Graphing Using Slope-Intercept Form

Learn how to quickly graph linear equations using the slope and y-intercept.

Intermediate25 minLesson
Printable worksheet

Definition

The slope-intercept form of a linear equation is:
y=mx+by = mx + b
where:
  • mm = slope (how steep the line is)
  • bb = y-intercept (where the line crosses the y-axis)
This form makes graphing easy because you can:
  1. 1.Start at the y-intercept (0,b)(0, b)
  2. 2.Move using the slope: rise over run
For example, in y=2x+3y = 2x + 3:
  • Slope m=2m = 2 means rise 2, run 1
  • Y-intercept b=3b = 3 means start at (0,3)(0, 3)

Try it now

In the equation y=4x+7y = 4x + 7, what is the slope?

Worked Examples

Graph y=2x+1y = 2x + 1
1

Identify the y-intercept

In y=2x+1y = 2x + 1, the y-intercept is b=1b = 1 → Start at (0,1)(0, 1)
2

Identify the slope

The slope is m=2=21m = 2 = \frac{2}{1} → Rise 2, run 1
3

Plot the y-intercept

Mark the point (0,1)(0, 1) on the y-axis → First point plotted
4

Use slope to find second point

From (0,1)(0, 1): move up 2, right 1 to get (1,3)(1, 3) → Second point: (1,3)(1, 3)
5

Draw the line

Connect the points and extend in both directions → Line complete

Common Mistakes

Confusing slope and y-intercept

Why it's wrong: In y=mx+by = mx + b, students sometimes think the first number is the y-intercept.

Correct: Remember: mm (multiplied by xx) is the slope. bb (the constant) is the y-intercept. In y=3x+5y = 3x + 5, slope is 3, y-intercept is 5.

Moving in the wrong direction for negative slope

Why it's wrong: Students may go up instead of down when the slope is negative.

Correct: A negative slope means the line falls from left to right. For m=−2m = -2: go down 2, right 1 (or up 2, left 1).

Inverting the slope fraction

Why it's wrong: Students confuse rise/run with run/rise.

Correct: Slope is always riserun\frac{\text{rise}}{\text{run}} (vertical change over horizontal change). For m=34m = \frac{3}{4}: rise 3, run 4.

Starting at the origin instead of y-intercept

Why it's wrong: Students default to starting at (0,0)(0, 0).

Correct: Always start at the y-intercept (0,b)(0, b). Only start at origin if b=0b = 0.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

How to Graph Linear Equations | Slope-Intercept, Point-Slope, and Standard Form

by Understand The Math

Watch on YouTube

Interactive Visual

Linear Function Explorer

y = 2x + 1
Slope (m)2
Y-Intercept (b)1
b
run
rise
xy
-2-3
-1-1
01
13
25

Interactive Sandbox

Interactive Grapher

-10-50510-10-50510xy

Tap or hover over the graph to read off values.

Try these examples:

2x + 1

m=2, b=1

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Practice Problems

15 problems
Problem 1 of 15
Easy
In the equation y=4x+7y = 4x + 7, what is the slope?

Why It Matters

Slope-intercept form is the most practical way to graph linear equations because:
  • Speed: Graph any line in seconds once you identify mm and bb
  • Prediction: Easily see how variables relate (every unit increase in xx causes mm units change in yy)
  • Real applications: Model costs (y=0.15x+20y = 0.15x + 20 for a phone plan), growth rates, and trends
  • Foundation: This form is the basis for analyzing data, making predictions, and understanding calculus
Scientists, economists, and engineers use this form daily to model relationships and make predictions.

Real World Applications

Phone Plan Costs

Mobile phone plans often have a base monthly fee plus a per-minute or per-gigabyte charge.

Example:

A plan costs 15 euros per month plus 0.05 euros per text. The equation y=0.05x+15y = 0.05x + 15 models the monthly cost, where xx is the number of texts.

1Try It Yourself

A data plan charges 20 euros per month plus 2 euros per gigabyte of data used.

Write the equation and find the cost for using 8 GB.

Step 1: Write the mathematical expression

Write as y=mx+by = mx + b:

Temperature Conversion

The relationship between Celsius and Fahrenheit is linear.

Example:

The formula F=1.8C+32F = 1.8C + 32 converts Celsius to Fahrenheit. The slope 1.8 means each degree Celsius equals 1.8 degrees Fahrenheit.

2Try It Yourself

Use F=1.8C+32F = 1.8C + 32 to find the Fahrenheit temperature when it's 25 degrees Celsius.

What is 25 degrees Celsius in Fahrenheit?

Step 1: Write the mathematical expression

Substitute C = 25:

Savings Growth

If you save a fixed amount regularly, your savings grow linearly.

Example:

Starting with 50 euros and saving 25 euros per week gives y=25x+50y = 25x + 50, where xx is weeks and yy is total savings.

3Try It Yourself

You have 100 euros saved and add 30 euros each week.

How much will you have after 12 weeks?

Step 1: Write the mathematical expression

Write the equation:

Key Takeaways

  • 1Slope-intercept form is y=mx+by = mx + b, where mm is slope and bb is y-intercept
  • 2To graph: start at (0,b)(0, b), then use slope m=riserunm = \frac{\text{rise}}{\text{run}} to find more points
  • 3Positive slope goes up from left to right; negative slope goes down
  • 4Convert equations to slope-intercept form by solving for yy
  • 5This method is faster than making a table of values

Frequently Asked Questions

Write it as a fraction: 3=313 = \frac{3}{1}. This means rise 3, run 1. You can also use 62\frac{6}{2} or −3−1\frac{-3}{-1} to find more points.
Write it as a fraction: 3=313 = \frac{3}{1}. This means rise 3, run 1. You can also use 62\frac{6}{2} or −3−1\frac{-3}{-1} to find more points.
This is y=0x+4y = 0x + 4. The slope is 0 (no rise), so it's a flat horizontal line passing through all points where y=4y = 4.
Vertical lines like x=3x = 3 cannot be written in slope-intercept form because their slope is undefined. They pass through all points where xx equals that value.

Glossary

Slope-intercept form
The equation format y=mx+by = mx + b where mm is slope and bb is the y-intercept
Slope
The steepness of a line, calculated as rise over run (ΔyΔx\frac{\Delta y}{\Delta x})
Y-intercept
The point where a line crosses the y-axis, written as (0,b)(0, b)
Rise
The vertical change between two points on a line
Run
The horizontal change between two points on a line
Today's challengeOne problem, three tries

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