Graphing Linear Functions

Slope-Intercept Form

The most common form of a linear equation:

y = mx + b

Where:

• m = slope (rate of change)
• b = y-intercept (where line crosses y-axis)

Example: Identify m and b

Equation: y = 3x + 2

m = 3 (slope) b = 2 (y-intercept)

Equation: y = −2x + 5

m = −2 (slope) b = 5 (y-intercept)

Equation: y = x − 4

m = 1 (slope, coefficient is 1) b = −4 (y-intercept)

What is the Y-Intercept?

The y-intercept is where the line crosses the y-axis.

At this point: x = 0

To find: Set x = 0 in the equation

Example: Find Y-Intercept

Equation: y = 2x + 7

When x = 0:

• y = 2(0) + 7 = 7

Y-intercept: (0, 7)

From equation: b = 7 tells us this directly!

Graphing Using Slope and Y-Intercept

Method: Start at y-intercept, use slope to find more points

Steps:

1. Plot y-intercept (0, b)
2. Use slope (rise/run) to find next point
3. Draw line through points

Example 1: Graph y = 2x + 1

Step 1: Identify slope and y-intercept

• m = 2 = 2/1 (rise 2, run 1)
• b = 1

Step 2: Plot y-intercept

• Point: (0, 1)

Step 3: Use slope from (0,1)

• Rise 2, run 1 → go up 2, right 1
• New point: (1, 3)

Step 4: Use slope again

• From (1,3): up 2, right 1
• New point: (2, 5)

Step 5: Draw line through points

Example 2: Graph y = −3x + 4

Slope: m = −3 = −3/1 (down 3, right 1) Y-intercept: b = 4

Steps:

1. Plot (0, 4)
2. From (0,4): down 3, right 1 → (1, 1)
3. From (1,1): down 3, right 1 → (2, −2)
4. Draw line

Example 3: Graph y = (1/2)x − 2

Slope: m = 1/2 (up 1, right 2) Y-intercept: b = −2

Steps:

1. Plot (0, −2)
2. From (0,−2): up 1, right 2 → (2, −1)
3. From (2,−1): up 1, right 2 → (4, 0)
4. Draw line

Negative Slopes

Slope is negative: Line goes down from left to right

Two ways to think about it:

• Rise is negative (go down), run is positive (go right)
• Rise is positive (go up), run is negative (go left)

Example: m = −2 = −2/1

Option 1: Down 2, right 1 Option 2: Up 2, left 1

Both work! Use whichever is easier.

Graphing from a Table

Make a table of x and y values, then plot points!

Example: Graph y = 3x − 1

Table:

Plot points and connect with a line

Finding Equation from Graph

To find y = mx + b:

Step 1: Find y-intercept (where line crosses y-axis)

• This gives b

Step 2: Find slope using two points

• m = rise/run

Example: Line Through (0,3) and (2,7)

Y-intercept: b = 3 (line crosses at (0,3))

Slope:

• From (0,3) to (2,7)
• Rise: 7 − 3 = 4
• Run: 2 − 0 = 2
• m = 4/2 = 2

Equation: y = 2x + 3

Special Cases

Horizontal Lines

Form: y = b (no x term)

Slope: m = 0 Example: y = 3

Graph: Horizontal line through (0,3)

Vertical Lines

Form: x = c (no y term)

Slope: Undefined Example: x = −2

Graph: Vertical line through (−2,0)

NOT a function! (Fails vertical line test)

Parallel Lines

Parallel lines have the same slope, different y-intercepts

Example: Parallel Lines

Line 1: y = 2x + 3 Line 2: y = 2x − 1

Both have m = 2 → Parallel!

Never intersect (different b values)

Perpendicular Lines

Perpendicular lines have slopes that are negative reciprocals

If one slope is m, the other is −1/m

Example: Perpendicular Lines

Line 1: y = 2x + 1 (m = 2) Line 2: y = −(1/2)x + 3 (m = −1/2)

Check: 2 × (−1/2) = −1 ✓

Lines are perpendicular (form 90° angle)

Finding X-Intercept

X-intercept: Where line crosses x-axis

At this point: y = 0

To find: Set y = 0 and solve for x

Example: Find X-Intercept

Equation: y = 2x − 6

Set y = 0:

0 = 2x − 6
6 = 2x
x = 3

X-intercept: (3, 0)

Real-World Applications

Earnings: y = 15x + 20

• m = 15 (hourly wage)
• b = 20 (signing bonus)
• Graph shows earnings over time

Temperature: F = (9/5)C + 32

• m = 9/5 (rate of change)
• b = 32 (freezing point offset)

Phone plan: y = 0.10x + 30

• m = 0.10 (per minute charge)
• b = 30 (base fee)

Practice