Middle school maths, year 2 · unit 5
Linear Functions
A money box starts with 3000 won and gets 1000 won every week. How much is in it after some weeks is fixed as soon as you choose the number of weeks. A relationship like this, where fixing one thing fixes another, is a function. We start with a machine that takes numbers in.
One in, one out
A board to arrangeHere is a machine that changes whatever number goes in according to its rule. Drag the blue chips into it and collect what comes out in the table.
Table to points, points to a line
A board to arrangeEach x, y pair in a table is one point on the plane. Drag each ordered-pair card to its place.
b moves it up and down
A board to adjustHow is y = 2x + 3 different from y = 2x? Drag the gold handle on the y-axis to move the line up and down.
a is the height of each step
A board to adjustThe slope is how far the line climbs for one square to the right. Drag the gold handle to make the line steeper or flatter.
Two points draw the line
A board to arrangeTwo points fix a straight line. Pin two points that fit the equation and draw the graph.
Try it yourself
Use what you found on the boards. The numbers change every time and the easy questions come first. Get three right and the lamp lights.
What to take away from this unit
- When choosing x fixes exactly one y, y is a function of x. If it has the form y = ax + b (a not 0), it is a linear function.
- The graph of a linear function is a straight line, because y changes by the same amount for every 1 in x.
- The graph of y = ax + b is the graph of y = ax translated by b in the y-direction. The y-coordinate where it meets the y-axis, b, is the y-intercept.
- Slope = (change in y) ÷ (change in x) = a. Positive, and the line runs uphill to the right; negative, and it runs downhill.
- The x-coordinate where the line meets the x-axis is the x-intercept. Find it by putting y = 0.
- To draw the graph, find two points that fit and join them: use the two intercepts, or start at the y-intercept and take one step of the slope.