Graphing Linear Equations

What is a Graph of a Linear Equation?

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)
• The second number is the y-coordinate (vertical position)

To plot (3, 4):

1. Start at origin
2. Move 3 units right (positive x)
3. Move 4 units up (positive y)
4. Mark the point

To plot (-2, 5):

1. Start at origin
2. Move 2 units left (negative x)
3. Move 5 units up (positive y)
4. 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

1. Plotting points incorrectly Remember: (x, y) not (y, x)!

2. Wrong direction for slope Positive slope: up-right Negative slope: down-right

3. Not extending line far enough Lines go on forever - add arrows!

4. Confusing slope and y-intercept In y = 3x + 2: slope is 3, intercept is 2

5. Calculating slope incorrectly Use (y₂ - y₁)/(x₂ - x₁), keep order consistent

Real-World Applications

Example 1: Phone Plan A phone plan costs $20/month plus$0.10 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 $30,000 depreciates$2,000/year. Equation: y = -2000x + 30000

• Slope: -$2,000/year (negative = decreasing)
• Y-intercept: $30,000 initial value

Problem-Solving Strategy

When given a word problem:

1. Identify rate of change (slope)
2. Identify starting value (y-intercept)
3. Write equation in y = mx + b form
4. Graph the equation
5. Use graph to answer questions

Quick Reference

| Form | Equation | Use | |------|----------|-----| | Slope-Intercept | y = mx + b | Easy to graph, see slope/intercept | | Standard | Ax + By = C | Find intercepts easily | | Point-Slope | y - y₁ = m(x - x₁) | Know slope and one point | | Horizontal | y = k | Slope is 0 | | Vertical | x = k | Undefined slope |

Practice Tips

• Always label axes and scale
• Plot at least 3 points to verify
• Use a ruler for straight lines
• Check your points in the equation
• Extend line with arrows
• Practice recognizing slope from graphs