Using Slope to Graph Equations of Lines and Linear Models

Slope-Intercept Form

• The slope-intercept form of a linear equation is written as $y = a + bx$.
• $a$ represents the $y$-intercept of the line, located at $(0, a)$.
• $b$ represents the slope of the line.

Graphing Linear Equations

• Step 1: Plot the $y$-intercept at $(0, a)$.
• Step 2: Use the slope $b$ (represented as $\frac{\text{rise}}{\text{run}}$) to find a second point. For example, if $b = \frac{2}{3}$, move $3$ units right and $2$ units up from the intercept.
• Step 3: Sketch a line passing through both points.

Finding an Equation from a Graph

• Identify the $y$-intercept $(0, a)$ from where the line crosses the vertical axis.
• Calculate the slope $b$ by determining the rise over run between two points on the line.
• Substitute the values of $a$ and $b$ into the standard form $y = a + bx$.

Linear Models and Predictions

• Linear models are created when there is a constant rate of change between two variables.
• Example Model (Ticket Sales): Total ticket sales $s$ (in billions) since 2000 is modeled by $s = 1.4 + 0.36t$.
• Extrapolation: Making predictions for values outside the known data range (e.g., predicting sales for the year 2020 based on earlier data). These predictions typically carry less certainty.
• Example Model (Enrollment): College enrollment $E$ (in thousands) since 2015 is modeled by $E = 20 + 2t$, where $2$ is the constant rate of change (slope) per year.