Understanding Linear Functions and Graphs

A linear function is a mathematical relationship that graphs as a straight line. The key feature of a linear function is its constant rate of change, which we call the slope.

Forms of Linear Equations

Linear functions can be written in three main forms:

1. Slope-Intercept Form: $y = mx + b$ Here, $m$ is the slope, and $b$ is the $y$-intercept (the point where the line crosses the $y$-axis).
2. Point-Slope Form: $y - y_1 = m(x - x_1)$ Use this form when you know the slope $m$ and a specific point $(x_1, y_1)$ on the line.
3. Standard Form: $Ax + By = C$ $A$, $B$, and $C$ are generally integers, and $A$ is traditionally positive.

Understanding Slope

The slope ($m$) measures the steepness of the line. You can find it using any two points $(x_1, y_1)$ and $(x_2, y_2)$ on the line: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Parallel Lines: Have the exact same slope ($m_1 = m_2$).
• Perpendicular Lines: Have slopes that are negative reciprocals of each other ($m_1 \cdot m_2 = -1$, or $m_1 = -\frac{1}{m_2}$).

Example Problems

Example 1: Write the equation of the line passing through $(2, 5)$ and $(4, 11)$.

First, calculate the slope: $m = \frac{11 - 5}{4 - 2} = \frac{6}{2} = 3$

Next, use the point-slope form with the point $(2, 5)$: $y - 5 = 3(x - 2)$

To convert this to slope-intercept form, distribute and isolate $y$: $y - 5 = 3x - 6$ $y = 3x - 1$

Example 2: Graph $y = -2x + 3$ and identify the slope and intercepts.

• Slope ($m$): $-2$. This means for every $1$ unit you move right, you move down $2$ units.
• $y$-intercept ($b$): $3$. The line crosses the $y$-axis at the point $(0, 3)$.
• $x$-intercept: Set $y = 0$ and solve for $x$: $0 = -2x + 3 \implies 2x = 3 \implies x = 1.5$ The line crosses the $x$-axis at $(1.5, 0)$.

To graph the line, start by plotting the $y$-intercept at $(0, 3)$. Then, use the slope to find the next point: from $(0, 3)$, go down $2$ units and right $1$ unit to land at $(1, 1)$. Draw a straight line extending through these points.