AP Pre-Calc Unit 1 Formulas (1.1 to 1.8)

f(x): This is just a way to say, “What does this function give us when we input x?”
x, y: These represent values on the graph, with x being horizontal and y being vertical.
m: The slope, which tells how steep a line is and whether it goes up or down.
b: The point where a line crosses the y-axis.
a, h, k: Numbers that adjust or move the graph around (shift, stretch, or compress).
f⁻¹(x): The function that "undoes" what f(x) does.
[a, b]: The interval includes both endpoints (a and b).
(a, b): The interval excludes the endpoints.
∘: Combining one function with another, like a sequence of steps.
|x|: The distance of x from zero, always positive.

A function, written as f(x) = y, connects each x (input) to exactly one y (output).
The domain is all x-values you’re allowed to use.
The range is all possible y-values.
Use the vertical line test: If a vertical line touches the graph more than once, it’s not a function.

[a, b]: The range includes the starting point (a) and the ending point (b).
(a, b): The range skips the starting and ending points.
Set builder notation: A fancy way of saying, “These x-values are between a and b, including or excluding endpoints.”

AROC compares how much y changes compared to x. Formula: AROC=Change in x/Change in y

Think of it as speed: how fast y changes per x.

Slope formula: Measures how steep a line is, calculated as: slope=Difference in y-valuesDifference in x-values\text{slope} = \frac{\text{Difference in y-values}}{\text{Difference in x-values}}
Slope-intercept form: y=mx+by = mx + b This means the line has a slope (m) and crosses the y-axis at b.
Point-slope form: y−y1=m(x−x1) y - y_1 = m(x - x_1) Use this when you know a specific point and the slope.

Plug one function into another like a two-step process: (f∘g)(x)=f(g(x))(f∘g)(x) = f(g(x)) First, find g(x), then use that result as the input for f(x).

To reverse a function, switch x and y in the equation and solve for y.
A function is reversible (invertible) if it passes the horizontal line test (each y-value is used only once).

Vertical shift: Add or subtract k to move the graph up or down.
Horizontal shift: Adjust h to move the graph left or right.
Stretch/compression: Multiply by a or b:
- Large values stretch it out.
- Small values squeeze it.
Reflection: Flip the graph over the x-axis (negative sign in front) or y-axis (negative sign inside).

Absolute value graphs look like a V-shape: f(x)=a∣x−h∣+kf(x) = a|x - h| + k
- Vertex: The point at the tip of the V is (h, k).
- On one side, the graph goes up with a slope of a. On the other side, it goes down with a slope of -a.