Linear patterns, constant rate, introduction to linear functions
Middle School Bridge • 6-8
A taxi charges $3 to get in and $2 for every mile driven. After 1 mile you pay $5, after 2 miles you pay $7, after 3 miles you pay $9. Notice the pattern: the cost increases by the same amount ($2) every mile. This is a linear relationship -- and when you graph it, the points form a perfectly straight line.
A relationship between two quantities is linear if it has a constant rate of change. This means that for every equal increase in the input, the output increases (or decreases) by the same amount.
Is this data linear?
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y | 5 | 8 | 11 | 14 | 17 |
Every linear relationship can be described by two key features:
These combine into the equation of a line:
Here m is the rate of change (slope) and b is the starting value (y-intercept).
Using the taxi example: $3 starting charge plus $2 per mile.
Check: when x = 4, y = 2(4) + 3 = 11. After 4 miles, the cost is $11.
To graph a linear equation, create a table of values, plot the points, and draw a straight line through them. Since a linear relationship produces a straight line, you technically need only two points -- but plotting three or more helps verify accuracy.
A plant is 2 cm tall and grows 1.5 cm per week.
| Week (x) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Height (y) | 2 | 3.5 | 5 | 6.5 | 8 |
If the rate of change is NOT constant, the relationship is non-linear. For example, in the sequence 1, 4, 9, 16, 25 the differences are 3, 5, 7, 9 -- not constant. This is a quadratic pattern (perfect squares), not a linear one. Always check for a constant difference before assuming linearity.
Students sometimes confuse the rate of change with the starting value. In y = 2x + 3, the rate of change is 2 (the coefficient of x), not 3. The starting value is 3 (the constant term). The rate of change tells you the steepness of the line, while the starting value tells you where the line crosses the y-axis.
1. Is this relationship linear? x: 1, 2, 3, 4 and y: 10, 7, 4, 1.
Differences in y: 7 - 10 = -3, 4 - 7 = -3, 1 - 4 = -3. Constant rate of change of -3, so yes, it is linear.
2. Find the rate of change and starting value: a gym charges a $25 sign-up fee plus $10 per month.
Rate of change: $10 per month (m = 10). Starting value: $25 (b = 25). Equation: y = 10x + 25.
3. Write the equation of the line passing through the points (0, 4) and (3, 13).
Rate of change = (13 - 4) / (3 - 0) = 9 / 3 = 3. Starting value = 4 (the y-value when x = 0). Equation: y = 3x + 4.
4. Using y = 5x - 2, find y when x = 6.
y = 5(6) - 2 = 30 - 2 = 28.
5. A candle is 30 cm tall and burns at a rate of 2 cm per hour. Write an equation for its height after x hours. When will the candle be completely gone?
Height: y = 30 - 2x (or y = -2x + 30). The candle is gone when y = 0: 0 = 30 - 2x, so 2x = 30, x = 15 hours.
A linear relationship has a constant rate of change. It can be described by the equation y = mx + b, where m is the rate of change (slope) and b is the starting value (y-intercept). To identify a linear pattern, check whether the differences in the output values are constant. Linear relationships produce straight-line graphs on the coordinate plane.