Comprehensive Study Notes: Precalculus Foundations, Linear, Exponential, and Trigonometric Modeling

Warming Up

• Problem-Solving Context: Precalculus is studied as an art and science through an "active apprenticeship" rather than passive reading. This involves assembling a toolkit of techniques to navigate between equations, data, and visual representations.
• Graphing and Modeling:
  • Graphing: The process of going from equations to pictures.
  • Modeling: The process of going from pictures (or raw data) to equations. This is typically harder and requires intuition.
• Units and Rates:
  • Numbers in modeling are rarely isolated; they usually have units attached.
  • Consistency is vital. Example: $(Total \ Change) = (Rate) \times (Time)$. Units must cancel correctly (e.g., $ft = (ft/sec) \times sec$).
  • Unit Conversions: Use conversion factors (e.g., $5,280 \ ft/mile$) as fractions to cancel unwanted units.
  • Density: Defined as $density = \frac{mass}{volume}$.
• Rate of Change:
  • Definition: $rate = \frac{\Delta \ quantity}{\Delta \ time}$.
  • Delta (\Delta): Shorthand for "change in."
  • Calculations: $\Delta \ quantity = (Value \ at \ Final \ Time) - (Value \ at \ Initial \ Time)$.
  • Constant Rate: When the rate is the same for all periods, $Total \ Change = Rate \times Time$.

Imposing Coordinates

• Coordinate Systems: A method to catalog points in a plane using pairs of real numbers $(x, y)$.
• The xy-Plane: Consists of a horizontal $x$-axis and a vertical $y$-axis intersecting at the Origin $(0,0)$.
• Quadrants: The plane is divided into four regions:
  • First: $(+, +)$
  • Second: $(-, +)$
  • Third: $(-, -)$
  • Fourth: $(+, -)$
• Scaling and Aspect Ratio:
  • Aspect Ratio: $\frac{length \ of \ one \ unit \ on \ vertical \ axis}{length \ of \ one \ unit \ on \ horizontal \ axis}$.
• Modeling Step: To "impose a coordinate system" is to choose an origin and axis orientation to solve a physical problem, such as tracking the motion of two runners on a runway.
• The Distance Formula:
  • For points $P = (x1, y1)$ and $Q = (x2, y2)$, the distance $d$ is: $d = \sqrt{(x<em>2 - x</em>1)^2 + (y<em>2 - y</em>1)^2}$.
  • Directed Distances: $\Delta x = x<em>2 - x</em>1$ and $\Delta y = y<em>2 - y</em>1$.

Three Simple Curves

• Lines:
  • Horizontal Line: Graph of $y = k$. Passes through $(0, k)$.
  • Vertical Line: Graph of $x = h$. Passes through $(h, 0)$.
• Circles:
  • Standard Form: $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.
  • Unit Circle: A circle with $r = 1$ centered at $(0, 0)$, defined by $x^2 + y^2 = 1$.
• Intersection of Curves: Found by solving the equations of both curves simultaneously. Example: To find where a circle intersects the $x$-axis, set $y = 0$ in the circle equation and solve for $x$.

Linear Modeling

• Two Points Determine a Line: If $P = (x1, y1)$ and $Q = (x2, y2)$ are known, the line is unique.
• Slope ($m$): $m = \frac{\Delta y}{\Delta x} = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$.
• Equations of Non-Vertical Lines:
  • Point-Slope: $y = m(x - x1) + y1$.
  • Two-Point: $y = (\frac{y<em>2 - y</em>1}{x<em>2 - x</em>1})(x - x<em>1) + y</em>1$.
  • Slope-Intercept: $y = mx + b$, where $b$ is the $y$-intercept.
• Parallel and Perpendicular Lines:
  • Parallel: Slopes are equal ($m1 = m2$).
  • Perpendicular: Slopes are negative reciprocals ($m<em>2 = -\frac{1}{m</em>1}$).
• Uniform Linear Motion: An object moving at constant speed. Position $(x, y)$ at time $t$ is modeled by parametric equations: $x = a + bt$ and $y = c + dt$.

Functions and Graphs

• Definition of a Function: A procedure matching a unique output to every allowable input.
• The Three Parts:
  1. Rule: The mathematical expression (e.g., $f(x) = x^2$).
  2. Domain: Set of allowed input $x$-values.
  3. Range: Set of possible output $y$-values.
• The Vertical Line Test: A curve is a function if every vertical line crosses it at most once.
• Multipart Functions: Functions defined by different rules on different parts of the domain. Notation uses a large brace with case-by-case conditions.
• Graphical Analysis:
  • Increasing/Decreasing: Walking uphill vs. downhill.
  • Local Extrema: Peaks (maxima) and valleys (minima).
  • Zeros/Roots: Inputs $x$ where $f(x) = 0$ (x-intercepts).
• Semicircles as Functions:
  • Upper: $y = k + \sqrt{r^2 - (x - h)^2}$.
  • Lower: $y = k - \sqrt{r^2 - (x - h)^2}$.

Quadratic Modeling

• Standard Parabola: Graph of a quadratic function $y = ax^2 + bx + c$ ($a \neq 0$).
• Vertex Form: $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex.
• Completing the Square: The process of converting $ax^2 + bx + c$ to vertex form.
  • $h = -\frac{b}{2a}$, $k = f(h)$.
• Visual Transformations:
  • $f(x - h)$: Horizontal shift by $h$.
  • $f(x) + k$: Vertical shift by $k$.
  • $af(x)$: Vertical dilation (stretch/compress).
  • $-f(x)$: Reflection across the $x$-axis.
• Optimization: The vertex provides the maximum (if $a < 0$) or minimum (if $a > 0$).

Composition

• The Formula: Replace every $x$ in $f(x)$ with the rule for $g(x)$. Result: $f(g(x))$.
• Domain Analysis: The input must be in the domain of $g$, and $g(x)$ must be in the domain of $f$.

Inverse Functions

• Definition: A function $f^{-1}(y)$ that reverses $y = f(x)$.
• One-to-One: A function must be one-to-one to have an inverse as a function. It must pass the Horizontal Line Test.
• Graphs: The graph of $f^{-1}$ is the reflection of the graph of $f$ across the line $y = x$.
• Composition Identity: $f(f^{-1}(y)) = y$ and $f^{-1}(f(x)) = x$.

Exponential and Logarithmic Functions

• Exponential Type: $A(x) = A0 b^x$. $A0$ is the initial value; $b$ is the base.
  • Growth: $b > 1$.
  • Decay: $0 < b < 1$.
• The Number $e$: Approximately $2.71828$. Defined by the limit as $n \to \infty$ of $(1 + \frac{1}{n})^n$.
• Compounding Interest:
  • Discrete: $P(t) = P_0(1 + \frac{r}{n})^{nt}$.
  • Continuous: $P(t) = P_0 e^{rt}$.
• Logarithmic Functions: The inverse of exponential functions.
  • Natural Log: $y = \ln(x)$ is the inverse of $y = e^x$.
  • Properties: $\ln(ab) = \ln a + \ln b$, $\ln(\frac{a}{b}) = \ln a - \ln b$, $\ln(a^t) = t \ln a$.
• Change of Base: $\log_b x = \frac{\ln x}{\ln b}$.
• Sound Intensity: $\beta = 10 \log<em>{10}(\frac{I}{I</em>0})$ decibels (db).

Linear-to-Linear Rational Functions

• Form: $f(x) = \frac{ax + b}{x + c}$.
• Asymptotes:
  • Vertical: $x = -c$ (where denominator is zero).
  • Horizontal: $y = a$ (limit as $x \to \infty$).

Trigonometry and Circular Motion

• Angle Measurement:
  • Degrees: $360^{\circ}$ in a full circle.
  • Radians: Based on arc length of unit circle. $180^{\circ} = \pi$ radians.
• Arc Length and Area:
  • $s = r\theta$ (length).
  • $A = \frac{1}{2}r^2\theta$ (area).
  • Note: $\theta$ must be in radians.
• Circular Motion:
  • Angular Speed (\omega): $\omega = \frac{\Delta \theta}{\Delta t}$. Units like $rad/sec$ or $RPM$.
  • Linear Speed (v): $v = r\omega$.
• The Circular Functions:
  • Sine (sin \theta): $y$-coordinate on unit circle. $\sin \theta = \frac{opposite}{hypotenuse}$.
  • Cosine (cos \theta): $x$-coordinate on unit circle. $\cos \theta = \frac{adjacent}{hypotenuse}$.
  • Tangent (tan \theta): $\frac{\sin \theta}{\cos \theta} = \frac{opposite}{adjacent}$. Equals the slope of the line from the origin.
• Sinusoidal Functions: $y = A \sin(\frac{2\pi}{B}(x - C)) + D$.
  • Amplitude (A): Peak deviation from mean.
  • Period (B): Duration of one cycle.
  • Phase Shift (C): Horizontal displacement.
  • Mean (D): Average value.
• Inverse Trigonometry:
  • $\sin^{-1}(z)$, $\cos^{-1}(z)$, $\tan^{-1}(z)$. Used to solve for angles in equations like $\sin \theta = c$.
  • Principal Solutions: Specific values within standard domains (e.g., $[-\pi/2, \pi/2]$ for $\sin^{-1}$).
  • General Solutions: Found by adding multiples of the period and using graph symmetry.