➕ Algebra 1 · Linear Functions
Slope-Intercept 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.
Head start versus speed
Maya has 30 dollars saved and adds 5 dollars each week. Her friend Jordan has 10 dollars saved and adds 10 dollars each week. Maya starts ahead, but Jordan saves faster. After 4 weeks they both have 50 dollars, and after that Jordan stays ahead.
How it connects: Maya: y = 5x + 30. Jordan: y = 10x + 10. The y-intercept is the head start, and the slope is how fast you gain. Check week 4: 5(4) + 30 = 50 and 10(4) + 10 = 50. The limit: the lines only describe the story while each friend keeps saving the same amount.
Pizza by the topping
A pizza shop charges 9 dollars for a plain pizza and 1.50 dollars for each topping. The cost in dollars is y = 1.5x + 9, where x is the number of toppings. Four toppings cost 1.5(4) + 9 = 15 dollars.
How it connects: The y-intercept b = 9 is the price with 0 toppings (where x = 0), and the slope m = 1.5 is the rate of change: dollars per topping. The limit: you can only order whole toppings, so the real graph is a row of dots, not a full line.
Behind the starting line
A runner starts 18 meters behind the starting line and runs 6 meters per second. Measure her position from the line, counting positions past the line as positive. After x seconds her position is y = 6x − 18. She reaches the line when y = 0, which happens at x = 3 seconds.
How it connects: A negative y-intercept (b = −18) means you start on the negative side of the reference point, here 18 meters behind the line. The slope 6 is her speed in meters per second. Check: 6(3) − 18 = 0. The limit: the equation fits only from x = 0 while she keeps a steady 6 meters per second.
Walking to the treasure chest
In a made-up game, your character appears 36 steps from a treasure chest and walks toward it at 3 steps per second. The distance left after x seconds is y = −3x + 36.
How it connects: The slope −3 is negative because the distance shrinks as time moves right: 3 fewer steps each second. The y-intercept 36 is the starting distance. The limit: the formula only fits until the distance reaches 0, at x = 12 seconds. After that it would give negative distances, which make no sense for this story.
A journey of a thousand miles
The proverb says, “A journey of a thousand miles begins with a single step.” Big journeys start with one small move.
How it connects: Start drawing at (0, b). Every step of 1 to the right adds exactly m, the same amount for the first step as for the thousandth, because a line has a constant rate of change. The limit: real steps vary, but the steps along a line never do. And (0, b) is only where you start drawing: the line itself keeps going to the left of it too.
From scratch
To “start from scratch” means to begin with nothing already done or prepared.
How it connects: In a story, starting from scratch means a starting value of 0: b = 0, so the line passes through the origin (0, 0), like y = 9x. A story with b > 0 has a head start, and one with b < 0 starts below zero.
The vertical line at the door
A vertical line walks up to the Slope-Intercept Club. The doorkeeper says, “Sorry, members need an m.” The vertical line sighs, “My slope is undefined.” So it goes to the club next door, where every member’s name looks like x = 4.
How it connects: A vertical line has an undefined slope, so it has no number m and cannot be written as y = mx + b. Its equation is x = a. Horizontal lines are fine: y = b is the case m = 0.
Memory help
m for move, b for begin
m tells how the line moves (the slope). b tells where you begin drawing: the point (0, b) on the y-axis. Say it as: “Move with m, begin at b.” (The line itself keeps going both ways.)
Times the x
“Times the x, the slope you chart. All alone, that’s where you start.” In y = 4 − 3x, the number multiplying x is −3 (the slope), and the number all alone is 4 (where you start drawing). Careful: x/4 means (1/4)x, so its slope is 1/4, not 4.
Read it like a receipt
Read y = mx + b as a sentence: “Start with b, then add m for every 1 of x.” For y = 6x + 3, that says: start with 3, then add 6 for every 1 of x. If m is negative, “add m” really means you lose that much.
Draw the staircase
Plot (0, b) first. Then draw a staircase from that point: across by the run, then up or down by the rise. Each stair corner is another point on the line. Label the stairs before you plot.
The sign stays with its number
In y = 4 − 3x, the slope is −3, not 3, and b is 4. In y = 5x − 7, b is −7. Rewrite the equation in the order y = mx + b before you read it.
Vertical lines are not members
x = a cannot be written as y = mx + b, because its slope is undefined. Do not try m = 0: that makes a horizontal line, which is a different line.