Slope-Intercept Form

Slope-intercept form is one of the most useful ways to write the equation of a straight line because it immediately tells us two important features of the graph:

y = mx + b

where:

• m = slope
• b = y-intercept

The slope tells us the line's steepness and direction, while the y-intercept tells us where the line crosses the y-axis.

For example:

y = 3x + 2

has:

slope = 3

y-intercept = 2, or the point (0, 2)

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Understanding y = mx + b

Consider:

y = 2x + 5

Compare this with:

y = mx + b

We can identify:

m = 2

b = 5

Therefore:

• the slope is 2
• the line rises from left to right
• y increases by 2 whenever x increases by 1
• the line crosses the y-axis at (0, 5)

Slope-intercept form gives us enough information to graph the line without making a complete table of values.

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The Slope

In:

y = mx + b

the coefficient of x is the slope.

For example:

y = 4x + 7

has:

m = 4

Since:

4 = 4/1

we can interpret the slope as:

right 1, up 4

The positive slope means the line rises from left to right.

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The y-Intercept

The y-intercept is the point where the graph crosses the y-axis.

Every point on the y-axis has:

x = 0

If:

y = mx + b

substitute x = 0:

y = m(0) + b

Therefore:

y = b

So the y-intercept is:

(0, b)

This explains why b represents the y-intercept.

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Identifying Slope and y-Intercept

Consider:

y = 5x − 3

The coefficient of x is:

5

Therefore:

m = 5

The constant is:

−3

Therefore:

b = −3

The y-intercept is:

(0, −3)

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Be Careful with Negative Signs

Consider:

y = −4x + 6

The slope is:

m = −4

The y-intercept is:

b = 6

Now consider:

y = 4x − 6

The slope is:

m = 4

The y-intercept is:

b = −6

The negative sign can belong to either m or b, so read the equation carefully.

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Equations with No Written Constant

Consider:

y = 3x

This can be written as:

y = 3x + 0

Therefore:

m = 3

b = 0

The y-intercept is:

(0, 0)

The line passes through the origin.

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Equations with No x-Term

Consider:

y = 7

This can be thought of as:

y = 0x + 7

Therefore:

m = 0

b = 7

The graph is a horizontal line crossing the y-axis at:

(0, 7)

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Fractional Slopes

Suppose:

y = (2/3)x + 1

Then:

m = 2/3

b = 1

The slope tells us:

rise = 2

run = 3

Starting from the y-intercept, we can move:

right 3, up 2

to locate another point.

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Negative Fractional Slopes

Consider:

y = −(3/4)x + 5

Then:

m = −3/4

b = 5

Starting at:

(0, 5)

move:

right 4, down 3

This gives another point on the line.

The negative slope means the line falls from left to right.

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Graphing from Slope-Intercept Form

To graph:

y = mx + b

use this method:

1. Identify m.
2. Identify b.
3. Plot the y-intercept (0, b).
4. Write the slope as a fraction.
5. Use the numerator as the rise.
6. Use the denominator as the run.
7. Plot another point.
8. Repeat the slope movement if useful.
9. Draw a straight line through the points.
10. Extend the line in both directions.

This method is usually much faster than creating a large table of values.

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Worked Example: Graphing a Positive Slope

Graph:

y = 2x + 1

Identify:

m = 2

b = 1

First plot the y-intercept:

(0, 1)

Write the slope as:

2/1

From (0, 1):

right 1

up 2

This gives:

(1, 3)

Repeat:

right 1

up 2

This gives:

(2, 5)

Draw a straight line through the points.

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Checking with Substitution

We can verify that our points satisfy:

y = 2x + 1

For (1, 3):

3 = 2(1) + 1

3 = 3

Correct.

For (2, 5):

5 = 2(2) + 1

5 = 5

Correct.

Substitution is a useful way to check points on a graph.

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Worked Example: Graphing a Negative Slope

Graph:

y = −3x + 4

Identify:

m = −3

b = 4

Plot:

(0, 4)

Write:

m = −3/1

From (0, 4):

right 1

down 3

This gives:

(1, 1)

Repeat:

right 1

down 3

This gives:

(2, −2)

The resulting line falls from left to right.

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Graphing a Fractional Slope

Graph:

y = (1/2)x − 3

Identify:

m = 1/2

b = −3

Start at:

(0, −3)

Use the slope:

right 2

up 1

This gives:

(2, −2)

Repeat:

right 2

up 1

This gives:

(4, −1)

Then draw the line.

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Moving Backward Along the Line

Slope can also be used in reverse.

Suppose:

m = 2/3

Normally we might move:

right 3

up 2

But we can also move:

left 3

down 2

because:

−2/−3 = 2/3

Both movements stay on the same line.

This is useful when the graph has limited space.

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Writing an Equation from a Graph

A graph can also be converted into slope-intercept form.

To write the equation:

1. Find the y-intercept.
2. Determine the slope.

3. Substitute both values into:

   y = mx + b

For example, suppose a graph crosses the y-axis at:

(0, 3)

and rises 2 units for every 1 unit moved right.

Then:

m = 2

b = 3

Therefore:

y = 2x + 3

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Finding Slope from Two Points on a Graph

Sometimes the slope is not immediately obvious.

Choose two points on the line.

Suppose:

(0, 2) and (4, 10)

Calculate:

m = (10 − 2)/(4 − 0)

m = 8/4

m = 2

The y-intercept is already visible:

b = 2

Therefore:

y = 2x + 2

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Worked Example: Equation from a Graph

Suppose a line crosses the y-axis at:

(0, 6)

and also passes through:

(2, 2)

Calculate slope:

m = (2 − 6)/(2 − 0)

m = −4/2

m = −2

The y-intercept is:

b = 6

Therefore:

y = −2x + 6

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When the y-Intercept Is Negative

Suppose a graph crosses the y-axis at:

(0, −4)

and has slope:

3

Then:

m = 3

b = −4

Therefore:

y = 3x − 4

Do not write:

y = 3x + 4

The sign of the y-intercept must match the graph.

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Finding an Equation from Two Points

If one point is the y-intercept, the process is especially easy.

Suppose a line passes through:

(0, −2) and (3, 4)

Calculate:

m = (4 − (−2))/(3 − 0)

m = 6/3

m = 2

Since:

(0, −2)

is the y-intercept:

b = −2

Therefore:

y = 2x − 2

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What if the y-Intercept Is Not Given?

Suppose a line passes through:

(2, 5) and (4, 9)

First find the slope:

m = (9 − 5)/(4 − 2)

m = 4/2

m = 2

Now use:

y = mx + b

Substitute one known point:

5 = 2(2) + b

5 = 4 + b

b = 1

Therefore:

y = 2x + 1

This equation can then be checked using the other point.

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Changing the Slope

Consider these equations:

y = x + 2

y = 2x + 2

y = 4x + 2

All have:

b = 2

Therefore, all three lines cross the y-axis at:

(0, 2)

But their slopes are different.

As the positive slope increases:

the line becomes steeper

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Positive Slope Changes

Compare:

m = 1/2

m = 1

m = 3

All are positive.

All lines rise from left to right.

However:

m = 3

is the steepest.

The larger the absolute value of the slope:

the steeper the line

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Negative Slope Changes

Compare:

m = −1

m = −3

m = −6

All lines fall from left to right.

The line with:

m = −6

is the steepest because:

|−6| = 6

The negative sign tells us direction.

The absolute value tells us steepness.

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Changing the Sign of the Slope

Compare:

y = 3x + 2

and:

y = −3x + 2

Both have the same y-intercept:

(0, 2)

Both have the same steepness because:

|3| = |−3|

But they move in opposite directions.

The first rises.

The second falls.

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When the Slope Is Zero

Consider:

y = 0x + 4

This simplifies to:

y = 4

The graph is horizontal.

Therefore:

m = 0

A zero slope means y does not change as x changes.

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Can a Vertical Line Be Written in Slope-Intercept Form?

A vertical line has an undefined slope.

For example:

x = 3

is a vertical line.

It cannot be written in the form:

y = mx + b

because there is no defined value of m.

Therefore, slope-intercept form describes nonvertical straight lines, including horizontal lines, but not vertical lines.

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Changing the y-Intercept

Now consider:

y = 2x + 1

y = 2x + 4

y = 2x − 3

All three lines have:

m = 2

Therefore, they have the same steepness and direction.

But they cross the y-axis at different locations.

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What the y-Intercept Does to the Graph

Changing b moves the line:

up or down

without changing its slope.

Increasing b shifts the line upward.

Decreasing b shifts the line downward.

For example:

y = 2x + 5

lies above:

y = 2x + 1

for every x-value.

The lines remain parallel because their slopes are equal.

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Same Slope, Different y-Intercepts

Consider:

y = 3x + 5

and:

y = 3x − 2

Both have:

m = 3

Therefore, they are distinct parallel lines.

Their y-intercepts are:

(0, 5)

and:

(0, −2)

Changing the y-intercept changes the vertical position of the line without changing its direction or steepness.

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Same y-Intercept, Different Slopes

Consider:

y = x + 4

and:

y = −2x + 4

Both cross the y-axis at:

(0, 4)

But:

first line: m = 1

second line: m = −2

Therefore, the lines have different directions and steepness.

They intersect at their shared y-intercept.

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Comparing m and b

The two parameters have different jobs.

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Parallel Lines and Slope-Intercept Form

Consider:

y = 4x + 1

y = 4x − 5

Both have:

m = 4

but different y-intercepts.

Therefore, the lines are:

parallel

For distinct nonvertical lines:

same slope + different y-intercepts → parallel lines

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Intersecting Lines

Consider:

y = 2x + 1

and:

y = 5x + 1

The slopes are different.

Therefore, the lines are not parallel.

They will intersect at exactly one point.

In this case they share the same y-intercept, so they intersect at:

(0, 1)

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Slope-Intercept Form in Real-World Situations

Slope-intercept form is especially useful for situations involving:

constant rates of change

The general model:

y = mx + b

can be interpreted as:

output = rate × input + starting value

This is one of the most useful interpretations of slope-intercept form.

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Real-World Example: Taxi Fare

Suppose a taxi charges:

• $4 starting fee
• $2 for every kilometre travelled

Let:

x = kilometres travelled

y = total cost

The equation is:

y = 2x + 4

Here:

m = 2

means:

$2 per kilometre

and:

b = 4

means:

$4 starting cost

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Real-World Example: Savings

Suppose a person already has:

$100

and saves:

$25 each week

Let:

x = number of weeks

y = total savings

Then:

y = 25x + 100

The slope:

25

means savings increase by $25 each week.

The y-intercept:

100

means the person started with $100 at week 0.

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Real-World Example: Water Tank

A tank initially contains:

500 L

Water enters at:

20 L/min

Let:

x = time in minutes

y = volume in litres

Then:

y = 20x + 500

Slope:

20 L/min

y-intercept:

500 L

The y-intercept represents the starting amount.

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Real-World Example: Decreasing Quantity

A tank begins with:

900 L

and loses:

30 L/min

The equation is:

y = −30x + 900

The negative slope indicates that the amount decreases.

The y-intercept represents:

900 L at time 0

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Interpreting the y-Intercept Carefully

Although b is always mathematically the value of y when x = 0, it may not always have a meaningful real-world interpretation.

Suppose x represents:

age of an adult

and the model is based only on ages 20–60.

The y-intercept would correspond to:

age 0

which is outside the model's intended domain.

Therefore, always ask:

Does x = 0 make sense in this situation?

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Writing an Equation from a Real-World Description

Suppose a gym charges:

$30 membership fee

plus:

$8 per visit

Let:

x = number of visits

y = total cost

Rate:

m = 8

Starting amount:

b = 30

Therefore:

y = 8x + 30

Slope-intercept form makes the relationship between the rate and starting value very clear.

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Writing an Equation from a Table

Consider:

The y-values increase by:

3

for every increase of 1 in x.

Therefore:

m = 3

When x = 0:

y = 5

Therefore:

b = 5

The equation is:

y = 3x + 5

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A Table Without x = 0

Consider:

Calculate slope:

m = (15 − 9)/(4 − 2)

m = 6/2

m = 3

Use one point in:

y = mx + b

9 = 3(2) + b

9 = 6 + b

b = 3

Therefore:

y = 3x + 3

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Checking an Equation Against a Graph

Suppose you think a graph represents:

y = −2x + 5

Check two features.

First:

Does the graph cross the y-axis at:

(0, 5)?

Second:

Does the graph move:

right 1, down 2?

If both are true, the equation is consistent with the graph.

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Checking an Equation with a Point

Suppose the proposed equation is:

y = 3x − 2

and the graph contains:

(4, 10)

Substitute:

10 = 3(4) − 2

10 = 12 − 2

10 = 10

Therefore, the point lies on the line.

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Graph Transformations Using m and b

The equation:

y = mx + b

can be thought of as a family of straight lines.

Changing m changes:

• steepness
• direction

Changing b changes:

• vertical position
• y-intercept

This allows us to predict graph changes without plotting many points.

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Comparing Several Equations

Consider:

A: y = 2x + 3

B: y = 5x + 3

C: y = −2x + 3

D: y = 2x − 4

Compare A and B:

same y-intercept, different slopes

B is steeper.

Compare A and C:

same y-intercept, opposite slope signs

They rise and fall in opposite directions.

Compare A and D:

same slope, different y-intercepts

They are parallel.

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From Graph to Equation: A Reliable Method

When writing an equation from a graph:

1. Locate the y-axis.
2. Find where the line crosses it.
3. Record that value as b.
4. Choose another clear point on the line.
5. Calculate rise and run.
6. Find m = rise/run.
7. Determine the sign of m.

8. Write:

   y = mx + b

9. Check the signs carefully.
10. Substitute a point to verify the equation.

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From Equation to Graph: A Reliable Method

When graphing:

y = mx + b

1. Identify m.
2. Identify b.
3. Plot (0, b).
4. Write m as a fraction.
5. Use the numerator for vertical movement.
6. Use the denominator for horizontal movement.
7. Plot a second point.
8. Repeat if useful.
9. Draw the straight line.
10. Check whether its direction matches the sign of m.

If m > 0:

the graph should rise.

If m < 0:

the graph should fall.

If m = 0:

the graph should be horizontal.

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Common Mistakes

Confusing m and b

In y = mx + b:

m = slope

b = y-intercept.

Giving only b instead of the y-intercept point

If b = 4, the y-intercept is:

(0, 4)

Plotting the y-intercept on the x-axis

The y-intercept must lie on the y-axis.

Ignoring the negative sign on the slope

m = −3 means the line falls from left to right.

Using slope backward

For m = 2/3:

rise = 2

run = 3

not rise = 3 and run = 2.

Thinking a negative y-intercept means negative slope

Slope and y-intercept are independent.

For example:

y = 4x − 7

has positive slope and negative y-intercept.

Thinking changing b changes steepness

Changing b moves the line vertically. It does not change slope.

Thinking changing m only moves the line up or down

Changing m changes the line's steepness and possibly its direction.

Thinking every straight line can be written in slope-intercept form

Vertical lines such as x = 4 have undefined slope and cannot be written as y = mx + b.

Assuming b always has a meaningful physical interpretation

In a model, x = 0 may lie outside the relevant domain.

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Key Terms

Slope-intercept form — form of a linear equation written as y = mx + b.

Slope (m) — rate of change and measure of a line's steepness and direction.

y-intercept (b) — y-value where the line crosses the y-axis.

y-intercept point — coordinate (0, b).

Rise — vertical change between points.

Run — horizontal change between points.

Positive slope — slope greater than zero; graph rises from left to right.

Negative slope — slope less than zero; graph falls from left to right.

Zero slope — slope of a horizontal line.

Undefined slope — slope of a vertical line.

Linear equation — equation whose graph is a straight line.

Linear relationship — relationship with a constant rate of change.

Rate of change — amount one variable changes relative to another.

Parallel lines — lines in the same plane that never meet; distinct nonvertical parallel lines have equal slopes.

Origin — point (0, 0).

Coefficient — numerical factor multiplying a variable.

Constant — number without a variable attached.

Initial value — value of the dependent variable when the independent variable is zero, when meaningful in context.

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Key Takeaways

• Slope-intercept form is y = mx + b.
• The value m represents slope.
• The value b represents the y-intercept.
• The y-intercept is the point (0, b).
• Positive slopes rise from left to right.
• Negative slopes fall from left to right.
• Larger absolute slope values produce steeper lines.
• A slope of zero produces a horizontal line.
• Vertical lines have undefined slope and cannot be written in slope-intercept form.
• To graph y = mx + b, plot the y-intercept first and then use the slope.
• A fractional slope can be interpreted using rise over run.
• A graph can be converted to an equation by finding its slope and y-intercept.
• If the y-intercept is not directly given, b can be calculated using a known point and slope.
• Changing m changes the line's steepness and direction.
• Changing b moves the line vertically without changing its slope.
• Distinct lines with the same slope and different y-intercepts are parallel.
• Lines with different slopes intersect at exactly one point.
• In real-world models, m often represents a rate.
• In real-world models, b often represents an initial or starting value.
• The y-intercept is not always meaningful if x = 0 lies outside the model's relevant domain.
• Slope-intercept form provides a direct connection between equations, graphs, tables, and real-world linear relationships.

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Check Your Understanding

1. In the equation y = 4x + 7, identify the slope and y-intercept.
2. Identify m and b in y = −3x − 5. State whether the line rises or falls from left to right.
3. Find the slope and y-intercept of y = (2/5)x + 4.
4. Describe how you would graph y = 3x − 2 using only its slope and y-intercept.
5. Describe how you would graph y = −(2/3)x + 5.
6. A line crosses the y-axis at (0, 4) and has slope 3. Write its equation in slope-intercept form.
7. A line crosses the y-axis at (0, −6) and passes through (2, 0). Calculate its slope and write its equation.
8. A line passes through (2, 7) and (4, 13). Calculate the slope, determine the y-intercept, and write the equation in slope-intercept form.
9. Compare y = 2x + 1 and y = 6x + 1. What remains the same, and what changes?
10. Compare y = −3x + 5 and y = −3x − 4. Describe the relationship between their graphs.
11. Explain what happens to the graph of y = mx + b when b increases while m remains constant.
12. Explain what happens as the absolute value of m increases while b remains constant.
13. A taxi charges a $5 starting fee and $3 per kilometre. Write an equation for total cost y after travelling x kilometres. Explain the meaning of both m and b.
14. Explain why the vertical line x = 6 cannot be written in slope-intercept form.
15. A student says, "Changing the y-intercept makes a line steeper because the line moves higher on the graph." Evaluate this statement. Explain the different effects of changing m and changing b.