Slopes and Equations of Lines: Study Notes for College Algebra

Graphs, Functions, and Models

Slopes of Lines

The slope of a line is a measure of its steepness, indicating how much the line rises or falls as you move from left to right. Slope is a fundamental concept in algebra and is used to describe linear relationships.

• Definition: The slope (m) between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

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

• "Rise over run": "Rise" is the change in y (vertical), and "run" is the change in x (horizontal).

• Order of points: The order in which you choose the points does not affect the slope, as long as you are consistent.

Example: Find the slope of the line passing through $(2,4)$ and $(1,2)$:

$m = \frac{4-2}{2-1} = \frac{2}{1} = 2$

Practice: Find the slope of the line containing $(-1,1)$ and $(4,3)$:

$m = \frac{3-1}{4-(-1)} = \frac{2}{5}$

Types of Slope

The slope of a line can be classified as positive, negative, zero, or undefined, depending on the direction and orientation of the line.

• Positive Slope: Line rises from left to right ($m > 0$).

• Negative Slope: Line falls from left to right ($m < 0$).

• Zero Slope: Horizontal line ($m = 0$), equation is $y = c$.

• Undefined Slope: Vertical line (division by zero), equation is $x = c$.

Table: Types of Slope

Type	Description	Equation
Positive	Line goes up from left to right	$m > 0$
Negative	Line goes down from left to right	$m < 0$
Zero	Horizontal line	$y = c$
Undefined	Vertical line	$x = c$

Slope-Intercept Form

The slope-intercept form of a linear equation is a convenient way to write the equation of a line using its slope and y-intercept.

• General form: $y = mx + b$

• m: Slope of the line

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

Example: For a line with slope $m = 1$ and y-intercept $b = -3$, the equation is:

$y = x - 3$

Practice: Given a graph, identify the slope and y-intercept, then write the equation in slope-intercept form.

Graphing Lines from Slope-Intercept Form

To graph a line given in slope-intercept form ($y = mx + b$):

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

2. Use the slope ($m$) to find another point (rise over run).

3. Draw a line through the two points.

Example: Graph $y = \frac{2}{3}x + 1$:

• y-intercept: $b = 1$

• Slope: $m = \frac{2}{3}$ (up 2 units, right 3 units)

Point-Slope Form

The point-slope form is useful when you know the slope and a point on the line (other than the y-intercept).

• General form: $y - y_1 = m(x - x_1)$

• $(x_1, y_1)$: A point on the line

• $m$: Slope of the line

Example: Write the equation of a line with slope $m = -\frac{2}{3}$ passing through $(1,3)$:

$y - 3 = -\frac{2}{3}(x - 1)$

Practice: Write the point-slope form for a line with slope $0$passing through$(2, -4)$:

$y + 4 = 0(x - 2)$

Comparing Slope-Intercept and Point-Slope Forms

Form	Equation	Use When
Slope-Intercept	$y = mx + b$	y-intercept is known
Point-Slope	$y - y_1 = m(x - x_1)$	Point and slope are known

Summary of Key Concepts

• Slope measures the steepness of a line and is calculated as $m = \frac{y_2 - y_1}{x_2 - x_1}$.

• Slope-intercept form ($y = mx + b$) is used when the y-intercept is known.

• Point-slope form ($y - y_1 = m(x - x_1)$) is used when a point and the slope are known.

• Horizontal lines have zero slope; vertical lines have undefined slope.

• Graphing lines involves plotting the y-intercept and using the slope to find additional points.