weaker relationship is indicated by more spread out data points.
• Extrapolating involves making predictions outside the known data set, a practice that should be approached with caution due to the potential for unreliable predictions.
• Interpolating refers to making predictions within the known data, offering a safer and more reliable approach to modeling and forecasting.

Glossary

coefficient of determination, [latex]R^2[/latex] or [latex]r^2[/latex]

a measure of the proportion of the variation of a response variable in linearly related bivariate data that can be explained by its relationship with the explanatory variable that ranges from [latex]0[/latex] and [latex]1[/latex]

common difference

the amount that the population changes each time [latex]n[/latex] increases by [latex]1[/latex]

correlation coefficient, [latex]r[/latex]

a value between [latex]-1[/latex] and [latex]1[/latex] that is returned by the least squares method and measures the correlation between the input and output variables of a model

linear regression

the process of finding the equation of the line that best “fits” the data

recursive relationship

a formula that relates the next value in a sequence to the previous values

Key Equations

explicit form of exponential growth

[latex]P_n= P_0(1+r)^{n}[/latex]

explicit form of linear growth

[latex]P_n = P_0 + d n[/latex]

recursive form of exponential growth

[latex]P_n = (1+r)P_{n-1}[/latex]

recursive form of linear growth

[latex]P_n = P_{n-1} + d[/latex]