What is Slope?

The slope of a line measures its steepness and direction. It represents the rate of change - how much y changes for every unit change in x.

Slope Formula:

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

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

Steps to Find Slope from Two Points:

1. Step 1: Identify the coordinates
   • Label the points as \( (x_1, y_1) \) and \( (x_2, y_2) \)
2. Step 2: Find the change in y (rise)
   • Subtract: \( y_2 - y_1 \)
3. Step 3: Find the change in x (run)
   • Subtract: \( x_2 - x_1 \)
4. Step 4: Divide rise by run
   • Calculate: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
   • Important: Keep the order consistent!

Types of Slopes:

Example 1: Find the Slope

Given points: \( (2, 3) \) and \( (6, 11) \)

Step 1: \( (x_1, y_1) = (2, 3) \) and \( (x_2, y_2) = (6, 11) \)

Step 2: Change in y: \( y_2 - y_1 = 11 - 3 = 8 \)

Step 3: Change in x: \( x_2 - x_1 = 6 - 2 = 4 \)

Step 4: \( m = \frac{8}{4} = 2 \)

Answer: The slope is 2 (positive, line rises)

Example 2: Slope from Equation

Given equation: \( y = -3x + 7 \)

In slope-intercept form \( y = mx + b \), the coefficient of x is the slope

Answer: The slope is \( m = -3 \) (negative, line falls)

Graphing Using Slope-Intercept Form:

The easiest way to graph a linear function is to use the slope-intercept form \( y = mx + b \)

Steps to Graph a Linear Function:

1. Step 1: Write in Slope-Intercept Form
   • Rearrange equation to \( y = mx + b \) if needed
   • Identify m (slope) and b (y-intercept)
2. Step 2: Plot the Y-Intercept
   • The y-intercept is the point \( (0, b) \)
   • Plot this point on the y-axis
3. Step 3: Use the Slope to Find Another Point
   • Write slope as a fraction: \( m = \frac{\text{rise}}{\text{run}} \)
   • From the y-intercept, move up/down (rise) and right/left (run)
   • Plot the second point
4. Step 4: Draw the Line
   • Connect the two points with a straight line
   • Extend the line in both directions
   • Add arrows at both ends

Understanding Slope as Rise/Run:

• Positive slope: Rise = up, Run = right
• Negative slope: Rise = down (or up negative), Run = right
• Example: \( m = \frac{3}{2} \) means rise 3, run 2
• Example: \( m = -\frac{2}{5} \) means down 2, right 5

Example 1: Graph \( y = 2x + 3 \)

Step 1: Identify m = 2 and b = 3

Step 2: Plot y-intercept at (0, 3)

Step 3: Slope = \( \frac{2}{1} \) → From (0, 3), rise 2, run 1 → Plot (1, 5)

Step 4: Draw a line through (0, 3) and (1, 5)

Example 2: Graph \( y = -\frac{3}{4}x + 1 \)

Step 1: Identify \( m = -\frac{3}{4} \) and b = 1

Step 2: Plot y-intercept at (0, 1)

Step 3: Slope = \( -\frac{3}{4} \) → From (0, 1), down 3, right 4 → Plot (4, -2)

Step 4: Draw a line through (0, 1) and (4, -2)

Forms of Linear Equations:

Method 1: Given Slope and Y-Intercept

Use Slope-Intercept Form: \( y = mx + b \)

1. Identify the slope (m) and y-intercept (b)
2. Substitute these values into \( y = mx + b \)
3. Simplify if necessary

Method 2: Given Two Points

1. Step 1: Find the slope
   • Use \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
2. Step 2: Use point-slope form
   • \( y - y_1 = m(x - x_1) \)
   • Choose either point as \( (x_1, y_1) \)
3. Step 3: Convert to slope-intercept form
   • Solve for y to get \( y = mx + b \)

Method 3: Given Slope and One Point

1. Step 1: Use point-slope form \( y - y_1 = m(x - x_1) \)
2. Step 2: Substitute the slope m and point \( (x_1, y_1) \)
3. Step 3: Simplify to slope-intercept form

Example 1: Given Slope and Y-Intercept

Given: Slope = 4, y-intercept = -2

Step 1: m = 4, b = -2

Step 2: Substitute into \( y = mx + b \)

Answer: \( y = 4x - 2 \)

Example 2: Given Two Points

Given points: \( (1, 3) \) and \( (4, 12) \)

Step 1: Find slope: \( m = \frac{12-3}{4-1} = \frac{9}{3} = 3 \)

Step 2: Use point-slope with (1, 3): \( y - 3 = 3(x - 1) \)

Step 3: Simplify: \( y - 3 = 3x - 3 \) → \( y = 3x \)

Answer: \( y = 3x \)

Example 3: Given Slope and One Point

Given: Slope = -2, Point (3, 5)

Step 1: Use point-slope: \( y - 5 = -2(x - 3) \)

Step 2: Distribute: \( y - 5 = -2x + 6 \)

Step 3: Solve for y: \( y = -2x + 11 \)

Answer: \( y = -2x + 11 \)

What is a Unit Interval?

A unit interval is an interval with length 1, typically represented as consecutive integers like [0, 1], [1, 2], [2, 3], etc.

For linear functions over unit intervals, the rate of change (slope) remains constant over each interval.

Key Concept:

For a linear function \( f(x) = mx + b \), the change in the function value over any unit interval equals the slope:

\( f(x+1) - f(x) = m \)

Understanding the Pattern:

• Constant Rate: Linear functions change by the same amount over each unit interval
• Equal Spacing: If you move 1 unit right, you move m units up/down
• Identifying Linearity: If differences over unit intervals are constant, the function is linear
• Non-linear functions: Have varying rates of change over unit intervals

Steps to Analyze Linear Functions Over Unit Intervals:

1. Step 1: Calculate differences
   • Find \( f(x+1) - f(x) \) for consecutive x-values
2. Step 2: Check for constant difference
   • If all differences are equal, the function is linear
3. Step 3: The constant difference is the slope
   • \( m = f(x+1) - f(x) \)
4. Step 4: Find y-intercept and write equation
   • Use any point and the slope to find b

Example 1: Verify Linear Function

Given: \( f(x) = 5x + 2 \)

Check unit intervals:

• \( f(1) - f(0) = 7 - 2 = 5 \)

• \( f(2) - f(1) = 12 - 7 = 5 \)

• \( f(3) - f(2) = 17 - 12 = 5 \)

Conclusion: Constant difference of 5 confirms linear function with slope m = 5

Example 2: Find Equation from Table

Given table:

Step 1: Calculate differences: 6-3=3, 9-6=3, 12-9=3

Step 2: Constant difference = 3, so slope m = 3

Step 3: Y-intercept: f(0) = 3, so b = 3

Answer: \( f(x) = 3x + 3 \)

Example 3: Non-Linear Function

Given table:

Calculate differences: 1-0=1, 4-1=3, 9-4=5

Conclusion: Differences are NOT constant (1, 3, 5), so this is NOT linear (it's \( f(x) = x^2 \))

• Find Slope: \( m = \frac{y_2 - y_1}{x_2 - x_1} \) or identify m in \( y = mx + b \)
• Graph Function: Plot y-intercept (0, b) → Use slope (rise/run) → Draw line
• Write Equation: Use \( y = mx + b \) with slope and y-intercept, or \( y - y_1 = m(x - x_1) \) with slope and point
• Unit Intervals: Linear functions have constant change over unit intervals equal to slope