Slope-Intercept Form: Reading and Graphing y = mx + b

If you only ever memorise one form of a line, make it this one. Slope-intercept form puts both of the numbers that describe a line — how steep it is and where it starts — in plain sight, which is why it is the form textbooks graph from and the form calculators expect.

The Two Numbers That Matter

• is the slope — the number multiplying . It controls steepness and direction.
• is the -intercept — the constant on its own. It gives the point .

The order matters when you read an equation. In the slope is , not — rewrite it as if the ordering confuses you. The slope is always whatever is attached to .

What Each Number Controls

Changing and changing do completely different things to the graph, and seeing them separately makes both stick.

Changing the slope pivots the line around its -intercept. Steeper positive slopes climb faster; negative slopes fall left to right; a slope of zero gives a flat line.

Changing the intercept slides the line up or down without tilting it. Lines with the same and different are parallel — they never meet.

Graphing Without a Table

This is the payoff. You need two points to draw a line, and hands you one for free.

1. Plot . That is the intercept, sitting on the -axis.
2. Step by the slope. Write as a fraction , then move that far from the first point.
3. Draw the line through the two points and extend it both ways.

For : start at , go up 2 and right 3 to , draw. A whole-number slope like is — up 4, right 1. A negative slope like is down 3, right 5.

Stepping a second time gives a third point, which is a free check: if all three do not line up, one of the steps was wrong.

Rearranging Into Slope-Intercept Form

You cannot read the slope off an equation until is alone. The rule is one line long: move everything else across, then divide by whatever multiplies .

Two things trip people here. First, divide every term, not just the -term. Second, watch the sign when the -coefficient is negative:

Dividing by flipped both signs on the right. Getting the slope’s sign backwards here is the single most common source of a wrong graph.

Reading m and b in Real Situations

Outside a maths class, and almost always mean the same two things: a rate and a starting amount.

A plumber charges a \$45 call-out fee plus \$20 an hour:

The intercept is what you owe for zero hours — the fee just for turning up. The slope is the cost of each additional hour. The same reading works for a phone plan (monthly fee plus per-gigabyte charge), a taxi fare (flag-fall plus per-kilometre), or a savings account with regular deposits (opening balance plus monthly amount).

A negative slope means the quantity is shrinking: a \$1,200 laptop depreciating \$300 a year is , and the -intercept tells you when it is nominally worthless.

Function Notation

A non-vertical line is a function, so it is often written . This is the same equation with a different label — replaces . The advantage is that it names the input: unambiguously means “the -value when ”.

For , , so the point is on the line.

The Line This Form Cannot Write

Every non-vertical line can be written as . Vertical lines cannot, because they have no slope — the run between any two points is zero, and dividing by zero is undefined.

A vertical line is written and needs standard form if you want it in a general template. Horizontal lines are fine: collapses to .

Worked Example A: Read and Graph

Graph .

Slope , intercept . Start at , go down 1 and right 2 to , then again to . The line falls gently to the right.

Worked Example B: Rearrange, Then Read

Find the slope and intercept of .

Slope , -intercept . Note both signs flipped when dividing by .

Worked Example C: Build It From a Point and a Slope

A line has slope and passes through . Write it in slope-intercept form.

Substitute the point into and solve for :

So . (Point-slope form gets there in one step if you prefer.)

Worked Example D: A Word Problem

A gym charges a \$60 joining fee and \$35 a month. Write the cost after months and find the cost of a year.

After 12 months: .

Worked Example E: Two Points

Write the line through and .

One of the points is already the intercept, so . The slope is . Therefore .

Common Mistakes to Avoid

• Reading the constant as the slope. In the slope is . Reorder to if that helps.
• Reading the slope before solving for . has slope , not .
• Only dividing one term. Going from to forgets the constant; it is .
• Sign errors when the -coefficient is negative. Dividing by a negative flips every sign on the right.
• Running before rising. The slope is — vertical over horizontal, not the other way round.
• Stepping left with a positive run. A positive run means right. To go left, both rise and run change sign.
• Trying to force a vertical line into . It has no slope; use .

Where This Form Leads Next

• Systems of equations — the substitution method is easiest when both equations are solved for .
• Linear inequalities. shades above the line, below.
• Parallel and perpendicular lines, both defined by what happens to .
• Lines of best fit. A regression line is reported as a slope and an intercept for exactly this reason.
• Calculus. The derivative at a point is the slope of the tangent, and the tangent is written in this form.

Practice Problems

Work each one before opening the answer.

Problem 1. State the slope and -intercept of .

Problem 2. State the slope and -intercept of .

Problem 3. Rearrange into slope-intercept form.

Problem 4. Rearrange and state the slope.

Problem 5. A line has slope and passes through . Find .

Problem 6. Write the equation of the line through with slope .

Problem 7. From , where does a slope of land after one step?

Problem 8. A phone plan costs \$15 a month plus \$4 per gigabyte. Write the cost equation and find the cost of 6 GB.

Problem 9. A car worth \$18,000 loses \$2,000 a year. Write the value equation and say when it reaches zero.

Problem 10. Is on the line ?

Problem 11. Write in slope-intercept form.

Problem 12. Two lines are and . Are they parallel?

Quick Reference

This is one of the three forms of a line. Use point-slope form when you have a slope and a point instead of an intercept, and standard form when you want the intercepts quickly. The underlying idea is slope, and there are more Algebra lessons to work through.