MTH1W Slope, y-Intercept and y = mx + b: Notes and Practice

A wheelchair ramp cannot be too steep, or it becomes dangerous to use. Accessibility rules limit ramps to a gentle steepness; a common rule, used in Ontario's building code for barrier-free ramps, allows at most 1 unit of height for every 12 units of horizontal length, and some standards ask for an even gentler slope. Roofers talk about a roof's pitch, and road signs warn of a "10% grade" before a steep hill. All of these describe the same idea, the slope of a line: how far it goes up or down for each step across. In the last lesson you met the rate of change of a linear relation. Slope is the same number, read straight off a graph. In this lesson you will calculate slope several ways and combine it with one more number, the $y$ -intercept, to write the equation of any non-vertical line.

A concrete ramp with metal railings that zigzags upward in several sections toward a footbridge, beside a railway track. — PhotoInstead of climbing straight up, this ramp to a footbridge in Auckland zigzags back and forth. Each section rises only a little over a long run, which keeps the slope gentle.Ingolfson, Public domain, via Wikimedia Commons

Slope is rise over run

Pick two points on a line. The rise is the vertical change from the first point to the second: positive if you go up, negative if you go down. The run is the horizontal change: positive if you go right. The slope, written $m$ , is the rise divided by the run. Because a line is straight, it does not matter which two points you pick: a bigger triangle has a bigger rise and a bigger run in the same proportion, so the quotient is always the same. If the points are $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ , the rise is $y_{2} - y_{1}$ and the run is $x_{2} - x_{1}$ .

m = \frac{rise}{run} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}, x_{2} \neq = x_{1}

Fig 1The line y = (2/3)x + 1 through A(−3, −1) and B(3, 3), with a slope triangle of run 6 and rise 4 and the y-intercept (0, 1) marked

Slope	What the line does (left to right)	Example
Positive	Rises	$y = 2 x + 1$
Negative	Falls	$y = - 3 x + 4$
Zero	Horizontal: the rise is 0	$y = 5$
Undefined	Vertical: the run is 0, and you cannot divide by 0	$x = 2$

What the sign of the slope tells you

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A line rises 6 units for every 2 units it runs to the right. What is its slope?

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Which line has an undefined slope?

VideoKhan Academy builds the idea of slope as the change in y divided by the change in x, with lines that rise and lines that fall.Watch on YouTube

Slope from two points

For $A (- 3, - 1)$ and $B (3, 3)$ : the rise is $3 - (- 1) = 4$ and the run is $3 - (- 3) = 6$ , so $m = \frac{4}{6} = \frac{2}{3}$ . Reduce the fraction, the way you would any fraction. You can also subtract the other way round, $\frac{- 1 - 3}{- 3 - 3} = \frac{- 4}{- 6} = \frac{2}{3}$ , and get the same slope, as long as you subtract in the same order on top and bottom. Mixing orders flips the sign. Be extra careful with negative coordinates: subtracting a negative means adding.

Fig 2The slope of the line through A(−3, −1) and B(3, 3) found both ways: from A to B the run is +6 and the rise is +4; from B to A the run is −6 and the rise is −4. Both give 2/3.

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Find the slope of the line through

(- 1, 4)

and

(2, - 5)

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Mia finds the slope through

(1, 2)

and

(4, 8)

\frac{8 - 2}{1 - 4}

. What happens?

The y-intercept and slope-intercept form

The y-intercept is where a line crosses the $y$ -axis. Every point on the $y$ -axis has $x = 0$ , so the $y$ -intercept is the point $(0, b)$ , and we usually just call the number $b$ the $y$ -intercept. A non-vertical line with slope $m$ and $y$ -intercept $b$ has the equation $y = m x + b$ , called slope-intercept form. This is the "rate of change and initial value" form from the last lesson: $m$ is the rate of change and $b$ is the initial value. For the line in the figure, $m = \frac{2}{3}$ and $b = 1$ , so $y = \frac{2}{3} x + 1$ . To graph $y = m x + b$ , plot $(0, b)$ , then use the rise and run of $m$ to reach a second point, and draw the line through both.

y = m x + b m = slope, b = y -intercept

Fig 3Graphing y = −(3/4)x + 6: plot the y-intercept (0, 6), then step right 4 and down 3 to (4, 3) and again to (8, 0), and draw the line through the points.

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What are the slope and the

y

-intercept of

y = - 5 x + 2

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To graph

y = \frac{1}{2} x + 3

, you plot

(0, 3)

, then move 2 right and 1 up. What is the

y

-coordinate of the point you reach?

VideoAn Ontario teacher reads the slope and y-intercept from y = mx + b and uses them to graph and write equations of lines.Watch on YouTube

Writing the equation from a graph, a table, two points or a description

From a graph: read $b$ where the line crosses the $y$ -axis, then pick two points exactly on grid corners and find rise over run. Check the scale of each axis first: one grid square is not always 1 unit.
From a table: the slope is the change in $y$ divided by the change in $x$ between rows. The $y$ -value in the row where $x = 0$ is $b$ ; if there is no such row, work backward with the slope.
From two points: find $m$ with the slope formula, then substitute one point and $m$ into $y = m x + b$ and solve for $b$ .
From a description: the amount added or lost per unit is $m$ (negative when it decreases), and the starting amount is $b$ .

Lines that break the pattern

A horizontal line has slope

0

, so

y = 0 x + b

, which is just

y = b

. A vertical line, such as

x = 2

, has a run of

0

between any two of its points, so its slope is undefined. A vertical line has no slope-intercept form at all: its equation is

x = k

, and it cannot be written as

y = m x + b

Graphing with a negative fractional slope

To graph

y = - \frac{3}{4} x + 2

, plot

(0, 2)

. Read the slope as rise

- 3

over run

4

: move 4 right and 3 down to

(4, - 1)

. Join the points. Check:

- \frac{3}{4} (4) + 2 = - 3 + 2 = - 1

✓.

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A line passes through

(1, 5)

and

(3, 9)

. Write its equation in the form y = mx + b.

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A horizontal line passes through

(4, - 3)

. What is its equation?

Slope, y-Intercept and y = mx + b

Key things to remember

Slope is rise over run

Slope from two points

The y-intercept and slope-intercept form

Writing the equation from a graph, a table, two points or a description

Worked examples

Where marks get lost

Flashcards 8

Practice