Read the notes, then try the practice. It adapts as you go.When you're ready.
Session Length
~17 min
Adaptive Checks
15 questions
Transfer Probes
8
AP Precalculus gets you ready for calculus by focusing on how functions work—and why.
You'll learn to shift, stretch, and flip graphs. You'll connect algebra to real growth (populations, money) and decay. You'll see how sine and cosine model things that repeat, like waves and seasons.
The big idea: understand what's happening, not just memorize steps. When you change what goes in vs what comes out, the graph moves in different ways. That distinction matters everywhere.
One step at a time.
Moving, stretching, or flipping a graph. Changes to the input (inside) move left/right. Changes to the output (outside) move up/down. That inside/outside split trips people up—but it's the key.
Example: $g(x)=-2(x+3)^2-1$ starts as $x^2$, then flips, stretches by 2, shifts left 3, and down 1.
Functions built from powers of x. The highest power tells you how the ends behave: odd degree = opposite ends; even degree = same ends.
Example: $f(x)=-x^3+4x$ rises left, falls right (odd degree, negative lead). Zeros at -2, 0, 2.
One polynomial divided by another. Vertical asymptotes happen where the bottom is zero. Holes happen when top and bottom share a factor.
Example: $f(x)=\frac{x^2-1}{x^2-4}$ has asymptotes at $x=\pm 2$ and crosses the x-axis at $\pm 1$.
Multiply by the same factor over equal steps. Growth when the base > 1; decay when it's between 0 and 1. Used for populations, money, radioactivity.
Example: Bacteria doubling every 3 hours: 500 to 1000 to 2000 to 4000. After 9 hours: 4000 cells.
Logs undo exponents. $\log_b(y)=x$ means "to what power do we raise b to get y?" Turns multiplication into addition.
Example: To solve $2^{t/3}=8$: $t/3=\log_2(8)=3$, so $t=9$.
Plug one function into another. Do the inner one first; use that result as the input to the outer one. Order matters.
Example: For $f(g(2))$: find g(2) first, then plug that into f. It's not f(2) times g(2).
A function that undoes another. $f^{-1}$ takes the output and gives back the input. Only works for one-to-one functions (each output comes from exactly one input).
Example: If $f(x)=2x+3$ doubles and adds 3, then $f^{-1}(x)=(x-3)/2$ reverses it.
Sine and cosine come from a circle of radius 1. cosine = x-coordinate, sine = y-coordinate. They repeat every $2\pi$—perfect for waves, tides, anything that cycles.
Example: At 60 degrees, the circle point is (1/2, sqrt(3)/2). So cos=1/2, sin=sqrt(3)/2.
More terms are available in the glossary.
Choose a different way to engage with this topic — no grading, just richer thinking.
Explore your way — choose one:
See how the key ideas connect. Nodes color in as you practice.
Walk through a solved problem step-by-step. Try predicting each step before revealing it.
Use these for paper practice or in-class checkpoints.
This is guided practice, not just a quiz. Hints and pacing adjust in real time.
Small steps add up.
What you get while practicing:
The best way to know if you understand something: explain it in your own words.
More ways to strengthen what you just learned.