AP PreCalc

1. Functions

1.1. Linear & Quadratic Functions

1.2. Polynomial Functions

Extrema: these are on the minimums and maximums of a function 1. relative extrema: switches between decreasing and increasing 2. absolute extrema: of all relative extrema, the greatest or least Points of inflection: where the graph switches from concave down to concave up or vice versa

2. Trig Values

2.1. Angles

Reference Angle: how far the angle is from the x-axis Coterminal Angle: any angle that ends up in the same spot as the original angle (+/- 360) Trig values memorization trick: SOH CAO TOA ![[Pasted image 20250920182128.png]]

2.2 Unit Circle

![[Pasted image 20250920182359.png]]

Pythagorean Identities: 1+cos²θ=csc²θ tan²θ+1=sec²θ

2.3. Graphing

Transformation:

f(x)=A(sin or cos)[B(x-C)]+D
A = change in amplitude
B = change in period (2pi/B=period)
C = phase shift
D = vertical shift

Inverse: the domain becomes the range could be written as sin^-1 or arcsin (could be any of the 6 trig values)

2.4. Trig Identities

![[Pasted image 20251007233539.png]] ![[Pasted image 20251007233622.png]]

Unit 3

3.1. Proofs

The goal with pre calc proofs is to show how the right side equals the left side, without touching the left side.

3.2. Parametric Equations

Definition Parametric equations define a relationship between x and y using a third variable, called a parameter, usually denoted by t. Instead of a single equation like y = f(x), you have a pair of equations that describe the coordinates of a point over time or another interval.

x = f(t) (describes horizontal position)
y = g(t) (describes vertical position)

The parameter t allows us to define not just the path of a curve, but also the direction of motion, or orientation, along that path. To graph a set of parametric equations, you can create a table of values.

Choose a range of values for the parameter t.
Plug these t-values into both the x and y equations to find the corresponding coordinates.
Plot the (x, y) points in order of their t-value.
Connect the points and use arrows to show the orientation of the curve as t increases.

Example: Graph x(t) = 2t - 1 and y(t) = t^2 + 1 for -1 \le t \le 2.

t	x = 2t - 1	y = t² + 1	(x, y)
-1	-3	2	(-3, 2)
0	-1	1	(-1, 1)
1	1	2	(1, 2)
2	3	5	(3, 5)

When plotted, these points form a segment of a parabola starting at (-3, 2), moving down to (-1, 1), and then up to (3, 5). Eliminating the Parameter This is the process of converting a pair of parametric equations into a single rectangular equation (an equation with only x and y).

Steps: 1. Choose one of the equations and solve for t. 2. Substitute the expression for t into the other equation. 3. Simplify the resulting equation.

Unit 4

4.1. Introduction to Vectors

Definition A vector is a mathematical object that has both magnitude (size or length) and direction. This is different from a scalar, which only has magnitude. - Vector Examples: Velocity (speed in a direction), Force, Displacement. Vector Representation Vectors can be represented in several ways: 1. Geometrically: As a directed line segment (an arrow). The length of the arrow is the magnitude, and the arrowhead points in the direction. It has an initial point (tail) and a terminal point (head). 2. Component Form: A vector v is written as <x, y>, where x is the horizontal component and y is the vertical component. Example: Find the component form of the vector with initial point (-2, 5) and terminal point (1, 7). $ v = <1 - (-2), 7 - 5> = <3, 2> $

4.2. Vector Properties & Operations

Magnitude The magnitude (or norm) of a vector is its length. It is denoted by $||v||$. For a vector $v = $, the magnitude is found using the Pythagorean theorem. Direction Angle The direction angle, $\theta$, is the angle the vector makes with the positive x-axis. Always check which quadrant the vector lies in to find the correct angle. Vector Operations Let $u = $ and $v = $, and let k be a scalar.

Operation	Rule	Geometric Interpretation
Vector Addition	$u + v = $	Place vectors tip-to-tail. The resultant vector goes from the tail of the first to the tip of the second.
Vector Subtraction	$u - v = $	Add the opposite vector: $u + (-v)$.
Scalar Multiplication	$k \cdot v = $	Stretches or shrinks the vector's magnitude. If $k < 0$, the direction is reversed.
#### 4.3. Polar Numbers
Definition
The polar coordinate system is an alternative to the rectangular/Cartesian `(x, y)` system for locating points on a plane. A point is defined by a distance and an angle.
- A point is represented by the coordinates `(r, θ)`.
- r: The radius, which is the directed distance from the origin (called the pole).
- θ: The angle, which is the directed angle measured counter-clockwise from the positive x-axis (called the polar axis).
Graphing Polar Points
1. Start at the pole.
2. Rotate by the angle `θ`.
3. Move out along the terminal ray by a distance of `r`.
- If `r` is negative, you move in the exact opposite direction (180° or π radians away) from the angle's terminal ray.
- Important: Polar coordinates are not unique. The point $(2, \frac{\pi}{3})$ is the same as $(2, \frac{7\pi}{3})$ and $(-2, \frac{4\pi}{3})$.

Coordinate Conversion We use these formulas to convert points and equations between the polar and rectangular systems.

Conversion	Formulas	Key Relationships
Polar to Rectangular	x = r \cos y = r \sin	These are derived directly from the unit circle definitions of sine and cosine.
Rectangular to Polar	r^2 = x^2 + y^2 tan=y/x	When finding $\theta$, always check the quadrant of the original `(x, y)` point to ensure you select the correct angle.
Converting Equations
The same substitution rules apply when converting entire equations between systems.
### Unit 5
#### 5.1. Polynomial Functions
End Behavior: the value of y as x approaches -∞ and ∞
- odd functions: end behavior will be opposite (-∞ and ∞)
- even functions: end behavior will be the same (∞ and ∞)
- Limit Notation:
![[Pasted image 20260118125657.png]]
Solving Polynomial Inequalities:
1. solve f(x)=0
2. create a sign chart with solutions
3. test values in each interval to see if the values in the interval are positive or negative
4. interpret the sign chart to answer the given inequality from the problem (using interval notation)
#### 5.2. Graphing Polynomials
D. H. RAT^2EY:
- D: domain
- H: holes
- R: roots (comes from the numerator set to 0)
- A: (vertical )Asymptotes (comes from the denominator set to 0)
- T^2: Tangency/Togetherness (factors with even multiplicities of 2, determines how the graph will look)
- E: End Behavior (horizontal asymptotes)(depends on the ratio of the first term degrees)
- Y: Y-Intercept (substitute 0 for x)
#### 5.3. Exponents
![[Pasted image 20260121221300.png]]
### 6. Logarithms
* Definition: A logarithm is the exponent to which a specified base must be raised to obtain a given number.
* If $b^x = y$, then $\log_b y = x$.
* Base (b): Must be positive and not equal to 1.
* Argument (y): Must be positive.

Common Logarithm: A logarithm with base 10. It is written as $\log y$ (the base 10 is implied).
Natural Logarithm: A logarithm with base $e$ (Euler's number, approximately 2.718). It is written as $\ln y$.
- $\log_e y = \ln y$
Properties of Logarithms:
- Product Rule: $\log_b (MN) = \log_b M + \log_b N$
  - The logarithm of a product is the sum of the logarithms of the factors.
- Quotient Rule: $\log_b (\frac{M}{N}) = \log_b M - \log_b N$
  - The logarithm of a quotient is the difference of the logarithms of the numerator and denominator.
- Power Rule: $\log_b M^p = p \log_b M$
  - The logarithm of a power is the exponent times the logarithm of the base.
- Change of Base Formula: $\log_b M = \frac{\log_c M}{\log_c b}$
  - This is useful for calculating logarithms with bases other than 10 or $e$ using a calculator. You can use common logs (base 10) or natural logs (base $e$) for $c$.
- Logarithm of the Base: $\log_b b = 1$
  - The exponent to which you raise $b$ to get $b$ is 1.
- Logarithm of 1: $\log_b 1 = 0$
  - The exponent to which you raise $b$ to get 1 is 0.
- Inverse Properties:
  - $b^{\log_b x} = x$
  - $\log_b b^x = x$
Solving Logarithmic Equations:
1. Isolate the logarithm: Get the logarithmic expression by itself on one side of the equation.
2. Convert to exponential form: If you have $\log_b y = x$, rewrite it as $b^x = y$.
3. Solve for the variable: Solve the resulting equation.
4. Check for extraneous solutions: Ensure that the arguments of all logarithms in the original equation are positive.
Solving Exponential Equations using Logarithms:
1. Isolate the exponential term: Get the term with the exponent by itself.
2. Take the logarithm of both sides: Use either the common logarithm (log) or the natural logarithm (ln).
3. Use the power rule: Bring the exponent down in front of the logarithm.
4. Solve for the variable: Isolate the variable.
Graphs of Logarithmic Functions:
- The graph of $y = \log_b x$ is the inverse of the graph of $y = b^x$.
- Domain: $(0, \infty)$ (x-values must be positive).
- Range: $(-\infty, \infty)$ (all real numbers).
- Vertical Asymptote: $x = 0$ (the y-axis).
- Key Point: $(1, 0)$ (since $\log_b 1 = 0$).
- Key Point: $(b, 1)$ (since $\log_b b = 1$).
- Transformations: Similar to other functions, you can apply transformations:
  - $y = a \log_b(x-h) + k$
    - a: Vertical stretch/compression and reflection over the x-axis.
    - h: Horizontal shift (shifts the vertical asymptote).
    - k: Vertical shift.

x. Test Review

Date:	Good:	Bad:	Results + why wrong:	Reminders:
09/22/25	- knew how to do all of the problems immediately - given formula sheet: guaranteed formulas were all right - had enough time to double check answers	- didn't know if a graph become steeper but is negative if the AROC was increasing or decreasing (it is decreasing got it wrong) - felt very rushed and made many silly mistakes (fixed them when checking second time)	90 - AROC decreasing and a graph decreasing are separate (practice online) - When choosing which graph is different, make sure to properly graph each one (or a faster way to plug in a variable)	- make sure to have all equations memorized (ideally week prior)
10/07/25			95

y. AP Specific

y.1. FRQ

use specific language during the response (consecutive integers, AROC, etc.)
use words from the question (if it asks for the slope of h say the slope of h in your response)
explain how you got your answer through a logical response, explaining each step