Graphs
Graphs of functions
Work out a few pairs of values, plot them, and join them up. A linear function always gives a perfectly straight line.
Building a table of values
Take the function y = x + 3. Choose some values of x and work out y for each: x = −4 → y = −4 + 3 = −1 x = −3 → y = 0 x = −2 → y = 1 x = 0 → y = 3 x = 2 → y = 2 + 3 = 5 A table keeps this tidy and makes the pattern visible.
Turning the table into coordinates
Each pair of values is a COORDINATE, written (x, y). From the table: (−4, −1), (−3, 0), (−2, 1), (0, 3), (2, 5). The x-value always comes first — along the corridor, then up the stairs.
Plotting and joining
Plot each point on a grid, then join them with a straight line using a ruler. These are LINEAR functions, so the points always lie in a perfectly straight line. If one of your points is off the line, you have made an arithmetic slip — go back and check it. That is a useful self-check no other topic gives you.
Reading values off the graph
Once the line is drawn you can read off values you never calculated. To find y when x = 1: go up from 1 on the x-axis to the line, then across to the y-axis. Extend the line and you can read values beyond your table too.
Worked examples
y = x + 3 when x = 2
- Substitute x = 2 into the rule.
- 2 + 3 = 5.
Answer: y = 5, giving the point (2, 5)
y = x + 3 when x = −4
- Substitute x = −4.
- −4 + 3 = −1.
Answer: y = −1, giving the point (−4, −1)
Building a table for y = 2x
- x = 0 → y = 0.
- x = 1 → y = 2.
- x = 3 → y = 6.
Answer: (0, 0), (1, 2), (3, 6)
Spotting a mistake
- Four points lie on a straight line but one is off it.
- A linear function must give a straight line.
- So that point was worked out wrongly.
Answer: Recalculate the point that is off the line
Where does y = x + 3 cross the y-axis?
- The y-axis is where x = 0.
- Substitute x = 0: y = 0 + 3.
Answer: It crosses at (0, 3)
Where you see this in real life
- A currency conversion graph lets you read off any amount without calculating.
- A mobile phone tariff drawn as a line shows the cost for any number of minutes.
Key words
- Linear function
- a function whose graph is a straight line
- Table of values
- a table pairing values of x with values of y
- Coordinate
- a pair of numbers giving a position, written (x, y)
- Plot
- to mark a point on a grid
- Axis
- one of the two number lines on a grid