Precalculus Study Guide: Functions, Trigonometry, and Analytic Geometry

Functions and Their Graphs

Relations and Functions

n the ranUnderstanding the concept of a function is foundational in precalculus. A function is a relation that assigns each element in the domain to exactly one elemet ige.

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

• Function notation: $f(x)$ represents the output when $x$ is the input.

• Vertical Line Test: A graph represents a function if no vertical line intersects the graph at more than one point.

Example: $f(x) = x^2$ is a function; for each $x$, there is one $f(x)$.

Types of Functions

• Polynomial Functions: Functions of the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_0$.

• Rational Functions: Functions expressed as the ratio of two polynomials, $f(x) = \frac{P(x)}{Q(x)}$.

• Piecewise Functions: Functions defined by different expressions over different intervals.

Example: $f(x) = \begin{cases} x^2 & x < 0 \\ x+1 & x \geq 0 \end{cases}$

Transformations of Functions

Transformations change the position or shape of a graph.

• Translations: Shifting the graph horizontally or vertically.

• Reflections: Flipping the graph over a line (e.g., x-axis or y-axis).

• Stretches/Compressions: Changing the graph's width or height.

Example: $f(x) = (x-2)^2 + 3$ shifts $y = x^2$ right by 2 and up by 3.

Polynomial and Rational Functions

Linear and Quadratic Functions

Linear functions have the form $f(x) = mx + b$; quadratic functions have the form $f(x) = ax^2 + bx + c$.

• Graph of a linear function: Straight line; slope $m$ and y-intercept $b$.

• Graph of a quadratic function: Parabola; opens upward if $a > 0$, downward if $a < 0$.

• Vertex: The highest or lowest point of a parabola, at $x = -\frac{b}{2a}$.

Example: $f(x) = 2x + 1$ (linear), $f(x) = x^2 - 4x + 3$ (quadratic).

Polynomial Functions

• Degree: The highest power of $x$ in the polynomial.

• Zeros/Roots: Values of $x$ where $f(x) = 0$.

• Factoring: Expressing a polynomial as a product of lower-degree polynomials.

Example: $f(x) = x^3 - 2x^2 + x$ has degree 3.

Rational Functions

• Vertical Asymptotes: Values of $x$ where the denominator is zero.

• Horizontal Asymptotes: Behavior as $x \to \infty$ or $x \to -\infty$.

Example: $f(x) = \frac{1}{x-2}$ has a vertical asymptote at $x=2$.

Exponential and Logarithmic Functions

Exponential Functions

An exponential function has the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

• Growth: If $a > 1$, the function increases rapidly.

• Decay: If $0 < a < 1$, the function decreases rapidly.

Example: $f(x) = 2^x$ (growth), $f(x) = (1/2)^x$ (decay).

Logarithmic Functions

A logarithmic function is the inverse of an exponential function: $f(x) = \log_a x$.

• Properties: $\log_a(ab) = \log_a a + \log_a b$; $\log_a(a^x) = x$.

• Domain: $x > 0$.

Example: $f(x) = \log_{10} x$.

Trigonometry

Right Triangle Trigonometry

Trigonometric functions relate the angles of a triangle to the lengths of its sides.

• Sine: $\sin x = \frac{\text{opposite}}{\text{hypotenuse}}$

• Cosine: $\cos x = \frac{\text{adjacent}}{\text{hypotenuse}}$

• Tangent: $\tan x = \frac{\text{opposite}}{\text{adjacent}}$

Reciprocal identities:

• $\sin x = \frac{1}{\csc x}$, $\csc x = \frac{1}{\sin x}$

• $\cos x = \frac{1}{\sec x}$, $\sec x = \frac{1}{\cos x}$

• $\tan x = \frac{1}{\cot x}$, $\cot x = \frac{1}{\tan x}$

Trigonometric Identities

• Pythagorean Identity: $\sin^2 x + \cos^2 x = 1$

• Sum and Difference Formulas:

  • $\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y$

  • $\cos(x \pm y) = \cos x \cos y \mp \sin x \sin y$

• Double Angle Formulas:

  • $\sin 2x = 2 \sin x \cos x$

  • $\cos 2x = \cos^2 x - \sin^2 x$

• Negative Angle Identities:

  • $\sin(-x) = -\sin x$

  • $\cos(-x) = \cos x$

Law of Sines and Law of Cosines

• Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

• Law of Cosines: $c^2 = a^2 + b^2 - 2ab \cos C$

Example: Use the Law of Sines to solve for an unknown side in a triangle when two angles and one side are known.

Graphs of Trigonometric Functions

• Periodicity: Sine and cosine functions repeat every $2\pi$.

• Amplitude: The maximum value from the midline.

• Phase Shift: Horizontal shift of the graph.

Example: $y = 3 \sin(x - \frac{\pi}{4})$ has amplitude 3 and phase shift $\frac{\pi}{4}$.

Analytic Trigonometry

Inverse Trigonometric Functions

Inverse trigonometric functions allow you to find angles from known ratios.

• arcsin: $y = \arcsin x$ means $x = \sin y$

• arccos: $y = \arccos x$ means $x = \cos y$

• arctan: $y = \arctan x$ means $x = \tan y$

Example: $\arcsin(0.5) = \frac{\pi}{6}$

Applications of Trigonometry

Solving Triangles

Trigonometry is used to solve for unknown sides and angles in triangles, especially in navigation, physics, and engineering.

• Use the Law of Sines and Law of Cosines for non-right triangles.

• Apply trigonometric ratios for right triangles.

Systems of Equations and Inequalities

Linear Systems

Systems of equations involve finding values that satisfy multiple equations simultaneously.

• Graphical Method: Plot each equation and find intersection points.

• Algebraic Methods: Substitution and elimination.

• Matrix Method: Use matrices to solve systems, especially for larger systems.

Example: Solve $\begin{cases} x + y = 5 \\ 2x - y = 3 \end{cases}$

Conic Sections

Types of Conics

Conic sections are curves formed by the intersection of a plane and a double-napped cone.

• Circle: All points equidistant from a center. $x^2 + y^2 = r^2$

• Ellipse: Sum of distances from two foci is constant.

• Parabola: Set of points equidistant from a focus and a directrix.

• Hyperbola: Difference of distances from two foci is constant.

Example: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (ellipse)

Summary Table: Key Trigonometric Identities

Identity	Formula
Reciprocal	$\sin x = \frac{1}{\csc x}$, $\cos x = \frac{1}{\sec x}$, $\tan x = \frac{1}{\cot x}$
Pythagorean	$\sin^2 x + \cos^2 x = 1$
Sum/Difference	$\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y$ $\cos(x \pm y) = \cos x \cos y \mp \sin x \sin y$
Double Angle	$\sin 2x = 2 \sin x \cos x$ $\cos 2x = \cos^2 x - \sin^2 x$
Law of Sines	$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
Law of Cosines	$c^2 = a^2 + b^2 - 2ab \cos C$

Additional info: These notes are based on the syllabus and textbook outline for a Precalculus course, including major topics such as functions, trigonometry, analytic geometry, and conic sections. The formulas and identities are essential for success in Precalculus and are frequently used in problem-solving and applications.