Linear Functions Change by the Same Amount Every Step

Hey there! I'm Lumi, and today we're going on a mental hike. Imagine you walk at a perfectly steady pace — 3 meters every second, no speeding up, no slowing down. After 1 second you've gone 3 m. After 2 seconds, 6 m. After 3 seconds, 9 m. Every time the seconds go up by 1, the distance goes up by exactly 3. That equal jump is called the **rate of change**. When a function always changes by the same amount for each equal step in the input, we call it a **linear function**. If you plot those (time, distance) pairs on a grid, they land perfectly on a straight line — that's where the word 'linear' comes from (think: line). The rate of change is the same thing as the **slope** of that line. Slope tells you: for every 1 unit you move to the right on the graph, how many units do you move up (or down)? In our hike, slope = 3. You can find slope from a table by picking any two rows and dividing: **slope = (change in output) ÷ (change in input)** This works even when the input steps are not 1. For example, if x goes from 0 to 2 and y goes from 5 to 9, then slope = (9 − 5) ÷ (2 − 0) = 4 ÷ 2 = 2. Always divide the output change BY the input change — don't just read the output difference alone, and don't use an x-value as if it were the input gap. **Predicting with slope:** Once you know the slope, you can find any output. The value of y when x = 0 is called the **starting value** — in algebra this is also known as the **y-intercept**, because it is where the line crosses the vertical axis. Use this rule: output = starting value + (slope × input). Say slope = 3 and the starting value (y-intercept) is 4. To find the output at x = 5: 4 + (3 × 5) = 19. If you get stuck on a slope calculation, try writing out two rows side by side: top row minus bottom row for outputs (that's your Δy), and top row minus bottom row for inputs (that's your Δx). Then divide: slope = Δy ÷ Δx. Writing it out prevents the two most common mix-ups — using an input value itself instead of the gap, or forgetting to divide at all. The big idea: **constant rate of change = linear function = straight-line graph**. Hint: if the rate of change is not the same between every pair of rows in the table, the function is NOT linear. Watch for that in the activity!