the whole idea, in one sentence
Every line is the story of a starting point (y-intercept) and a rate (slope). Once you know those two, you know every point on the line.
No wrong choice. Try one. Switch if it's not clicking.
You sign up for a phone plan. It costs $30 every month. Plus a one-time $50 activation fee on day one.
Month 1 total cost: $50 + $30 = $80. Month 2: $50 + $60 = $110. Month 3: $140. Every month the total goes up by exactly $30.
Two things define this story:
Starting point: $50 (what you paid before any months passed). This is the y-intercept. We call it b.
Rate: $30 per month (how fast the total grows). This is the slope. We call it m.
Put them together and you get a formula that gives you the total cost (y) for any month (x):
y = 30x + 50
That's it. That's a linear function. m is how fast it changes (slope). b is where it started (intercept).
Any time you see "a fixed amount + a steady rate" — gym memberships, monthly bills, hourly pay with a sign-on bonus — it's a line. And it'll look like y = mx + b.
A line on a graph has two numbers that tell its whole story.
1. The y-intercept (b). Where does the line cross the y-axis? That's the line's starting point — what y is when x = 0.
2. The slope (m). How steep is the line? Slope is rise over run:
slope = rise / run = (change in y) / (change in x)
Picture it as stairs. To get from one point on the line to the next:
y
↑
| ● ← go RIGHT 1, then UP 2
| /
| ●
| /
| ●
|————————→ x
slope = 2/1 = 2 · for every 1 step right, line goes up 2
If the line goes up as you move right → positive slope. Down as you move right → negative slope. Flat → slope of 0. Straight vertical → undefined.
To find slope from two points (x₁, y₁) and (x₂, y₂):
m = (y₂ − y₁) / (x₂ − x₁)
Example: from (1, 3) to (4, 9): m = (9-3) / (4-1) = 6/3 = 2.
The equation of a line is almost always written like this:
y = mx + b
That's slope-intercept form. Two letters do all the work:
m = slope (the rate · how fast y changes when x changes)
b = y-intercept (the starting point · y when x = 0)
To read a line: Look at y = 3x + 5. m = 3 (slope is 3). b = 5 (line crosses y-axis at 5).
To graph a line:
Step 1. Plot the y-intercept (b). That's your first dot.
Step 2. From that dot, use the slope to find the next one. If m = 2, go RIGHT 1 and UP 2. (m = 2 means 2/1 = rise/run.)
Step 3. Connect the dots with a straight line. Done.
To write the equation from a description: Find m, find b, plug into y = mx + b.
Example: "$10 per hour + $5 tip" → m=10, b=5 → y = 10x + 5. That's it.
5 problems. Wrong answers are how brains learn. No grading. Take your time.