Precalculus Study Notes: Functions, Trigonometry, and Analytic Geometry

Functions and Their Graphs

Relations and Functions

Understanding relations and functions is foundational in precalculus. A relation is a set of ordered pairs, while a function is a relation in which each input has exactly one output.

• 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 of function f for input x.

• Vertical Line Test: A graph represents a function if no vertical line intersects it more than once.

Example: The function $f(x) = x^2$ has domain $(- , 1e)$ and range $[0, 1e)$.

Piecewise Functions and Transformations

Piecewise functions are defined by different expressions over different intervals. Transformations shift, stretch, or reflect graphs.

• Vertical shift: $f(x) + k$ shifts up/down.

• Horizontal shift: $f(x - h)$ shifts right/left.

• Reflection: $-f(x)$ reflects over x-axis; $f(-x)$ reflects over y-axis.

• Stretch/Compression: $af(x)$ stretches vertically if $|a| > 1$, compresses if $0 < |a| < 1$.

Example: $f(x) = |x|$ is a piecewise function: $f(x) = x$ if $x 3e 0$, $f(x) = -x$ if $x < 0$.

Polynomial and Rational Functions

Linear and Quadratic Functions

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

• Vertex of a quadratic: $x = -\frac{b}{2a}$

• Axis of symmetry: Vertical line through the vertex.

• Roots/Zeros: Solutions to $f(x) = 0$.

Example: $f(x) = 2x^2 - 4x + 1$ has vertex at $x = 1$.

Polynomial Functions

Polynomials are sums of terms of the form $ax^n$. The degree is the highest exponent.

• End behavior: Determined by leading term.

• Real zeros: Values of x where $f(x) = 0$.

• Factoring: Used to find zeros and simplify expressions.

Rational Functions

Rational functions are ratios of polynomials, $f(x) = \frac{P(x)}{Q(x)}$.

• Domain: All real numbers except where $Q(x) = 0$.

• Vertical asymptotes: Values where denominator is zero.

• Horizontal asymptotes: Determined by degrees of numerator and denominator.

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions have the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

• Growth/Decay: If $a > 1$, function grows; if $0 < a < 1$, function decays.

• Domain: $(-\infty, \infty)$

• Range: $(0, \infty)$

Logarithmic Functions

Logarithmic functions are inverses of exponential functions: $f(x) = \log_a x$.

• Domain: $(0, \infty)$

• Range: $(-\infty, \infty)$

• Properties: $\log_a(ab) = \log_a a + \log_a b$; $\log_a(\frac{a}{b}) = \log_a a - \log_a b$

Systems of Equations and Inequalities

Solving Systems

Systems of equations can be solved by substitution, elimination, or using matrices.

• Substitution: Solve one equation for a variable, substitute into the other.

• Elimination: Add/subtract equations to eliminate a variable.

• Matrices: Use matrix methods for larger systems.

Example: Solve $\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}$ by adding equations.

Trigonometry

Right Triangle Trigonometry

Trigonometric ratios relate the angles and sides of right triangles.

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

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

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

• Cosecant: $\csc x = \frac{1}{\sin x}$

• Secant: $\sec x = \frac{1}{\cos x}$

• Cotangent: $\cot x = \frac{1}{\tan x}$

Trigonometric Identities

Identities are equations true for all values in the domain.

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

• Reciprocal Identities: $\sin x = \frac{1}{\csc x}$, $\cos x = \frac{1}{\sec x}$, $\tan x = \frac{1}{\cot x}$

• Even-Odd Identities: $\sin(-x) = -\sin x$, $\cos(-x) = \cos x$

• Double Angle: $\sin 2x = 2 \sin x \cos x$

• Sum and Difference: $\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y$

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

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

Graphs of Trigonometric Functions

Trigonometric functions are periodic and have characteristic graphs.

• Period: The length of one cycle. For $\sin x$ and $\cos x$, period is $2\pi$.

• Amplitude: Maximum value from the midline.

• Phase shift: Horizontal shift of the graph.

Applications of Trigonometry

Trigonometry is used in solving triangles, modeling periodic phenomena, and in analytic geometry.

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

• Modeling: Trigonometric functions model sound waves, tides, and other periodic events.

Analytic Geometry

Conic Sections

Conic sections include circles, parabolas, ellipses, and hyperbolas, each defined by a specific equation.

• Circle: $(x - h)^2 + (y - k)^2 = r^2$

• Parabola: $y = a(x - h)^2 + k$

• Ellipse: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$

• Hyperbola: $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$

Systems of Equations: Matrices

Matrices are used to solve systems of linear equations efficiently.

• Matrix notation: $A \vec{x} = \vec{b}$

• Row reduction: Used to find solutions.

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	$\sin^2 x + \cos^2 x = 1$
Even-Odd	$\sin(-x) = -\sin x$, $\cos(-x) = \cos x$
Double Angle	$\sin 2x = 2 \sin x \cos x$
Sum/Difference	$\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y$
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 course syllabus and textbook outline for a college-level Precalculus course, including major topics such as functions, trigonometry, analytic geometry, and systems of equations. The formulas and identities are standard for Precalculus and are essential for exam preparation.