1.2.1 Lines: slope and intercept

What a slope measures

A straight line changes at a steady rate. Every step of 1 to the right changes $y$ by the same amount, and that amount is the line's slope. Its units are units of $y$ per unit of $x$ .

A candle is 24 cm tall when it is lit, and it burns down 1.5 cm every hour. Here is its height $h$ after $t$ hours.

$t$ (hours)	0	1	2	3	4
$h$ (cm)	24	22.5	21	19.5	18

Each hour takes off the same 1.5 cm, so the points lie on a straight line. Its slope is $-1.5$ cm per hour, negative because the height falls. The starting height, 24 cm at $t = 0$ , is where the line meets the vertical axis.

Slope from two points: rise over run

Usually you are not told the rate. You have two points on the line instead. Going from the first point to the second, the run is the change in $x$ and the rise is the change in $y$ .

Rule: Slope from two points

m = \frac{y_2 - y_1}{x_2 - x_1}

The slope $m$ of the line through $(x_1, y_1)$ and $(x_2, y_2)$ is the rise divided by the run. Any two points of the line give the same slope, in either order, as long as top and bottom use the same order.

For $(1, 3)$ and $(4, 9)$ , the run is $4 - 1 = 3$ and the rise is $9 - 3 = 6$ , so $m = 6 \div 3 = 2$ .

Two other points of the same line, $(2, 5)$ and $(3, 7)$ , give $2 \div 1 = 2$ again. In reverse order, $\frac{3 - 9}{1 - 4} = \frac{-6}{-3} = 2$ .

Rising, falling and flat

The sign of the rise, over a positive run, gives the sign of the slope.

Rise over run on three lines, each with a run of 3. The dot marked $c$ is where each line crosses the $y$ -axis.

Positive: the line climbs from left to right. From $(1, 3)$ to $(4, 9)$ the slope is 2.
Negative: the line falls. From $(1, 6)$ to $(4, 3)$ the rise is $-3$ , so the slope is $-3 \div 3 = -1$ .
Zero: the line is flat. From $(1, 4)$ to $(4, 4)$ the rise is 0, so the slope is 0.

The size of the slope is the steepness. A slope of $-3$ is as steep as 3, but falls. A slope of $\frac{1}{2}$ climbs 1 for every 2 steps to the right.

Vertical lines have no slope

Through $(2, 1)$ and $(2, 5)$ the run is $2 - 2 = 0$ , so rise over run would divide by zero. A vertical line has no slope at all, which is not the same as a slope of 0. Its equation is $x = 2$ , and it fails the vertical line test of lesson 1.1.1, so it is not the graph of a function.

Try it slope is the rule above, with each point written as a pair (x, y) . The cell checks every case in this section, including both orders for the first line. The vertical line raises ZeroDivisionError, which the try block catches. Each slope prints as a decimal (2.0, not 2) because / in Python always gives one.

def slope(p, q): (x1, y1), (x2, y2) = p, q return (y2 - y1) / (x2 - x1) print(slope((1, 3), (4, 9)), slope((4, 9), (1, 3))) print(slope((1, 6), (4, 3))) print(slope((1, 4), (4, 4))) try: print(slope((2, 1), (2, 5))) except ZeroDivisionError as err: print("vertical line:", err)

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The equation of a line

Every line that is not vertical can be written as

y = mx + c

where $m$ is the slope and $c$ is the intercept: the value of $y$ where the line crosses the $y$ -axis, at $x = 0$ . Lesson 1.1.1 called it $f(0)$ , and it is in the units of $y$ . The candle is $h = -1.5t + 24$ : slope $-1.5$ cm per hour, intercept 24 cm.

Rule: Line through two points

\begin{gathered} m = \frac{y_2 - y_1}{x_2 - x_1} \\ c = y_1 - m\,x_1 \end{gathered}

Find the slope first. Then put one point into $y = mx + c$ and solve for the intercept: $y_1 = m x_1 + c$ gives $c = y_1 - m x_1$ .

Example 1, by a table and by algebra. Take $(1, 3)$ and $(4, 9)$ again, with slope 2. Each step of 1 in $x$ adds 2 to $y$ , so one step back from $(1, 3)$ gives $y = 1$ at $x = 0$ . The table steps forward from there and lands on $(4, 9)$ :

$x$	0	1	2	3	4
$y$	1	3	5	7	9

The algebra agrees: $c = 3 - 2 \times 1 = 1$ , so the line is $y = 2x + 1$ . Check it with the other point: $2 \times 4 + 1 = 9$ .

Example 2, a falling line. Through $(2, 7)$ and $(6, -1)$ , the slope is $\frac{-1 - 7}{6 - 2} = \frac{-8}{4} = -2$ . Then $c = 7 - (-2)(2) = 7 + 4 = 11$ , so $y = -2x + 11$ . Check: $-2 \times 6 + 11 = -1$ .

From one point and a slope

Sometimes you know one point $(x_1, y_1)$ and the slope $m$ . Then you can write the line down at once, without finding $c$ first.

Rule: Point and slope

y = y_1 + m(x - x_1)

At $x = x_1$ the bracket is 0, so $y = y_1$ : the line passes through the point. Each step of 1 to the right adds $m$ . Multiplying out gives $y = mx + (y_1 - m x_1)$ , the same line as before.

Slope 3 through $(2, 5)$ gives $y = 5 + 3(x - 2)$ , which multiplies out to $y = 3x - 1$ . Write $\Delta x$ (read "delta $x$ ") for a change in $x$ . A change of $\Delta x$ moves $y$ by $m\,\Delta x$ : from $x = 2$ to 2.5, $y$ rises $3 \times 0.5 = 1.5$ , to 6.5.

Lines written another way

A line may come as $3x + 2y = 6$ , and its slope is not 3. Make $y$ the subject first, as in lesson 0.3.2: $2y = -3x + 6$ , so $y = -1.5x + 3$ , with slope $-1.5$ and intercept 3.

An equation with no $y$ in it, such as $2x = 4$ , is the vertical line $x = 2$ .

x

import numpy as np def line_through(p, q): (x1, y1), (x2, y2) = p, q m = (y2 - y1) / (x2 - x1) return m, y1 - m * x1 print(line_through((1, 3), (4, 9))) print(line_through((2, 7), (6, -1))) print(np.polyfit([2, 6], [7, -1], 1)) print(np.polyfit([0, 1, 2, 3], [1.1, 2.9, 5.2, 6.8], 1))

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Where a line crosses the $x$ -axis

On the $x$ -axis, $y = 0$ . So the crossing, the line's zero from lesson 1.1.1, solves $mx + c = 0$ , which gives $x = -\frac{c}{m}$ .

For $y = -2x + 11$ the crossing is at $x = 5.5$ . For the candle, $-1.5t + 24 = 0$ gives $t = 16$ : the candle burns out after 16 hours.

A flat line breaks the formula, because $m = 0$ would put zero on the bottom. The line $y = 4$ never meets the $x$ -axis, and $y = 0$ is the axis itself.

-\frac{c}{m}

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Where two lines meet

Two lines meet where they give the same $y$ for the same $x$ . So set the two right-hand sides equal and solve, as in lesson 0.3.2. For $y = 2x + 1$ and $y = -x + 7$ :

\begin{aligned} 2x + 1 &= -x + 7 \\ 3x &= 6 \\ x &= 2 \end{aligned}

Then $y = 2 \times 2 + 1 = 5$ , so they meet at $(2, 5)$ . The other line agrees: $-2 + 7 = 5$ .

Lines with different slopes cross at one point. Lines with the same slope stay the same distance apart, here 3, so they never meet.

Rule: Where two lines meet

m_1 x + c_1 = m_2 x + c_2

Different slopes: exactly one meeting point. Same slope and different intercepts: parallel lines, which never meet. Same slope and same intercept: one line written twice, so every point is shared.

For the parallel lines $y = 2x + 1$ and $y = 2x - 2$ , the equation is $2x + 1 = 2x - 2$ . Taking $2x$ from both sides leaves $1 = -2$ , which is false for every $x$ , so there is no meeting point.

m_1 x + c_1 = m_2 x + c_2

def meet(m1, c1, m2, c2): if m1 == m2: return None x = (c2 - c1) / (m1 - m2) return x, m1 * x + c1 print(meet(2, 1, -1, 7)) print(meet(2, 1, 2, -2))

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Lesson 1.6.1 joins data points with lines, and lesson 2.2.1 uses rise over run to measure how fast a curve changes. Case study 1.2.4 meets a line in finance.

Common mistakes

Use the same order on top and bottom. $\frac{y_2 - y_1}{x_1 - x_2}$ has the wrong sign.
Rise over run, not run over rise. For $(1, 3)$ and $(4, 9)$ , $\frac{3}{6} = 0.5$ is upside down; the slope is 2.
Watch the signs in $c = y_1 - m x_1$ . With $m = -2$ and the point $(2, 7)$ , $c = 7 + 4 = 11$ , not $7 - 4 = 3$ .
The intercept $c$ is on the $y$ -axis, at $x = 0$ . The crossing of the $x$ -axis is $-\frac{c}{m}$ .
Make $y$ the subject before reading the slope. The slope of $3x + 2y = 6$ is $-1.5$ , not 3.
Flat and vertical are different. A flat line has slope 0. A vertical line has no slope and cannot be written as $y = mx + c$ .
Equal slopes mean parallel lines. Solving gives a false statement such as $1 = -2$ , so the lines never meet.

Lines

0 of 12 correct

1.
Find the slope of the line through $(2, 5)$ and $(6, 13)$
2.
Find the slope of the line through $(-1, 4)$ and $(3, -4)$
3.
Find the slope of the line through $(-3, 5)$ and $(4, 5)$
4.
A candle is 30 cm tall and 18 cm tall 4 hours later. What is the slope of its height, in cm per hour?
5.
A line has slope 3 and passes through $(1, 7)$ . Find its intercept $c$
6.
Write the line through $(-1, 1)$ and $(3, 9)$ in the form $mx + c$
7.
A line has slope $-2$ and passes through $(3, 1)$ . Write it in the form $mx + c$
8.
On $y = 4x - 2$ , by how much does $y$ change when $x$ goes from 1 to 3.5?
9.
Where does $y = -2x + 8$ cross the $x$ -axis? Give $x$
10.
What is the slope of $3x + 2y = 6$ ?
11.
$y = 3x - 4$ and $y = x + 2$ meet at one point. Give its $x$
12.
Write the line parallel to $y = -3x + 2$ through $(1, 4)$ in the form $mx + c$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

m = \frac{y_2 - y_1}{x_2 - x_1}

Lines: slope and intercept

What a slope measures

Slope from two points: rise over run

Rule: Slope from two points

Rising, falling and flat

Vertical lines have no slope

The equation of a line

Rule: Line through two points

From one point and a slope

Rule: Point and slope

Lines written another way

Where a line crosses the $x$ -axis

Where two lines meet

Rule: Where two lines meet

Common mistakes

Lines

Summary card

Free further reading

What a slope measures

Slope from two points: rise over run

Rule: Slope from two points

Rising, falling and flat

Vertical lines have no slope

The equation of a line

Rule: Line through two points

From one point and a slope

Rule: Point and slope

Lines written another way

Where a line crosses the xxx-axis

Where two lines meet

Rule: Where two lines meet

Common mistakes

Summary card

Free further reading

Where a line crosses the $x$ -axis