➕ Algebra 1 · Linear Functions
Graphing Linear Equations: fun ways and memory help
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
A cookie price list
A bakery sells cookies for 2 dollars each and charges 1 dollar for the bag. The cost y of x cookies is y = 2x + 1. The price list is a table of values: 1 cookie costs 3 dollars, 2 cookies cost 5, 3 cookies cost 7, and 4 cookies cost 9.
How it connects: Each row of the price list is a point: (1, 3), (2, 5), (3, 7), (4, 9). They line up because the cost rises by the same 2 dollars for every cookie, which is the slope. The 1-dollar bag is the starting charge: the y-intercept is 1, and the line crosses the y-axis at (0, 1). Limit: you probably buy whole cookies only, so the story uses just some points on the line. The math graph includes every x.
The scout on the map
In a strategy game, a scout starts at map position (0, 2). Each move takes it 3 squares right and 1 square up, so it stands at (3, 3), then (6, 4), then (9, 5).
How it connects: Repeating the same rise-and-run step keeps you on one straight line. Here the slope is 1/3 and the y-intercept is 2 (the scout starts at (0, 2)), so all three positions satisfy y = (1/3)x + 2. That is exactly how you graph from the slope and the y-intercept.
The jogger’s graph
A jogger runs at a steady pace, so her distance (y) rises along a straight line as time (x) passes. Then she stops at a water station. Time keeps going, but her distance stays the same, so that stretch of the graph is perfectly flat.
How it connects: A flat stretch is like y = b: the same y-value for every x, with slope 0 (HOY). A vertical stretch would mean the distance changes while no time passes, which no runner can do. That is the shape of x = a, whose slope is undefined (VUX). Limit: on a motion graph, time always moves left to right. The equation x = a is perfectly good math, but it cannot describe a runner.
The metronome
A metronome set to 80 beats per minute clicks 80 times each minute. After x minutes, the total is y = 80x clicks: 80 after 1 minute, 160 after 2, and 240 after 3.
How it connects: The line passes through the origin (0, 0), because 0 minutes means 0 clicks, and its slope is 80 clicks per minute. Both intercepts are 0, so the line crosses both axes at the same point, (0, 0), and the intercept method gives only one point. Use the slope or a table to get a second one. Limit: a metronome cannot run for negative minutes, so the story uses only the part of the line to the right of the y-axis.
Measure twice, cut once
A carpenter’s proverb says to check your measurements twice before you make a cut you cannot undo.
How it connects: Two points are enough to draw a line, but plot a third before you pick up the ruler. If the three points do not line up, there is a slip in your arithmetic or your plotting, and you catch it before the line is drawn.
X marks the spot
On a treasure map, a mark shows the exact place where the treasure is buried. We say “X marks the spot” when we point out an exact location.
How it connects: On a coordinate grid, a spot needs two numbers, (x, y). An equation is the clue: a spot is on the line only if its two numbers make the clue true, so substitute to test it. One number alone, like x = 4, is not a single spot. It is a whole vertical line of spots.
The stubborn line
Why is the line x = 4 so stubborn? No matter what y says, x refuses to change.
How it connects: In x = 4, every point has x-coordinate 4, while y can be anything. That makes a vertical line through (4, 0) with an undefined slope.
Memory help
Walk, then climb
In (x, y), x comes first: walk along the hallway (right or left) before you climb the stairs (up or down). The point (3, −2) means walk 3 to the right, then go 2 down.
SIT down and pick a method
When you need points for a line, SIT: Slope-intercept (plot (0, b), then rise and run), Intercepts (set one letter to 0), Table (choose x, find y).
The axis rhyme
On the y-axis wall, x is zero, that’s all. On the x-axis floor, y is zero, no more. So for the y-intercept, make x zero. For the x-intercept, make y zero.
Draw the staircase
From your starting point, sketch a small staircase. The sideways step is the run and the up-or-down step is the rise. Write the signs on them before you plot. For slope −3/4, the stair goes right 4 and down 3.
A negative slope goes down as you move right
A slope of −3/4 means down 3 and right 4 (or up 3 and left 4). Going up 3 and right 4 gives the slope +3/4, a different line that tilts the other way. Always read a line from left to right.
The rope and the nails
Pull a rope tight between two nails. A tight rope is straight, so two nails are all it takes to fix it in place. Now hold a third nail against the rope. If the rope touches it, all three nails are lined up. If it misses, one nail is in the wrong place. In the same way, two points fix a line, and a third point checks your work.