➕ Algebra 1 · Linear Functions
Point-Slope Form: 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.
The mid-ride checkpoint
A cyclist starts a fitness timer partway along a route. When the timer shows 40 minutes, a sign says she has gone 11 kilometers along the route. She rides at a steady 0.25 kilometers per minute. At the 60-minute mark she has gone 11 + 0.25 × 20 = 16 kilometers.
How it connects: Let t be the minutes on the timer and d the kilometers along the route. The checkpoint is the point (40, 11) and the slope is 0.25, so d − 11 = 0.25(t − 40). The factor t − 40 is the time since the checkpoint, and 0.25(t − 40) is the distance gained since then. The limit: the equation only works while the pace stays the same. Real riders speed up and slow down.
The glider’s descent
In a made-up game, you glide down toward the ground. At the 20-second mark your altitude is 300 meters, and you lose 15 meters every second. At the 30-second mark you are at 300 − 15(10) = 150 meters.
How it connects: Let x be the seconds and y the altitude in meters. The slope is −15 and the known point is (20, 300), so y − 300 = −15(x − 20). The minus sign in front of the 15 belongs to the slope (the altitude is falling). The signs inside the parentheses come only from the point. The limit: the formula stops fitting at ground level, where y = 0, which is at x = 40 seconds.
Pick up where you left off
To “pick up where you left off” means to continue something from the exact place where you stopped.
How it connects: In point-slope form, (x1, y1) is where you left off, and the slope says how to continue from there: y − y1 = m(x − x1) starts at y1 and adds the slope times how far x has moved. You never have to walk back to the y-axis first. The limit: a line also extends backward from the known point, so the equation describes the points before it as well as the points after it.
You can’t judge a book by its cover
This proverb says that how something looks on the outside does not tell you everything about what is inside.
How it connects: In x + 3, the plus sign makes it look like the point is about positive 3. Look inside: x + 3 = x − (−3), so x1 = −3. Rewrite it as a subtraction before you read the point. The limit: the cover is not random. In x + 3 or y − 5, the sign you see is the opposite of the coordinate’s sign, so one fixed rule (flip the sign) reads it correctly.
A cooling evening
On a clear evening the temperature is 18 degrees Celsius at 2 pm and 10 degrees Celsius at 6 pm. Use a 24-hour clock, so 2 pm is t = 14 and 6 pm is t = 18. At 8 pm (t = 20) the temperature is 6 degrees Celsius.
How it connects: Two readings give two points, (14, 18) and (18, 10), so find the slope first: (10 − 18) ÷ (18 − 14) = −2 degrees per hour. Then use either point: T − 18 = −2(t − 14). Using (18, 10) gives T − 10 = −2(t − 18), the same line. At t = 20 both give 6: 18 − 2(6) = 6 and 10 − 2(2) = 6. The limit: real temperatures do not fall at a perfectly steady rate all night, so the line is only a rough model for a few hours.
The point and the slope
A point says, “I know where I am, but not where I’m going.” A slope says, “I know where I’m going, but not where I am.” They team up, and now there is a line.
How it connects: One point fits endless lines through it, and one slope fits endless parallel lines. A point plus a slope pins down exactly one line, which is why y − y1 = m(x − x1) needs both. The limit: this works when the slope is a number, so it leaves out vertical lines.
Memory help
Flip the sign
“Flip the sign by the x and the y, and the point is easy to spy.” (This is for the form y − y1 = m(x − x1).) For y + 4 = 3(x − 6), the y + 4 flips to y1 = −4, and the x − 6 flips to x1 = 6. The point is (6, −4). The slope 3 is not flipped: it keeps the sign in front of it.
Slope, Pick, Plug
With two points, do three things in order: find the Slope, Pick one point, Plug into y − y1 = m(x − x1). Say “Slope, Pick, Plug.” If you skip the first step, there is no m to plug in.
Since the checkpoint
Picture (x1, y1) as a checkpoint on a trail. The factor x − x1 is how far you have moved sideways since the checkpoint. Multiply by the slope m and you get how much you climbed (or dropped) since then. That is exactly y − y1. So the equation says: climb since the checkpoint = slope × sideways move since the checkpoint.
A pin and an arrow
Draw a pin at (x1, y1) and a small staircase for the slope: across by the run, then up or down by the rise. The line goes through the pin along the staircase, in both directions.
The letters x and y stay
In y − 5 = 2(x − 3), do not put 3 in place of x or 5 in place of y. Those letters stand for every point on the line. Only the subscripted x1 and y1 get numbers. If you replace x and y too, you just get 0 = 0, which is true but is no longer an equation of a line.
The slope keeps its own sign
In y + 2 = −3(x + 5), the slope is −3. The plus signs inside make x1 and y1 negative, but they do not change the slope.