Lesson 11-3: Equations for Lines of Fit

Property

A trend line (or regression line) models the relationship between two variables, coming as close as possible to all data points. To find its equation, pick two points on the drawn line, which do not need to be original data points.

• First, calculate the slope ($m$) using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, which represents the vertical change divided by the horizontal change.
• Next, identify the y-intercept ($b$), which is the point where the line crosses the y-axis and occurs when $x$ is zero.
• Finally, substitute $m$ and $b$ into the slope-intercept form, $y = mx + b$.

Examples

• Find the slope between $(2, 3)$ and $(7, 9)$ using the formula: $m = \frac{9 - 3}{7 - 2} = \frac{6}{5}$.
• A regression line passes through $(5, 1.25)$ and $(25, 3.35)$. The slope is $m = \frac{3.35 - 1.25}{25 - 5} = 0.105$. The equation simplifies to $y = 0.105x + 0.725$.
• If the calculated slope of a line of fit is $m = \frac{2}{3}$ and the y-intercept is $b = -4$, the complete equation is $y = \frac{2}{3}x - 4$.

Explanation

The regression line is a straight line that best summarizes the trend in a scatterplot. The slope formula is a way to calculate the steepness of this line without relying solely on a visual graph. Once you calculate the slope and identify the y-intercept, substituting your specific numerical values into $y = mx + b$ gives you a final linear model that represents the overall trend of the data.

Property

In a real-world linear model written in the form $y = mx + b$, the mathematical variables have specific, practical meanings:

• The slope ($m$) represents the rate of change. It describes how much the dependent variable ($y$) changes for every one-unit increase in the independent variable ($x$).
• The y-intercept ($b$) represents the initial value or starting point. It is the value of the dependent variable ($y$) when the independent variable ($x$) is 0.

Examples

• An equation for manatee deaths is $y = 4.7 + 2.6t$, where $t$ is years since 1975. This line models an increasing trend, with deaths increasing by about 2.6 per year.
• A plumber charges a fee based on $C = 75h + 50$, where $C$ is total cost and $h$ is hours worked. The slope is 75, meaning the cost increases by 75 dollars for each hour of work. The y-intercept is 50, meaning there is a 50 dollar initial fee before any work begins.
• The amount of water $V$ in a tank after $t$ minutes is modeled by $V = -10t + 300$. The slope is -10, meaning the water decreases by 10 gallons each minute. The y-intercept is 300, meaning the tank initially contained 300 gallons.

Explanation

When a real-world situation is modeled by a linear function, the slope and y-intercept are no longer just abstract numbers. The slope tells you the exact rate at which a quantity is changing over time or per unit. The y-intercept tells you the starting amount or fixed fee before that change even begins.

Property

To make a prediction using a line of fit, substitute a given $x$-value into your linear equation and solve for $y$.
Extrapolation is the process of using the model to make predictions for $x$-values that fall completely outside the range of the original data.

Examples

• Valid Prediction: A line of fit for plant growth is $y = 1.5x + 2$, where $x$ is weeks (with data collected from 1 to 10 weeks). To predict the height at 6 weeks: $y = 1.5(6) + 2 = 11$ inches.
• Unreliable Extrapolation: A line of fit for a car's value is $y = -2000x + 25000$, where $x$ is the car's age (with data collected from 1 to 5 years). Predicting the value at 20 years gives $y = -2000(20) + 25000 = -15000$. This extrapolation is unreliable because a car's value cannot physically be negative.

Explanation

Once you have the equation for a line of fit, you can use it to estimate unknown values by substituting a specific $x$-value to calculate the corresponding $y$-value. When this value falls within the range of your original data, the prediction is usually reliable. However, extrapolation can be risky and unreliable because you are assuming the mathematical trend will continue indefinitely, which is rarely true in real-world scenarios.