Functions and Coordinate Geometry Basics: Linear Functions and Proportion
What you must be able to do
In the coordinate plane, a point is written P(x, y): x is horizontal, y is vertical. Know all four quadrants and their sign patterns.
A linear function has the form y = kx + b. The graph is a straight line; k is the slope (rise over run) and b is the y-intercept, where the line crosses the y-axis. A line rising from left to right has k > 0; falling, k < 0; horizontal, k = 0. Steeper lines have larger |k|. Two lines are parallel exactly when their slopes are equal.
Direct proportionality y = kx is the special case b = 0: the graph passes through the origin, and the ratio y/x is constant. Inverse proportionality y = k/x is a curve, not a line, and its graph never touches the axes.
To find the equation of a line through two points, compute k = (y₂ − y₁)/(x₂ − x₁), then substitute one point to find b.
Traps that appear in questions
- Swapping the coordinates and plotting (y, x).
- Computing slope as Δx/Δy instead of Δy/Δx.
- Forgetting that a proportional relationship must pass through the origin.
- Confusing the slope with the intercept when reading a formula.
- Sign errors when subtracting negative coordinates.
Rule of thumb
To read a graph, find two lattice points on the line, count rise and run, and check the intercept against the formula.
Worked example
A line passes through A(1, 5) and B(3, 11). Slope k = (11 − 5)/(3 − 1) = 3. Using A: 5 = 3·1 + b, so b = 2. The line is y = 3x + 2.