Linear Equations in Two Variables and Functions: Graphs, Slope, and Rate of Change

Linear Equations in Two Variables

Standard Form of a Linear Equation

Linear equations in two variables are fundamental in algebra and are typically written in the standard form:

• Standard Form: $Ax + By = C$ where A and B are real numbers, and not both zero.

• This form is useful for quickly identifying intercepts and for certain algebraic manipulations.

• To be in true standard form, the coefficient of x should be positive, and there should be no fractions.

Graphing Linear Equations and Identifying Intercepts

To graph a linear equation, it is helpful to find the x-intercept (where the line crosses the x-axis) and the y-intercept (where the line crosses the y-axis). These points can be found by setting the other variable to zero and solving for the remaining variable.

• x-intercept: Set $y = 0$ and solve for $x$.

• y-intercept: Set $x = 0$ and solve for $y$.

Slopes of Lines

Definition and Calculation of Slope

The slope of a line measures its steepness and direction. It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line:

• Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Positive slope: Line rises from left to right.

• Negative slope: Line falls from left to right.

• Zero slope: Horizontal line.

• Undefined slope: Vertical line.

Special Cases of Slope

• Horizontal Line: $y = k$ (where $k$ is a constant), slope $m = 0$.

• Vertical Line: $x = k$ (where $k$ is a constant), slope is undefined.

Slope-Intercept Form of a Line

Definition and Use

The slope-intercept form of a line is especially useful for graphing and quickly identifying the slope and y-intercept:

• Form: $y = mx + b$

• m: Slope of the line

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

Graphing Using Slope and Intercept

To graph a line using the slope-intercept form:

1. Plot the y-intercept $(0, b)$ on the y-axis.

2. Use the slope $m = \frac{rise}{run}$ to find another point on the line.

3. Draw a straight line through the points.

Types of Slope

Classification of Slope

• Positive Slope: Line rises from left to right.

• Negative Slope: Line falls from left to right.

• Zero Slope: Horizontal line.

• Undefined Slope: Vertical line.

Average Rate of Change

Definition and Formula

The average rate of change of a function $f$ over the interval $[x_1, x_2]$ is the slope of the secant line passing through the points $(x_1, f(x_1))$ and $(x_2, f(x_2))$ on the graph of $f$.

• Formula: $\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

• This concept generalizes the idea of slope to nonlinear functions.

Example: Calculating Average Rate of Change

Given a function $f(x)$ and two points, such as $(-1.5, 0)$ and $(0, 2)$, the average rate of change is:

• $\frac{2 - 0}{0 - (-1.5)} = \frac{2}{1.5} = 1.33$

Summary Table: Types of Lines and Their Slopes

Type of Line	Equation	Slope
Slanted (positive/negative)	$y = mx + b$	m (real number)
Horizontal	$y = k$	0
Vertical	$x = k$	Undefined