Look at y=2x+3 and at first it seems like just a string of letters and numbers, but graph it on a coordinate plane and a sloped line appears instantly. I think linear functions are the unit where students first really feel, for the first time, that an "equation" and a "picture" in math are exactly connected.
In the linear function y=ax+b, a is called the slope and b the y-intercept. The slope a decides how steep the graph is, and whether it rises to the upper right (a positive) or falls to the lower right (a negative). The y-intercept b sets the point where the graph meets the y-axis — in other words, the height the graph starts at. The core idea of a linear function is that knowing just these two numbers lets you draw the line perfectly.
What students get mixed up on most often is the sign and size of the slope. They understand that a bigger slope means steeper, but they get confused about which way the graph tilts when the slope is negative. Here's how I explain it: "if y rises as x moves to the right, the slope is positive; if y falls as x moves to the right, the slope is negative." And if you get them to picture the slope number as the height of a stair step, it clicks intuitively that a bigger number means a steeper staircase.
Linear functions come up constantly for expressing "situations that change at a constant rate" in daily life — calculating a taxi fare, a cell phone plan, how fast a water tank fills up. When comparing which of two rate plans is the better deal, overlaying both graphs and finding where they cross is also a classic use of linear functions.
Just as important as the y-intercept on a linear function's graph is the x-intercept. The x-intercept is the point where the graph meets the x-axis, and at that point the y-value is always 0. Working from the equation, you can find the x-intercept by plugging 0 in for y in y=ax+b and solving for x. This x-intercept, as it happens, is exactly the same value as the solution to the linear equation ax+b=0. In other words, finding the point where the line crosses the x-axis on the graph, and solving the equation by hand, give you the exact same answer. Once you understand this connection, you realize that equations and functions aren't separate units at all — they're two different ways of looking at the same problem. Going back and forth between checking it visually on the graph and working it out by hand with the equation builds a much sturdier understanding.
Learn the slope and y-intercept as pure numbers and they're easy to forget quickly, but adjust them yourself with sliders and watch the graph move in real time, and the intuition sticks around much longer. On our activity page, you can freely change both the slope and the y-intercept with sliders and immediately see how the line changes. Try predicting what will happen if you set the slope to 0, or make it negative, before checking with the slider — approach it this way, and you're not just looking at a graph, you're practicing genuine mathematical thinking: predicting and then verifying. Follow that up by drawing two lines at once and finding where they intersect, and when you get to systems of equations later, the image of "the point where two graphs meet" will already come to mind, making the new topic click much more easily.