Linear functions

A linear function takes a number, multiplies it by something, then adds something. That's it.

We write it as $f(x) = mx + b$, where:

• $x$ is your input — the thing you know.
• $m$ is the slope — how much the output changes per unit of input.
• $b$ is the y-intercept — the value of the function when $x = 0$.

Reading a linear function in real life: $f(x) = 3x + 7$ might mean for every hour you rent a bike ($x$), the cost goes up by $3 ($m$), and there's a $7 fee just to start ($b$).

Two operations show up on the test over and over:

1. Plug in a value. If $f(x) = 3x + 7$ and the question asks for $f(4)$, you replace $x$ with 4 and compute: $f(4) = 3(4) + 7 = 19$. That's it. No tricks.

2. Build the function from a word problem. When you see "$2.25 per mile", that's the slope — it's a per-thing rate. When you see "$3.50 flat fee", that's the y-intercept — a one-time amount paid regardless of how big $x$ gets.

A few other things to know:

• The x-intercept is where the function equals zero. Set $f(x) = 0$ and solve.
• A negative slope ($m < 0$) means the output goes DOWN as the input goes up — useful for decay, depreciation, distance remaining.
• A steeper slope (larger $|m|$) means the output changes faster per unit of input.