How to Graph Lines by Using Slope–Intercept Form? (+FREE Worksheet!)

How to Graph Lines by Using Slope–Intercept Form? (+FREE Worksheet!)
Algebra 1

How to Graph Lines Using Slope-Intercept Form

Slope-intercept form, \(y = mx + b\), is the friendliest way to graph a line: \(b\) tells you where to start and \(m\) tells you how to step. Plot one point, walk the slope, and you’ve got the whole line. Let’s make it second nature, solver, drills, and a worksheet maker are a tap away.

Tutor-style math help

Graph Lines by Using Slope, Intercept Form: what to notice and how to work it

Linear skill
Linear topics are about constant rate of change. The slope tells how fast y changes for each 1-unit change in x, and an intercept anchors the line on an axis.

What to notice first

Intercepts are axis-crossing points. Set y = 0 to find an x-intercept and set x = 0 to find a y-intercept.

Common student mistake

Do not mix up x-intercepts and y-intercepts. At an x-intercept, y = 0; at a y-intercept, x = 0.

Key formulas and cues

\(x\text{-intercept: set }y=0\)
\(y\text{-intercept: set }x=0\)
\(y=mx+b\)
runrise yx

A reliable path

  1. Find slopeUse two points, a table, or the coefficient of x in slope-intercept form.
  2. Find an anchorUse a point or intercept so the line is in the right location.
  3. Check directionPositive slope rises left to right; negative slope falls left to right.

Worked examples

Find slope from two points

Example: \((1,4)\) and \((3,10)\)
  1. Change in y is 10 – 4 = 6.
  2. Change in x is 3 – 1 = 2.
  3. Divide rise by run.
Answer: \(m=3\)

Write slope-intercept form

Example: slope 3 and y-intercept -2
  1. Use y = mx + b.
  2. Put m = 3 and b = -2.
  3. Write the line.
Answer: \(y=3x-2\)
Open pop-up practice
Try one before moving on
Try: Find the y-intercept of \(y=-3x+7\).
Answer: \(7\), so the point is \((0,7)\).
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.
Step-by-Step SolverWorksheet Maker✏Practice DrillsFlashcards
Illustration of students learning How to Graph Lines Using Slope-Intercept Form

To graph a line using slope-intercept form, you really need just two things, and the equation \(y = mx + b\) hands you both. Once a line is in this form, graphing it is almost mechanical, in a good way. The number \(b\) is your starting point on the \(y\)-axis, and the slope \(m\) is the step you repeat to find the next points. Plot, step, step, draw. Let’s turn that into a habit you can do without thinking.

The big idea

What Is Slope-Intercept Form?

Slope-intercept form is \(y = mx + b\), where \(m\) is the slope (rise over run) and \(b\) is the \(y\)-intercept (where the line crosses the \(y\)-axis). It’s built for graphing because it hands you a point and a direction for free.

How to graph a line from \(y=mx+b\) (3 steps):

  1. Plot the \(y\)-intercept \((0, b)\).
  2. From there, use the slope as rise over run to step to a second point.
  3. Draw the straight line through both points.
Tutor tip: Write the slope as a fraction so “rise over run” is obvious. A slope of \(2\) is \(\tfrac21\) (up 2, right 1); \(\tfrac14\) is up 1, right 4; and \(-\tfrac12\) is down 1, right 2.
Worked on the grid

Graphing \(y = 2x – 1\)

Start at the \(y\)-intercept \((0,-1)\). The slope is \(2 = \tfrac21\), so step up 2 and right 1 to \((1,1)\), again to \((2,3)\), and draw. That’s the whole line, try your own equation in the solver.

Graph a line
y = 2x − 1(0, -1)

Reading the Two Pieces: Slope and Intercept

The intercept \(b\)

Where to start

The lone constant. Plot it on the \(y\)-axis first.

In \(y=2x-1\), \(b=-1\), so begin at \((0,-1)\).
The slope \(m\)

How to step

The coefficient of \(x\). Rise over run from the intercept.

In \(y=2x-1\), \(m=2=\tfrac21\): up 2, right 1.
Direction

Up or down?

Positive slope rises; negative slope falls left-to-right.

\(y=-\tfrac12x+3\) falls: down 1, right 2.

Worked Examples

Plot the intercept, step the slope, draw, each line below is graphed exactly that way.

Example A, Positive slope

Graph \(y = 2x – 1\).

  1. Plot the \(y\)-intercept \((0,-1)\).
  2. Slope \(2 = \tfrac21\): step up 2, right 1 to \((1,1)\), again to \((2,3)\).
  3. Draw the straight line through the points.

Answer: line through \((0,-1)\) and \((1,1)\)

y = 2x − 1(0, -1)

Example B, Negative, fractional slope

Graph \(y = -\tfrac12 x + 3\).

  1. Plot the intercept \((0,3)\).
  2. Slope \(-\tfrac12\): step down 1, right 2 to \((2,2)\), then \((4,1)\).
  3. Connect them, a gently falling line.

Answer: falling line through \((0,3)\)

y = −½x + 3(0, 3)

Example C, A horizontal line

Graph \(y = 7\).

  1. Here \(m = 0\), so there’s no rise, every point has \(y = 7\).
  2. The graph is a flat horizontal line through \((0,7)\).
  3. Note: a vertical line like \(x = 7\) can’t be written as \(y = mx + b\). Its slope is undefined.

Answer: horizontal line at \(y = 7\)

y = 7(0, 7)

Example D, Through the origin

Graph \(y = -2x\).

  1. Since \(b = 0\), the intercept is the origin \((0,0)\).
  2. Slope \(-2 = \tfrac{-2}{1}\): step down 2, right 1 to \((1,-2)\).
  3. Draw the steep, falling line through the origin.

Answer: line through \((0,0)\) and \((1,-2)\)

y = −2x(0, 0)

Slope-Intercept in the Wild

This form is everywhere a quantity starts somewhere and changes at a steady rate. A phone plan that’s $20 plus $0.10 a minute is \(y = 0.10x + 20\): the \(20\) is the starting cost (the intercept), the \(0.10\) is the rate (the slope). Graphing it shows the bill climbing minute by minute, same picture, real money.

Slip-Ups That Cost Easy Points

  • Plotting the intercept on the wrong axis. \(b\) lives on the \(y\)-axis: the point is \((0, b)\), not \((b, 0)\).
  • Stepping the slope the wrong way. A negative slope goes down as you move right. Keep the sign attached to the rise.
  • Reading slope before solving for \(y\). If the equation isn’t \(y = mx + b\) yet, rearrange first. You can’t read \(m\) off \(2x + y = 5\) until it’s \(y = -2x + 5\).
  • Only plotting one point. A line needs two. Step the slope at least once before you draw.
  • Skipping the check. Step the slope once more for a third point, if all three don’t line up, you misread the rise or run.

Your Turn: Find the Slope and \(y\)-Intercept

For each equation, state the slope, the \(y\)-intercept, and a second point you’d plot by stepping the slope. Reveal answers to check.

  1. \(y = 3x – 5\)
  2. \(y = -x + 2\)
  3. \(y = \tfrac14 x + 1\)
  4. \(y = -2x\)
  5. \(y = 7\)
  6. \(y = \tfrac23 x – 4\)
Show answers
  1. \(\color{blue}{m=3,\ b=-5;\ \text{next point }(1,-2)}\)
  2. \(\color{blue}{m=-1,\ b=2;\ (1,1)}\)
  3. \(\color{blue}{m=\tfrac14,\ b=1;\ (4,2)}\)
  4. \(\color{blue}{m=-2,\ b=0;\ (1,-2)}\)
  5. \(\color{blue}{m=0,\ b=7;\ (1,7)}\)
  6. \(\color{blue}{m=\tfrac23,\ b=-4;\ (3,-2)}\)
Keep practicing

Make Your Own Graphing Worksheet

Generate fresh lines to graph, with a full answer key, print or save as a PDF.

New lines every click, never the same sheet twice
Step-by-step answer key so you can self-check
Open the Worksheet Maker ✏Quick Practice Drills

Frequently Asked Questions

What do \(m\) and \(b\) stand for in \(y = mx + b\)?

\(m\) is the slope (how steep the line is and which way it tilts) and \(b\) is the \(y\)-intercept (the \(y\)-value where the line crosses the \(y\)-axis, at the point \((0,b)\)).

How do I graph a line in slope-intercept form?

Plot the \(y\)-intercept \((0,b)\), use the slope as rise over run to step to a second point, then draw the line through both. Two points are all you need.

What if the equation isn’t solved for \(y\)?

Rearrange it into \(y = mx + b\) first. For example, \(2x + y = 5\) becomes \(y = -2x + 5\), so the slope is \(-2\) and the intercept is \(5\).

How do I graph a horizontal or vertical line?

\(y = c\) is a horizontal line through \((0,c)\) with slope 0. A vertical line is \(x = c\). It can’t be written in slope-intercept form because its slope is undefined.

Related Topics

Finding Slope Writing Equations from Graphs Graphing Linear Inequalities Systems of Equations ← Algebra 1 Hub

Continue Your Study

Ready for the next step? Pick up right where this lesson leaves off:

Up nextHow to Solve Systems of Equations Word ProblemsContinue with your studiesOpen lesson →Up nextHow to Write Linear Equations From GraphsContinue with your studiesOpen lesson →Up nextHow to Model Real-World Situations Using FunctionsContinue with your studiesOpen lesson →
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