Linear Functions and Equations of Lines: Forms, Properties, and Applications

Linear Functions - Equations of Lines

Introduction to Linear Functions

Linear functions are fundamental in algebra and are used to model relationships with a constant rate of change. The general form of a linear function is:

• Linear Function Form (Slope & y-Intercept): $y = f(x) = mx + b$

• m: Slope of the line (rate of change)

• b: y-intercept (value of y when $x = 0$)

The graph of a linear function is always a straight line.

Understanding Slope

The slope of a line measures its steepness and direction. It is defined as the ratio of the change in y-values to the change in x-values between any two points on the line.

• Slope Formula:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

• If $m > 0$, the line rises from left to right.

• If $m < 0$, the line falls from left to right.

• If $m = 0$, the line is horizontal.

Properties of Linear Functions

Consider the linear function $y = f(x) = -\frac{3}{2}x + 4$. The following are key properties:

• Domain: $\text{Dom } f = \mathbb{R}$ (all real numbers)

• Intercepts:

  • y-intercept: Set $x = 0$ $f(0) = -\frac{3}{2}(0) + 4 = 4$ Point: $(0, 4)$

  • x-intercept: Set $y = 0$ $0 = -\frac{3}{2}x + 4$ $\frac{3}{2}x = 4$ $x = \frac{8}{3}$ Point: $(\frac{8}{3}, 0)$

• Slope: $m = -\frac{3}{2}$

• Range: $\text{Range } f = \mathbb{R}$ (all real numbers)

Graphical Note: A negative slope means the line is slanted downward.

Forms of Linear Equations

There are several standard forms to represent the equation of a straight line:

• Standard Form: $Ax + By = C$

• Slope & y-Intercept Form: $y = mx + b$

• Point & Slope Form: $y - y_1 = m(x - x_1)$

• Two Point Form: $y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$

Each form is useful depending on the information given (points, slope, intercepts).

Worked Examples

Example 1: Converting Standard Form to Linear Function Form

Given: $4x + 7y = 24$

Step	Equation	Comments
0	$4x + 7y = 24$	Given
1	$7y = 24 - 4x$	Isolate $y$
2	$y = \frac{24}{7} - \frac{4}{7}x$	Solve for $y$
3	$y = -\frac{4}{7}x + \frac{24}{7}$	Linear function form

• Domain: $\mathbb{R}$

• y-intercept: $(0, \frac{24}{7})$

• x-intercept: $x = 6$, so $(6, 0)$

• Slope: $m = -\frac{4}{7}$

• Range: $\mathbb{R}$

Example 2: Using Point-Slope Form

Given: Slope $m = \frac{5}{3}$ and point $P(-4, 7)$

Step	Calculation	Comments
0	$y - 7 = \frac{5}{3}(x + 4)$	Point-slope form
1	$y = \frac{5}{3}x + \frac{41}{3}$	Simplify

• Domain: $\mathbb{R}$

• y-intercept: $(0, \frac{41}{3})$

• x-intercept: $x = -\frac{41}{5}$, so $(-\frac{41}{5}, 0)$

• Slope: $m = \frac{5}{3}$

• Range: $\mathbb{R}$

Example 3: Line Through Two Points

Given: $P_1(-3, 2)$ and $Q(5, 9)$

• Slope: $m = \frac{9 - 2}{5 - (-3)} = \frac{7}{8}$

Step	Calculation	Comments
0	$y - 2 = \frac{7}{8}(x + 3)$	Point-slope form
1	$y = \frac{7}{8}x + \frac{37}{8}$	Simplify

• Domain: $\mathbb{R}$

• y-intercept: $(0, \frac{37}{8})$

• x-intercept: $x = -\frac{37}{7}$, so $(-\frac{37}{7}, 0)$

• Slope: $m = \frac{7}{8}$

• Range: $\mathbb{R}$

Summary Table: Forms of Linear Equations

Form	Equation	When to Use
Standard Form	$Ax + By = C$	General equations, integer coefficients
Slope & y-Intercept Form	$y = mx + b$	Known slope and y-intercept
Point & Slope Form	$y - y_1 = m(x - x_1)$	Known slope and a point
Two Point Form	$y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$	Two known points

Key Takeaways

• Linear functions have constant slope and are represented by straight lines.

• The slope indicates the direction and steepness of the line.

• Intercepts are found by setting $x = 0$ (y-intercept) and $y = 0$ (x-intercept).

• Multiple forms exist for the equation of a line, each suited to different given information.