Unit circle + reciprocal trig
made by Michel for studying. it runs offline, nothing to install.
Start with a circle of radius 1. Put a point on it and turn it by some angle. The point's x is the cosine of the angle and its y is the sine. That's really all it is. Grab the blue dot and drag it around.
Keyboard: click the dot, then ← → move 1°, hold Shift for 5°, PgUp/PgDn jump to the next nice angle.
Find the 5 nice angles 0 / 5
These five angles in the first quadrant have exact values you can write down. Visit each one (drag, or hit a jump button) and it gets checked off.
stuff to notice
- cos is the dot's x position. Left of center it's negative.
- sin is the dot's y position. Below center it's negative.
- tan = sin ÷ cos. On the circle it's the length of the yellow line at x = 1. Near 90° the ray never hits that line, so tan blows up.
The table you have to memorize
Trick so you don't forget it: the sin row is √0, √1, √2, √3, √4 all over 2. The cos row is the same list backwards.
Reference angles
Here's how you find a reference angle. Any angle, even a huge or negative one, ends up sitting above some small acute angle measured from the x-axis. That small angle is the reference angle. You only need the table for 0° to 90°, and the reference angle brings you back to it. The sizes stay the same, only the sign can change, and the quadrant tells you which.
4 rules for the reference angle
θ is your angle after wrapping it into one turn (0° to 360°). If it lands exactly on 0°, 90°, 180° or 270° it's on an axis, so there's nothing to fold.
All Students Take Calculus
Go around starting at quadrant I. The first letter of each word tells you what's positive there.
- I All: sin, cos, tan are all +
- II Students: only sin (and csc) +
- III Take: only tan (and cot) +
- IV Calculus: only cos (and sec) +
Fold one in each quadrant 0 / 3
Fold an angle that lands in quadrant II, then III, then IV.
Flipping: sec, csc, cot
The other three functions are just the first three upside down: sec θ = 1 ÷ cos θ, csc θ = 1 ÷ sin θ, cot θ = 1 ÷ tan θ. Pick a pair and drag across the graph. Watch what happens when the bottom of the fraction gets close to zero.
Try all three pairs 0 / 3
Pick each pair and move the angle at least once.
Solver
Pick sec, csc or cot and type any angle. It does the same steps every time: rewrite as 1 ÷ something, wrap, find the quadrant, reference angle, table value and sign, flip. Step through it and see the working.
Finish 2 solutions 0 / 2
Go all the way to the answer (or press “Show everything”) two times, with different angles.
Practice problems
11 problems. For some you place the dot, for some you put the steps in order, and for some you work out a value. The clues turn blue as they come true, so move the dot around before you lock it in.
0 of 11 done
Transformations of |x| and x²
lesson 1.7, my notes turned into stuff you can play with.
There are two parent functions, |x| and x². Both get changed by the same four steps, and always in the same order: H.T. (slide left or right), H.D. (squish or stretch sideways), V.D. (squish or stretch up and down), V.T. (slide up or down). Inside first, then outside.
g(x) = a|bx − h| + kg(x) = a(bx − h)² + k
(In sections 1 to 5, b is always positive. a can be negative, and that flips the graph over.)
Open the lab every transformation on one graph: drag, play the four steps, take the challenges
Parent functions
Both have their vertex at (0, 0). These are the key points we follow through every step, so memorize them.
Build your own g(x)
Move the sliders (or type a number, fractions like 1/2 work). The thin dashed curve is the parent, the thick one is your g(x), and the dots are the key points after all four steps.
What the graph does
Key points after all four steps
One step at a time
Pick one of the examples from my notes (or use whatever you built above) and step through it. The faint curve is where you just were, the thick one is where you are now.
g(x) = ...
Why the order matters
Try the steps in the wrong order and you land on a point that isn't on the graph. Hit the buttons and then check each answer by plugging in.
Quick check with factoring
For b > 0 you can factor b out of the inside: |bx − h| = b|x − h/b| and (bx − h)² = b²(x − h/b)². So the vertex is always (h/b, k). Example: |2x − 4| = 2|x − 2|, and (2x − 4)² = 4(x − 2)².
Practice
Four sets. Everything is checked by the same code that draws the graphs, and the answers match my answer key.
Find g(x) from the graph 0 / 6
Step 1: read the vertex (h, k). Step 2: look at the second point to get a. For |x| a is the slope. For x², moving 1 right from the vertex changes y by a.
Order the transformations 0 / 10
First tap the four steps in the right order. Then say what each step does for the function, and give the vertex.
Find the mistake 0 / 3
Challenges 0 / 4
The lab
All of lesson 1.7 on one graph. Each control does exactly one transformation. The faint curve is the parent, the bold blue one is g(x). Drag the handles on the graph, or use the controls (fractions like 1/3 work).
Play the four steps
Always in this order: H.T. → H.D. → V.D. → V.T. The dots are the key points, the thin lines are their trails.
Match the graph 0 / 0
Slide the controls or drag the handles until your blue g(x) lies right on the thick yellow target. It notices by itself when they match. The parent is set by the round.
Try these (each one resets the rest)
Describe it
The equation
Key points through each step
November test: functions, graphs, and a boss
units 1.1 to 1.4 (+ 1.7 again). made for the test on Mon 2 Nov.
Same deal as the other parts: a tiny bit of reading, then you play with a graph. Every predict box wants your guess before it shows the answer. The practice questions are random every time and every one has a worked solution. When you feel ready, fight the boss at the end (a timed mixed test).
removable · jump · infinitef(−x) = f(x) even · f(−x) = −f(x) odd(f(b) − f(a)) / (b − a)
Skip to the boss fight timed, mixed 1.1 to 1.4 and 1.7, review + redo the wrong ones
Functions & continuity
A function is continuous at x = c when you can draw through c without lifting your pencil: the limit from the left, the limit from the right and f(c) all agree. When they don't, the break has a name.
removable
A hole (or one point moved away). limx→c f(x) exists but isn't f(c). Fix one point and it's fixed.
jump
Left and right limits both exist but are different. The graph steps up or down.
infinite
|f(x)| grows without bound near c. Vertical asymptote at x = c.
Mystery graph 0 right
Predict first: what kind of point is it?
Domain & range, two ways to write them
Interval notation uses brackets: [ ] includes the end, ( ) doesn't, and ∞ always gets ( ). Set-builder notation says the same thing with an inequality: [−2, ∞) = {x | x ≥ −2, x ∈ ℝ}.
Linearity, intercepts & symmetry
Even: f(−x) = f(x), the graph is a mirror across the y-axis. Odd: f(−x) = −f(x), spin it 180° about the origin and it lands on itself. Most functions are neither. Test it by plugging in −x.
Symmetry tester 0 right
Predict first: even, odd or neither?
Intercepts: where the graph crosses the axes
x-intercepts (the zeros): set f(x) = 0 and solve. y-intercept: put x = 0. A linear function f(x) = mx + b has a constant rate of change m, and its y-intercept is b.
Extrema & end behavior
A relative maximum is a peak (higher than its neighbours), a relative minimum a valley. The absolute max/min is the highest/lowest point of the whole graph. Increasing and decreasing are read left to right and written with open intervals of x.
Trace it
End behavior in limit notation
Far out to the left and right only the leading term a·xⁿ matters. Even n: both ends point the same way. Odd n: opposite ways. a > 0: the right end goes up.
Predict where the ends go:
Average rate of change = slope of the secant
Sketching graphs & comparing functions
To sketch: zeros, y-intercept, end behavior, then a test point between the zeros to see if the graph is above or below. To compare two functions given in different forms (equation, table, graph), turn both into the same number: a y-intercept, a max, an average rate of change.
Sketch it in 5 steps 0 / 5
Mock exam boss fight
A timed mixed test: 1.1, 1.2, 1.3, 1.4 and 1.7. Every right answer hits the boss. One try per question, like the real test (you can skip). Afterwards you get every worked solution and can redo just the ones you got wrong.
Pick your fight
Use paper for working. Type answers like 5/2, (2, -1), [-3, inf) or -2, 4. The clock is 1.5 minutes per question.