AESOP PRE-CALCULUS
Final Review
Section 1 — Conceptual Sweep
50 minutes
TODAY'S ROADMAP
Block 1
15 min
Block 2
15 min
Block 3
Functions & Their
Behavior
Geometric
Edge Behaviors
What is a function
Transformations
Function families
Unit circle
Trig graphs & identities
Complex numbers
Sequences & Series
Limits
The common thread
+ 5 min opening warm-up · 5 min flex buffer
10 min
WARM-UP
Try these on your own — we'll discuss together.
Q1
Given f(x) = 2x² − 3, evaluate f(−1) and state the domain of f.
Q2
Which of these is NOT a function: y = x², y = ±√x, y = |x| ?
How can you tell?
Opening · 5 min
Block 1 · 15 min
Functions & Their Behavior
What is a function · Transformations · Function families
What Is a Function?
One input → exactly one output
Domain — all valid inputs (x-values)
Range — all possible outputs (y-values)
Block 1 · 3 min
DISCUSS
Is a circle
a function?
Why or why not?
Function notation
f(x) = x² + 1
f(3) = 3² + 1 = 10
Hint: use the
vertical line test.
Transformations — The Universal Language
Block 1 · 5 min
These rules apply to every function family.
y = a · f ( b(x − h) ) + k
a
Vertical stretch / reflect
|a| > 1 stretch · 0 < |a| < 1 compress · a < 0 flips over x-axis
b
Horizontal stretch / reflect
Period scales by 1/|b| (especially important for trig functions)
h
Horizontal shift
Shift RIGHT by h
k
Vertical shift
Shift UP by k
Example: y = −2(x + 3)² − 1
→
(x − h): so x + 3 means shift LEFT 3
a = −2, b = 1, h = −3, k = −1
Function Families — Side by Side
Polynomial
Block 1 · 7 min
Exponential & Log
General shape
Smooth curves; U-shape (quadratic) or
S-shape (cubic); defined for all x
Exponential: rapid growth or decay curve
Log: slow, ever-increasing curve
Key feature
End behavior set by degree & leading
coefficient; zeros = x-intercepts
Exponential: horizontal asymptote (y = 0)
Log: vertical asymptote (x = 0)
Solve technique
Factor · Quadratic formula ·
Polynomial long division
Match bases · Apply log properties
log(Mⁿ) = n · log(M)
Tip: for each family know the shape, the key feature, and one solving technique — that covers most exam questions.
Block 2 · 15 min
Geometric
Unit circle · Trig graphs · Identities · Complex numbers
The Unit Circle — Home Base
cos θ = x-coordinate
Block 2 · 4 min
·
sin θ = y-coordinate
·
radius = 1
Angle (rad)
Angle (deg)
cos θ
sin θ
0
0°
1
0
π/6
30°
√3/2
1/2
π/4
45°
√2/2
√2/2
π/3
60°
1/2
√3/2
π/2
90°
0
1
π
180°
−1
0
3π/2
270°
0
−1
2π
360°
1
0
tan θ = sin θ / cos θ
(undefined when cos θ = 0, i.e. at θ = π/2, 3π/2)
Reading Trig Graphs — 4 Things That Matter
y = A · sin ( Bx − C ) + D
A
C/B
Amplitude
Block 2 · 5 min
(same structure for cosine)
B
Period
Formula: |A|
Formula: 2π / |B|
Height from midline to peak
(always positive)
Length of one full cycle
Phase Shift
D
Vertical Shift
Formula: C / B
Formula: +D
Horizontal shift
(right if positive)
Moves the midline up or down
Example: y = 2sin(3x − π) + 1 → Amplitude = 2, Period = 2π/3, Phase shi = π/3 right, D = 1
The Key Identity & Solving Trig Equations
Block 2 · 4 min
sin²x + cos²x = 1
The Pythagorean Identity — the one to always remember.
Also useful:
sin²x = 1 − cos²x
[ WORKED EXAMPLE — live on board ]
Solve: 2sin²x − sin x = 0 on [0, 2π]
Hint: factor first, then use the unit circle to find solutions.
cos²x = 1 − sin²x
Bridge: Trig → Complex Numbers
Block 2 · 2 min
Complex numbers live on a 2D plane — just like coordinates.
Cartesian Plane
Complex Plane
Axes:
x (horizontal) & y (vertical)
Axes:
Real (horizontal) & Imaginary
(vertical)
Point:
(x, y)
Point:
a + bi
Distance:
r = √(x² + y²)
Modulus:
r = √(a² + b²)
Angle:
θ = arctan(y / x)
Polar form:
r(cos θ + i · sin θ)
If you remember the unit circle, you already understand polar form.
Block 3 · 10 min
Edge Behaviors
Sequences & Series · Limits · The common thread
Sequences & Series
Block 3 · 4 min
Arithmetic
Geometric
Add a constant d
Multiply by a constant r
2, 5, 8, 11, 14 ... (d = 3)
3, 6, 12, 24, 48 ... (r = 2)
nth term:
nth term:
aₙ = a₁ + (n − 1)d
Sum of n terms:
aₙ = a₁ · rⁿ⁻¹
Sum of n terms:
Sₙ = n/2 · (a₁ + aₙ)
Sₙ = a₁(1 − rⁿ) / (1 − r)
Key insight: a geometric series converges to a finite sum when |r| < 1 — this naturally leads us to limits.
Limits — What Is the Function Approaching?
Block 3 · 4 min
"What value is the function approaching,
even if it never actually gets there?"
Two important cases:
Limit EXISTS
Limit DOES NOT EXIST
Function approaches the same value from both sides.
Function jumps, oscillates, or goes to ±∞.
Write: lim f(x) = L as x → a
Left-side limit ≠ right-side limit.
[ LIVE EXAMPLE — build a table of values approaching x = 2 ]
What does f(x) = (x² − 4)/(x − 2) approach as x → 2? (Why can't we just plug in x = 2?)
The Common Thread
Block 3 · 2 min
"What happens at the edge?"
Sequences
→
What is the nth term
as n grows large?
Series
→
Does the running sum
converge or diverge?
Limits
What value does
f(x) approach near a?
Recognizing this through-line helps organize the whole course — and helps you recall formulas under pressure.
COMING UP NEXT
Section 2
Practice Exam + Collaborative Debrief
8–10 problems · one per concept family · breadth over difficulty
This is a self-assessment — not a test.