Algebra I: Function Notation & Linear Models

In math class, parentheses usually mean multiply, like $2(3)=6\text{.}$ But $f(x)$ does not mean $f$ times $x\text{.}$

• $f$ is the name of the blender, like "Smoothie Machine #1."

• $x$ is what you drop in the top, the input.

• $f(x)$ is what pours out the bottom, the output. It plays the same role as $y\text{.}$

• Say it out loud as "f of x."

Before you do any math, ask: "Did they hand me the ingredient, or the finished smoothie?"

• Handed the input $x\text{?}$ Plug it in and go forward. This is called evaluating.

• Handed the output $f(x)\text{?}$ You know how it ended, so rewind: solve backward to find the $x$ that went in.

A taxi charges $\$8$ just to get in, plus $\$2.50$ for every mile. Its rule is $f(x)=2.5x+8\text{,}$ where $x$ is the miles driven. Every straight-line story works like this taxi bill:

total = cost per mile × miles + starting fee

• Start where the ride starts: 0 miles costs $\$8\text{.}$ That point, $(0, 8)\text{,}$ is $b\text{,}$ the $y$-intercept.

• Step: every mile to the right, the price climbs $\$2.50\text{.}$ That climb is $m\text{,}$ the slope.

• Piggy bank (linear): you add the same amount every day: $10\text{,}$ $13\text{,}$ $16\text{,}$ $19\text{.}$ Always $+3\text{.}$

• Magic candy box (exponential): the candy doubles every day: $10\text{,}$ $20\text{,}$ $40\text{,}$ $80\text{.}$ Always $\times 2\text{.}$

You start a new game with 40 coins and earn 3 coins for every level you beat. Your coins after $L$ levels are $C(L)=3L+40\text{.}$

• $b=40$ is what you have before you start, at $L=0\text{:}$ the starting amount.

• $m=3$ is what you get for each level: a rate of 3 coins per level.

Slope tells you how steep a line is, like a staircase.

• Rise is how far up you go.

• Run is how far across you go.

• Slope $=\dfrac{\text{rise}}{\text{run}}\text{:}$ up on top, across on the bottom.

Read a line from left to right, like skiing across the page.

• Positive: going uphill.

• Negative: going downhill.

• Zero: flat ground. The rise is $0\text{,}$ so $m=0\text{.}$

• Undefined: a wall, straight up and down. The run is $0\text{,}$ and you can't divide by $0\text{.}$

A straight line is like a perfectly straight road: it has the same steepness everywhere. If the steepness changes between two points, the road bent, so the points can't all sit on one straight line.

Pilots run a checklist before every flight, even after years of flying. Run this one before you hand in a slope.