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Linear Functions

Middle schoolMathematicsExact graph of a linear function

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About this experiment

A linear function has a straight-line graph, and two numbers decide everything about it: the slope, which says how steep the line is, and the y-intercept, which says where it crosses the vertical axis. Drag the points or move the sliders and watch the equation, the slope triangle and the table of values change together. The notation switch matches the letters used in your country, whether that is y = mx + b, y = mx + c, y = ax + b or y = kx + b.

Learning goals

  • Identify the slope and y-intercept from the equation of a line
  • Calculate the slope as rise over run from any two points on the line
  • Describe how the sign and size of the slope change the graph
  • Connect the constant difference in a table of values to the slope

How to use it

  1. 1Drag the black point up and down. Which part of the equation changes, and what happens to the line?
  2. 2Drag the blue point to make the line steeper, flatter and then falling. Read the rise and run each time.
  3. 3Turn on the table of values. By how much does y change each time x goes up by 1?
  4. 4Set the slope to 0, then to a negative value, and describe the line in words.

The model

Slope-intercept form. Other countries write y = mx + c, y = ax + b or y = kx + b.
Slope as rise over run between any two points
Where the line crosses the x-axis, when m is not zero
The slope is the tangent of the angle the line makes with the x-axis

Assumptions

  • The graph is drawn exactly from the equation. Dragged points snap to half units so the numbers stay easy to read.
  • Vertical lines such as x = 3 are not functions of x and cannot be written in this form, so the slope is limited to the range from -5 to 5.

Try this in class

  1. Q1A line has slope 2 and passes through (0, -3). Write its equation and check it on the graph.
  2. Q2Two lines have the same slope but different intercepts. What do you notice about them?
  3. Q3Why does the slope come out the same whichever two points of the line you use?
  4. Q4A taxi charges a fixed fee plus a price per kilometre. Which number is the slope and which is the intercept?

Frequently asked questions

What does a negative slope mean?

The line falls from left to right: as x increases, y decreases. The bigger the number without its sign, the steeper the fall.

Why do different countries use different letters?

The idea is the same everywhere. The US and Germany usually write y = mx + b, the UK writes y = mx + c, France, Japan and Korea use y = ax + b, and China uses y = kx + b.

Is y = 3 a linear function?

Yes. It is a horizontal line with slope 0 and y-intercept 3. Every x gives the same y.