- Create a linear model that describes a real-world situation
- Use linear regression to analyze a data set and find the best-fit line
- Calculate and use the coefficient of determination to determine how well a linear model fits the data
Modeling Linear Growth
Situations involving growth (or decay) in which the output quantity changes by a constant amount per unit of input are said to exhibit linear growth. That is, from one input to the next, there is a common difference between the output values.
For example, let’s say you inherit a collection of [latex]47[/latex] silver dollars minted between 1878 and 1935. After doing some research, you learn that they are worth between $14 and $30 each. Rather than just cashing them in, you decide to grow the collection by purchasing an additional silver dollar each month. You can write a mathematical model to quickly predict how many silver dollars you’ll have in the collection at any time in the future assuming you increase your collection by [latex]12[/latex] dollars per year. We can write a mathematical model to describe this situation from the information given about the starting amount and the constant quantity of change.
Let the input variable (the independent variable) [latex]x[/latex] represent the number of years you’ve owned the collection of silver dollars. The output variable (the dependent variable) [latex]y[/latex] will represent the total number of dollars in the collection in any year [latex]x[/latex].
| The number of dollars in the collection | is obtained from | the starting number, 47 | together with | 12 more for each year you’ve owned it |
| [latex]y[/latex] | [latex]=[/latex] | [latex]47[/latex] | [latex]+[/latex] | [latex]12\ast x[/latex] |
[latex]y=47+12x[/latex]
Equivalently, in the slope-intercept form of a linear equation,
[latex]y=mx+b[/latex]
[latex]y=12x+47[/latex]
- [latex]\displaystyle \text{Slope }=\frac{\text{rise}}{\text{run}}[/latex] and [latex]\displaystyle m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}[/latex] where [latex]m=\text{slope}[/latex] and [latex]\displaystyle ({{x}_{1}},{{y}_{1}})[/latex] and [latex]\displaystyle ({{x}_{2}},{{y}_{2}})[/latex] are two points on the line.
- The slope-intercept form of the equation of line is [latex]y=mx+b[/latex] where [latex]m[/latex] represents the slope of the line, [latex]b[/latex] represents the [latex]y[/latex]-intercept, and [latex]x[/latex] and [latex]y[/latex] may be substituted by the coordinates of any point contained on the graph of the equation to satisfy it.