Linear functions: the straight line y = ax + b

Say a taxi charges 20 SRD to get in. After that you pay 5 SRD per kilometre. A ride of 7 km then costs 20 + 5 × 7 = 55 SRD. Every extra kilometre costs exactly 5 SRD more. Plot the price against the distance and you get a straight line.

A formula that gives a straight line is called a linear function. In this article you see what the numbers in such a formula do. Then you set up a formula like that yourself.

Two numbers: a and b

You can write every linear function like this:

y = ax + b

• b is the y-intercept. Fill in x = 0: then y = a × 0 + b = b. So the line goes through (0, b).
• a is the slope. It tells you how steep the line is. Take 1 step to the right and you go a steps up.

Slide a and b below. The graph starts at y = 2x − 1. The dashed step starts at (0, b): 1 step across, then a steps up.

Slide a and b

a = 2b = −1

y = 2x − 1

• Graph
• 1 across, a up
• y-intercept

Slope

a = 2

y-intercept

(0, −1)

The line

rises

Show the numbers

x	y
−3	−7
−2	−5
−1	−3
0	−1
1	1
2	3
3	5

This is what you see:

• If a is positive, the line rises from left to right.
• If a is negative, the line falls. One step to the right then takes you down.
• If a = 0, the line is flat. The formula is then y = b.
• The further a is from 0, the steeper the line.
• b does not change the slope. b moves the whole line up or down.

From a table to a line

For y = 2x − 1 you make a table. Pick a few values for x and work out y:

x = 0: y = 2 × 0 − 1 = −1
x = 1: y = 2 × 1 − 1 = 1
x = 2: y = 2 × 2 − 1 = 3

Plot the points (0, −1), (1, 1) and (2, 3) and draw a line through them. Two points are enough for a straight line. The third point is a check. If it is not on the line, there is a mistake somewhere.

Now look at the y-values: −1, 1 and 3. Each time 2 is added. That is no accident. When x goes up by 1, y goes up by exactly a. On a straight line that step is always the same size.

It works the other way round too. This is how you find the formula for a table:

• The step in y, when x goes up by 1 each time, is a.
• The y at x = 0 is b.

Pick a table below and look at the row of steps.

Equal steps give a straight line

Table ATable BTable C

• Line
• Points from the table

x	0	1	2	3	4
y	−1	1	3	5	7
step		+2	+2	+2	+2

The same step every time, so a straight line: y = 2x − 1

Table A has step +2 every time, and x = 0 goes with y = −1. That gives y = 2x − 1. Table B has step −0.5 every time and starts at 3. That gives y = −0.5x + 3. Table C has the steps −1, +1, +3 and +5. Those steps are not equal. The points lie on a curve, not on a line.

The line through two points

A test often asks this: give the formula of the line through A(1, 4) and B(3, 10). That takes two steps.

Step 1: work out a. See how far you go across and how far up. Across is called Δx and up is called Δy. You say delta x and delta y.

Δx = 3 − 1 = 2
Δy = 10 − 4 = 6
a = Δy ÷ Δx = 6 ÷ 2 = 3

Step 2: work out b. Fill in a and one point in y = ax + b. Take A(1, 4):

4 = 3 × 1 + b
4 = 3 + b
b = 1

The formula is y = 3x + 1. Check with the other point: 3 × 3 + 1 = 10. That is right.

Drag A and B below to other places. The formula follows.

Drag the two points

y = 3x + 1

• Line through A and B
• Δx across, Δy up

A

(1, 4)

B

(3, 10)

Δx

3 − 1 = 2

Δy

10 − 4 = 6

Slope

a = 6 ÷ 2 = 3

y-intercept

b = 4 − 3 × 1 = 1

A falling line works the same way. Take A(−1, 5) and B(2, −1):

Δx = 2 − (−1) = 3
Δy = −1 − 5 = −6
a = −6 ÷ 3 = −2

Fill in A: 5 = −2 × (−1) + b. Then 5 = 2 + b, so b = 3. The formula is y = −2x + 3. The line falls, so a is negative.

A formula from a story

Back to the taxi. You pay 20 SRD to get in and 5 SRD per km. Call the distance x, in km. Call the price y, in SRD:

y = 5x + 20

The fixed amount at the start is b = 20. The amount per kilometre is a = 5. This works for every story with a fixed start and a fixed step. The starting amount is b. What is added per step is a.

The price of a taxi ride

Distance: 7 km

• Price of the ride
• Getting in: 20 SRD

Price

20 + 5 × 7 = 55 SRD

Show the numbers

Distance	Price
0 km	20 SRD
2 km	30 SRD
4 km	40 SRD
6 km	50 SRD
8 km	60 SRD
10 km	70 SRD
12 km	80 SRD

It works the other way round too. Say you have 60 SRD. Then you solve this equation:

5x + 20 = 60
5x = 40
x = 8

You get 8 km. Set the slider to 8 km to check.

Where two lines cross

Taxi B charges 8 SRD to get in and 7 SRD per km. That gives y = 7x + 8. Taxi B is cheaper to get in, but more expensive per kilometre. At one distance they cost the same. That is where the two lines cross.

Set the two formulas equal to each other:

5x + 20 = 7x + 8
20 − 8 = 7x − 5x
12 = 2x
x = 6

At 6 km both cost 5 × 6 + 20 = 50 SRD. The point where they cross is (6, 50). For shorter rides taxi B is cheaper. For longer rides taxi A is cheaper.

Change the prices of taxi B below.

Taxi A against taxi B

Taxi B, getting in: 8 SRDTaxi B, per km: 7 SRD

• Taxi A: y = 5x + 20
• Taxi B: y = 7x + 8
• Crossing point

Equation

5x + 20 = 7x + 8

Crossing point

(6, 50)

Up to 6 km, Taxi B is cheaper. After that, Taxi A is cheaper.

Show the numbers

Distance (km)	Taxi A	Taxi B
0	20 SRD	8 SRD
2	30 SRD	22 SRD
4	40 SRD	36 SRD
6	50 SRD	50 SRD
8	60 SRD	64 SRD
10	70 SRD	78 SRD
12	80 SRD	92 SRD

Set taxi B to 5 SRD per km. Then both lines have the same a. They are parallel and never cross. You can see that in the equation too. Take a starting price of 8 SRD:

5x + 20 = 5x + 8
20 = 8

That is impossible. There is no solution, so there is no crossing point either.

Lines that are perpendicular to each other follow a fixed rule too. Multiply the two slopes and you get −1. Take y = 2x + 1 and y = −0.5x + 4. Then 2 × (−0.5) = −1. These two lines are perpendicular.

Try it yourself