Quarry School

Draw a line by plotting points

Explain it like I am five

Picture a city map. A location needs two instructions: how far across, then how far up. A coordinate pair, such as (2, 5), records those two instructions in that order.

A function is a rule that gives one output for each input. Its graph is a picture of all the input and output pairs. You put the input on the horizontal x-axis and the output on the vertical y-axis.

A linear function gives a straight line. You can mark a few addresses that obey its rule and draw the line through them. Two different addresses locate the line. A third address helps you notice a calculation mistake before you draw.

−2−11234−3−2−1123456789(0, 0)(−0, 0)(0, 0)(1, 2)(2, 4)
The coordinate grid acts like a map: every marked point has an across position and a height.
Reminder
  • Coordinate pairs. (−2, 3) means left 2, then up 3; the input is first.
  • Function notation. f(3) asks for the output at input 3; if f(x) = 2x, then f(3) = 6.
  • Order of operations. Multiply before adding: 2 × 3 + 1 = 7.
  • Fraction multiplication. 23 × 6 = 123 = 4; multiply the top, then divide.
  • Signed addition. −4 + 5 = 1 because five forward steps undo four backward steps.
Why it works. For f(x) = mx + b, changing x by any amount changes the output by m times that amount. Every step right 1 changes height by m. Repeating that same step traces a straight line, so the graph never changes its slant. Two distinct points determine one straight line. A third calculated point must fall on it if the arithmetic is correct. Connecting only the marked points would leave out other accepted inputs, so an unrestricted linear function extends in both directions.
RuleRule: Graph a linear function by plotting at least two distinct coordinate pairs (x, f(x)) and drawing the entire straight line through them. A third point checks the arithmetic.
The same idea, five ways
Say it

Say: put an input in, find its output, and mark their address.

Write it

The graph contains every point whose output obeys the function rule.

In math
  • y = f(x)
  • (x, f(x))
  • f(x) = mx + b
Like

A street map locates an address with two directions.

See it
−224−22468(0, 0)(−0, 0)(0, 0)(1, 2)(2, 4)
The coordinate grid acts like a map: every marked point has an across position and a height.
The same idea, other ways
As a map

The point (1, 2) means one unit across and two up. For f(x) = 2x, the address (2, 4) also fits, because twice 2 is 4.

2246(0, 0)(−0, 0)(1, 2)(2, 4)
The marked points give addresses on the same straight line.
As a machine

Each chosen input makes one output. Plotting is recording the machine's answers on a map.

3multiply by 26inputoutput
The machine's answer makes the point (3, 6).
Strategy: step by step
  1. 1. Choose input values. If the slope contains a fraction, try multiples of its denominator. Multiples are results of multiplying by whole numbers: for a denominator of 3, use 0 × 3 = 0, 1 × 3 = 3 and 2 × 3 = 6.
  2. 2. Substitute each input into the function to find its output.
  3. 3. Record the pairs in a values picture, then plot each input across and output up.
  4. 4. Draw a straight line through the points and extend it both ways.
  5. 5. If a third point misses the line, recalculate before drawing.
Strategy
Strategy: graph from a formula by choosing points
1
Does the coefficient of x have a fractional denominator?
YesChoose multiples of that denominator to keep the products whole.
NoChoose small inputs, including 0.
↓
2
Do all three calculated points lie on one straight line?
YesDraw the line through them.
NoRecheck signs and substitution in each output.
  1. Choose convenient inputs.
  2. Compute one output per input using multiplication before addition.
  3. Plot at least two distinct points and check a third.
  4. Extend the straight line through the points.
Worked examplePlot the verified line

You need to draw all the input and output pairs of f(x) = −23x + 5. Use convenient inputs to locate its line.

input xoutput f(x)053361
Each column keeps one input paired with the output found from it.
−4−22468−22468(0, 5)(7.5, 0)(0, 5)(3, 3)(6, 1)
The marked points give addresses on the same straight line.
  1. Choose inputs 0, 3 and 6. Look at their columns in the values picture.Each is a multiple of 3, which cancels the bottom of the slope fraction when multiplied.
  2. f(0) = −23 × 0 + 5 = 0 + 5 = 5, giving (0, 5).Substitution replaces x with 0. Any number times 0 is 0.
  3. f(3) = −23 × 3 + 5 = −2 + 5 = 3, giving (3, 3).The 3 cancels the bottom 3; do the multiplication before adding 5.
  4. f(6) = −23 × 6 + 5 = −4 + 5 = 1, giving (6, 1).Six contains two groups of 3, so the product is −4.
  5. Plot those three points and draw one straight line extending through them in both directions.Every allowed input needs a point, including inputs between and beyond the three chosen ones.
Answer
The line passes through (0, 5), (3, 3) and (6, 1).
Check The two moves each go right 3 and down 2. They have the same change, confirming that the third point fits the line.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: for f(x) = 2x, use (2, 2) because the input is 2.
The second coordinate must be the output. Substituting 2 gives f(2) = 4.
✓ Instead: Plot (2, 4). The point (2, 2) belongs to a different rule.
Tips and tricks
  • Tip: Choose 0 as one input when possible. Multiplication by 0 leaves the starting value.
Trap. Plotting (3, 3) as output first can hide the mistake because both numbers match. Practice with (0, 5): across 0, then up 5. The input always comes first.