The graph of a linear equation is a straight line on the coordinate plane. Every point (x, y) on the line is a solution to the equation.
For example, the equation y = 2x + 1 is graphed as a line where every point satisfies the equation. The point (2, 5) is on the line because 5 = 2(2) + 1.
The Coordinate Plane
The coordinate plane (also called the Cartesian plane) has:
A horizontal axis called the x-axis
A vertical axis called the y-axis
The point where they meet is the origin (0, 0)
The plane is divided into four quadrants:
Quadrant I: (+, +) upper right
Quadrant II: (-, +) upper left
Quadrant III: (-, -) lower left
Quadrant IV: (+, -) lower right
Ordered Pairs
Points are written as (x, y) called ordered pairs:
The first number is the x-coordinate (horizontal position)
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⚠️ Common Mistakes: Graphing Linear Equations
Avoid these 3 frequent errors
🌍 Real-World Applications: Graphing Linear Equations
See how this math is used in the real world
📝 Worked Example: Solving a Quadratic by Factoring
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The second number is the y-coordinate (vertical position)
To plot (3, 4):
Start at origin
Move 3 units right (positive x)
Move 4 units up (positive y)
Mark the point
To plot (-2, 5):
Start at origin
Move 2 units left (negative x)
Move 5 units up (positive y)
Mark the point
Graphing Linear Equations Using a Table
Method: Make a table of values, plot points, draw a line
Example: Graph y = 2x - 3
Step 1: Create a table (choose x-values, calculate y)
x
y = 2x - 3
(x, y)
-1
2(-1) - 3 = -5
(-1, -5)
0
2(0) - 3 = -3
(0, -3)
1
2(1) - 3 = -1
(1, -1)
2
2(2) - 3 = 1
(2, 1)
3
2(3) - 3 = 3
(3, 3)
Step 2: Plot these points on a coordinate plane
Step 3: Draw a straight line through the points
Step 4: Add arrows at both ends (line extends forever)
Tip: You only need 2 points to draw a line, but use 3 as a check!
Slope-Intercept Form
The most common form for graphing is slope-intercept form:
y = mx + b
Where:
m is the slope (steepness of line)
b is the y-intercept (where line crosses y-axis)
Example: y = 3x + 2
Slope (m) = 3
Y-intercept (b) = 2
Understanding Slope
Slope measures the steepness and direction of a line.
Slope = rise/run = change in y / change in x = (y₂ - y₁)/(x₂ - x₁)
Types of Slope:
Positive Slope (m > 0):
Line goes upward from left to right
Example: y = 2x + 1 (slope = 2)
Negative Slope (m < 0):
Line goes downward from left to right
Example: y = -3x + 5 (slope = -3)
Zero Slope (m = 0):
Horizontal line
Example: y = 4 (slope = 0)
Undefined Slope:
Vertical line
Example: x = 3 (slope undefined)
Graphing Using Slope-Intercept Form
Method: Start at y-intercept, use slope to find next points
Example: Graph y = (2/3)x - 1
Step 1: Identify slope and y-intercept
Slope m = 2/3 (rise 2, run 3)
Y-intercept b = -1
Step 2: Plot y-intercept
Plot point (0, -1)
Step 3: Use slope to find next point
From (0, -1):
Rise 2 (up 2 units)
Run 3 (right 3 units)
Plot point (3, 1)
Step 4: Continue or go backwards
From (0, -1):
Fall 2 (down 2)
Run left 3 (left 3)
Plot point (-3, -3)
Step 5: Draw line through points
Finding Slope from Two Points
Given two points (x₁, y₁) and (x₂, y₂):
slope = (y₂ - y₁)/(x₂ - x₁)
Example: Find slope through (1, 3) and (4, 9)
m = (9 - 3)/(4 - 1)
m = 6/3
m = 2
The Y-Intercept
The y-intercept is where the line crosses the y-axis.
At this point, x = 0
Written as point (0, b)
To find y-intercept from an equation:
Set x = 0 and solve for y
Example: Find y-intercept of 2x + 3y = 12
Set x = 0:
2(0) + 3y = 12
3y = 12
y = 4
Y-intercept: (0, 4)
The X-Intercept
The x-intercept is where the line crosses the x-axis.
At this point, y = 0
Written as point (a, 0)
To find x-intercept from an equation:
Set y = 0 and solve for x
Example: Find x-intercept of 2x + 3y = 12
Set y = 0:
2x + 3(0) = 12
2x = 12
x = 6
X-intercept: (6, 0)
Graphing Using Intercepts
Method: Find x-intercept and y-intercept, draw line through them
Example: Graph 3x + 2y = 12
Step 1: Find y-intercept (set x = 0)
3(0) + 2y = 12
y = 6
Point: (0, 6)
Step 2: Find x-intercept (set y = 0)
3x + 2(0) = 12
x = 4
Point: (4, 0)
Step 3: Plot both intercepts
Step 4: Draw line through them
Step 5: Check with a third point
Standard Form
Standard form: Ax + By = C
Where A, B, C are integers and A is positive
Example: 2x + 3y = 12
To graph from standard form:
Find x and y intercepts, OR
Convert to slope-intercept form
Converting to slope-intercept form:
2x + 3y = 12
3y = -2x + 12
y = (-2/3)x + 4
Special Lines
Horizontal Lines:
Form: y = k (constant)
Slope = 0
Parallel to x-axis
Example: y = 3
Vertical Lines:
Form: x = k (constant)
Undefined slope
Parallel to y-axis
Example: x = -2
Parallel and Perpendicular Lines
Parallel Lines:
Have the SAME slope
Never intersect
Example: y = 2x + 1 and y = 2x - 3 (both have slope 2)
Perpendicular Lines:
Slopes are NEGATIVE RECIPROCALS
Intersect at 90° angle
If slope₁ = m, then slope₂ = -1/m
Example: y = 2x + 1 (slope = 2) and y = (-1/2)x + 3 (slope = -1/2)
Product of perpendicular slopes = -1
Common Mistakes to Avoid
Plotting points incorrectly
Remember: (x, y) not (y, x)!
Wrong direction for slope
Positive slope: up-right
Negative slope: down-right
Not extending line far enough
Lines go on forever - add arrows!
Confusing slope and y-intercept
In y = 3x + 2: slope is 3, intercept is 2
Calculating slope incorrectly
Use (y₂ - y₁)/(x₂ - x₁), keep order consistent
Real-World Applications
Example 1: Phone Plan
A phone plan costs 20/monthplus0.10 per text.
Equation: y = 0.10x + 20
Slope: $0.10 per text
Y-intercept: $20 base fee
Example 2: Temperature
Converting Fahrenheit to Celsius: C = (5/9)(F - 32)
This is a linear relationship.
Example 3: Car Depreciation
A car worth 30,000depreciates2,000/year.
Equation: y = -2000x + 30000