1. What is Slope?

Definition: Slope is a measure of the steepness of a line. It describes how much a line rises or falls as you move from left to right.

Slope Formula (Rise Over Run):

\( m = \frac{\text{rise}}{\text{run}} = \frac{\text{change in } y}{\text{change in } x} = \frac{\Delta y}{\Delta x} \)

Using Two Points:

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

where \( (x_1, y_1) \) and \( (x_2, y_2) \) are two points on the line

Key Terms:

• Rise: Vertical change (change in y-values)
• Run: Horizontal change (change in x-values)
• Slope (m): Rate of change of y with respect to x

Important Notes:

• Slope is also called gradient or rate of change
• The symbol for slope is usually \( m \)
• Slope can be positive, negative, zero, or undefined
• The order of points doesn't matter as long as you're consistent

2. Four Types of Slope

Detailed Explanations:

1. Positive Slope: As x increases, y increases. Example: \( m = 2 \), \( m = \frac{3}{4} \)

2. Negative Slope: As x increases, y decreases. Example: \( m = -3 \), \( m = -\frac{1}{2} \)

3. Zero Slope: Horizontal line where y-value stays constant. Rise = 0. Example: \( y = 5 \)

4. Undefined Slope: Vertical line where x-value stays constant. Run = 0, division by zero. Example: \( x = 3 \)

3. Find the Slope from a Graph

Method 1: Using Rise and Run

1. Choose two clear points on the line (preferably where line crosses grid intersections)
2. Count the vertical change (rise) between the two points
3. Count the horizontal change (run) between the two points
4. Apply the formula: \( m = \frac{\text{rise}}{\text{run}} \)

Direction Rules:

• Up = positive rise
• Down = negative rise
• Right = positive run
• Left = negative run

Method 2: Using Two Points

1. Identify coordinates of two points: \( (x_1, y_1) \) and \( (x_2, y_2) \)
2. Substitute into formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
3. Simplify the fraction if possible

Examples:

Example 1: A line passes through points (1, 2) and (4, 8) on a graph. Find the slope.

Using rise/run: From (1,2) to (4,8): Rise = 6 (up), Run = 3 (right)

\( m = \frac{6}{3} = 2 \)

Example 2: A line goes through (0, 5) and (3, 1). Find the slope.

Rise = -4 (down 4), Run = 3 (right 3)

\( m = \frac{-4}{3} = -\frac{4}{3} \)

4. Find the Slope from Two Points

Formula:

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

Steps:

1. Label the two points as \( (x_1, y_1) \) and \( (x_2, y_2) \)
2. Subtract the y-coordinates: \( y_2 - y_1 \)
3. Subtract the x-coordinates in the same order: \( x_2 - x_1 \)
4. Divide: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
5. Simplify if possible

Important Tips:

• Be consistent with the order (don't mix up which point is first)
• It doesn't matter which point you call (x₁, y₁) as long as you're consistent
• Always subtract in the same order: second minus first

Examples:

Example 1: Find the slope through points (2, 3) and (5, 9).

Let \( (x_1, y_1) = (2, 3) \) and \( (x_2, y_2) = (5, 9) \)

\( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 3}{5 - 2} = \frac{6}{3} = 2 \)

Example 2: Find the slope through points (-1, 4) and (3, -2).

\( m = \frac{-2 - 4}{3 - (-1)} = \frac{-6}{4} = -\frac{3}{2} \)

Example 3: Find the slope through points (0, 5) and (4, 5).

\( m = \frac{5 - 5}{4 - 0} = \frac{0}{4} = 0 \)

Zero slope (horizontal line)

Example 4: Find the slope through points (3, 1) and (3, 7).

\( m = \frac{7 - 1}{3 - 3} = \frac{6}{0} = \text{undefined} \)

Undefined slope (vertical line)

5. Find the Slope from a Table

Steps:

1. Choose any two rows from the table
2. Use the values as coordinates: (x₁, y₁) and (x₂, y₂)
3. Apply the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
4. The slope should be the same for any two pairs you choose

Example 1:

Find the slope from this table:

Using first two rows: (1, 5) and (3, 11)

\( m = \frac{11 - 5}{3 - 1} = \frac{6}{2} = 3 \)

Verify with rows 2 and 3: (3, 11) and (5, 17)

\( m = \frac{17 - 11}{5 - 3} = \frac{6}{2} = 3 \) ✓

Example 2:

Using (0, 8) and (2, 4):

\( m = \frac{4 - 8}{2 - 0} = \frac{-4}{2} = -2 \)

6. Find a Missing Coordinate Using Slope

When to Use This:

When you know the slope and one complete point, plus one coordinate of another point, you can find the missing coordinate.

Steps:

1. Write the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
2. Substitute all known values (slope and coordinates)
3. Solve for the unknown variable
4. Check your answer by calculating the slope

Examples:

Example 1: A line has slope \( m = 3 \) and passes through points (2, 5) and (4, y). Find y.

Step 1: Write formula: \( 3 = \frac{y - 5}{4 - 2} \)

Step 2: Simplify: \( 3 = \frac{y - 5}{2} \)

Step 3: Multiply both sides by 2: \( 6 = y - 5 \)

Step 4: Solve: \( y = 11 \)

Example 2: A line has slope \( m = -2 \) and passes through (1, 7) and (x, 3). Find x.

\( -2 = \frac{3 - 7}{x - 1} \)

\( -2 = \frac{-4}{x - 1} \)

\( -2(x - 1) = -4 \)

\( -2x + 2 = -4 \)

\( -2x = -6 \)

\( x = 3 \)

Example 3: Find the missing y-coordinate if the slope is \( \frac{1}{2} \) and the points are (0, 4) and (6, y).

\( \frac{1}{2} = \frac{y - 4}{6 - 0} \)

\( \frac{1}{2} = \frac{y - 4}{6} \)

\( 6 \cdot \frac{1}{2} = y - 4 \)

\( 3 = y - 4 \)

\( y = 7 \)

7. Graph a Line Using Slope

Method: Using Slope and a Point

1. Plot the given point on the coordinate plane
2. Write the slope as a fraction: \( m = \frac{\text{rise}}{\text{run}} \)
3. From the plotted point, move vertically by the rise
4. Then move horizontally by the run
5. Plot the new point
6. Draw a line through both points
7. Extend the line in both directions

Tips for Graphing:

• If slope is a whole number, write it as a fraction: \( 3 = \frac{3}{1} \)
• For negative slope, you can go down and right, or up and left
• For positive slope, go up and right, or down and left
• Use at least two points to draw the line accurately

Examples:

Example 1: Graph a line with slope \( m = 2 \) passing through point (1, 3).

Step 1: Plot (1, 3)

Step 2: Write slope as fraction: \( 2 = \frac{2}{1} \)

Step 3: From (1, 3), move up 2 and right 1 → reach (2, 5)

Step 4: Draw line through (1, 3) and (2, 5)

Example 2: Graph a line with slope \( m = -\frac{3}{4} \) passing through (0, 2).

Plot (0, 2)

Slope: rise = -3 (down 3), run = 4 (right 4)

From (0, 2), move down 3 and right 4 → reach (4, -1)

Draw line through both points

Example 3: Graph a line with slope 0 through point (2, 5).

Slope = 0 means horizontal line

Draw a horizontal line through y = 5

8. Real-World Applications of Slope

Where Slope is Used:

• Rate of change: Speed (miles per hour), cost per item
• Construction: Roof pitch, ramp steepness
• Roads: Grade of highway (percentage)
• Economics: Growth rates, trends
• Science: Temperature change over time

Examples:

Example 1: A car travels 150 miles in 3 hours. What is the slope (rate)?

Points: (0, 0) and (3, 150)

\( m = \frac{150 - 0}{3 - 0} = \frac{150}{3} = 50 \) mph

Example 2: A phone plan costs $30 for 0 GB plus $10 per GB. What is the slope?

The slope is $10 per GB (rate of change)

Quick Reference: Slope Formulas

Main Formula:

\( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x} \)

Types of Slope:

Special Cases:

• Horizontal line: \( m = 0 \) (equation: y = constant)
• Vertical line: \( m = \text{undefined} \) (equation: x = constant)
• Parallel lines: Same slope
• Perpendicular lines: Slopes are negative reciprocals

💡 Key Tips for Slope

• ✓ Slope = rise/run = change in y / change in x
• ✓ Formula: m = (y₂ - y₁)/(x₂ - x₁)
• ✓ Always subtract in the same order!
• ✓ Positive slope: line goes up from left to right ↗
• ✓ Negative slope: line goes down from left to right ↘
• ✓ Zero slope: horizontal line (y stays constant)
• ✓ Undefined slope: vertical line (x stays constant, division by zero)
• ✓ Steeper line = larger absolute value of slope
• ✓ When graphing: plot point, then use rise/run
• ✓ Check your work by calculating slope with found points
• ✓ Slope represents rate of change in real situations
• ✓ Remember: Up is positive, down is negative