Graphing Functions Reference

Key Concepts

Domain: All possible input values ($x$) for which the function is defined.

Range: All possible output values ($y$) the function can produce.

Linear Functions

Slope ($m$) measures the steepness and direction of a line. It represents the rate of change: rise over run.

\[m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\Delta y}{\Delta x}\]

Forms

Form	Formula	Use Case
Slope-Intercept	\(include::example$math/linear-slope-intercept.adoc[\)]	When you know slope and y-intercept
Standard	\(include::example$math/linear-standard.adoc[\)]	Finding intercepts
Point-Slope	\(include::example$math/linear-point-slope.adoc[\)]	When you know slope and one point

Form

Formula

Use Case

Slope-Intercept

$include::example$math/linear-slope-intercept.adoc[$]

When you know slope and y-intercept

Standard

$include::example$math/linear-standard.adoc[$]

Finding intercepts

Point-Slope

$include::example$math/linear-point-slope.adoc[$]

When you know slope and one point

Graph Characteristics

Shape: Straight line
Domain: $(-\infty, \infty)$
Range: $(-\infty, \infty)$
Slope: $m > 0$ rises, $m < 0$ falls, $m = 0$ horizontal

$Linear Function Graph$

Slope Variations

$Linear Slope Variations$

Parallel and Perpendicular Lines

Parallel: Same slope ($m_1 = m_2$)
Perpendicular: Negative reciprocal slopes ($m_1 \cdot m_2 = -1$)

$Parallel and Perpendicular Lines$

Quadratic Functions

Vertex is the turning point of a parabola. For $f(x) = ax^2 + bx + c$, the vertex is at:

\[\left( -\frac{b}{2a}, f\left(-\frac{b}{2a}\right) \right)\]

Forms

Form	Formula	Use Case
Standard	\(include::example$math/quadratic-standard.adoc[\)]	General form, finding y-intercept
Vertex	\(include::example$math/quadratic-vertex.adoc[\)]	When you know the vertex \((h,k)\)
Quadratic Formula	\(include::example$math/quadratic-formula.adoc[\)]	Finding x-intercepts (roots)

Form

Formula

Use Case

Standard

$include::example$math/quadratic-standard.adoc[$]

General form, finding y-intercept

Vertex

$include::example$math/quadratic-vertex.adoc[$]

When you know the vertex $(h,k)$

Quadratic Formula

$include::example$math/quadratic-formula.adoc[$]

Finding x-intercepts (roots)

Graph Characteristics

Shape: Parabola
Domain: $(-\infty, \infty)$
Range: $[k, \infty)$ if $a > 0$, $(-\infty, k]$ if $a < 0$
Opens: Up if $a > 0$, down if $a < 0$
Axis of symmetry: $x = -\frac{b}{2a}$

$Quadratic Function Graph$

Vertex Form and Direction

$Quadratic Vertex Form$

$Quadratic Direction$

Quadratic Transformations

$Quadratic Transformations$

Polynomial Functions

General Form

\[f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\]

Cubic Function

\[f(x) = ax^3 + bx^2 + cx + d\]

Graph Characteristics

Degree determines end behavior and max turning points
Even degree: both ends same direction
Odd degree: ends opposite directions
Max turning points: $n - 1$ for degree $n$

$Cubic Function Graph$

Polynomial Degrees and End Behavior

$Polynomial Degrees$

Roots and Multiplicity

$Polynomial Roots$

Turning Points

$Polynomial Turning Points$

Exponential Functions

Forms

Type	Formula
General	\(include::example$math/exponential-general.adoc[\)]
Natural	\(include::example$math/exponential-natural.adoc[\)]

Type

Formula

General

$include::example$math/exponential-general.adoc[$]

Natural

$include::example$math/exponential-natural.adoc[$]

Graph Characteristics

Shape: J-curve
Domain: $(-\infty, \infty)$
Range: $(0, \infty)$
Horizontal asymptote: $y = 0$
Growth: $b > 1$, Decay: $0 < b < 1$

$Exponential Function Graph$

Growth vs Decay

$Exponential Growth vs Decay$

Exponential Transformations

$Exponential Transformations$

Asymptote Shifts

$Exponential Asymptote Shifts$

Logarithmic Functions

Forms

Type	Formula
General	\(include::example$math/logarithmic-general.adoc[\)]
Natural	\(include::example$math/logarithmic-natural.adoc[\)]

Type

Formula

General

$include::example$math/logarithmic-general.adoc[$]

Natural

$include::example$math/logarithmic-natural.adoc[$]

Graph Characteristics

Shape: Inverse of exponential
Domain: $(0, \infty)$
Range: $(-\infty, \infty)$
Vertical asymptote: $x = 0$
Key point: $(1, 0)$ always on graph

$Logarithmic Function Graph$

Different Bases

$Logarithm Bases$

Logarithmic Transformations

$Logarithmic Transformations$

Exponential and Logarithm as Inverses

$Exp-Log Inverse Relationship$

Rational Functions

Asymptote is a line that a curve approaches but never touches.

Vertical asymptote: $x = a$ where the function is undefined
Horizontal asymptote: $y = b$ as $x \to \pm\infty$
Oblique asymptote: diagonal line approached at infinity

General Form

\[f(x) = \frac{p(x)}{q(x)}\]

Graph Characteristics

Vertical asymptotes: where $q(x) = 0$
Horizontal asymptote: compare degrees of $p(x)$ and $q(x)$
Holes: common factors in numerator and denominator

$Rational Function Graph$

Asymptote Types

$Rational Asymptotes$

Rational Transformations

$Rational Transformations$

Holes vs Vertical Asymptotes

$Holes vs Asymptotes$

Radical Functions

Square Root

\[f(x) = \sqrt{x}\]

Graph Characteristics

Shape: Half parabola on its side
Domain: $[0, \infty)$
Range: $[0, \infty)$
Starting point: $(0, 0)$

$Square Root Function Graph$

Radical Function Types

$Radical Types$

Radical Transformations

$Radical Transformations$

Cube Root Function

$Cube Root Function$

Absolute Value Functions

Standard Form

\[f(x) = |x|\]

Graph Characteristics

Shape: V-shape
Domain: $(-\infty, \infty)$
Range: $[0, \infty)$
Vertex: at the point where expression inside equals zero

$Absolute Value Function Graph$

Absolute Value Transformations

$Absolute Value Transformations$

Piecewise Functions

Definition

A piecewise function is defined by different expressions over different intervals.

Step Functions

$Step Functions$

Floor and Ceiling Functions

Floor $\lfloor x \rfloor$: Greatest integer less than or equal to $x$
Ceiling $\lceil x \rceil$: Least integer greater than or equal to $x$

$Floor and Ceiling Functions$

Custom Piecewise Functions

$Piecewise Functions$

Trigonometric Functions

Basic Trigonometric Functions

$Basic Trig Functions$

Sine Transformations

General form: $y = A\sin(B(x - C)) + D$

A: Amplitude (vertical stretch)
B: Affects period (period = $\frac{2\pi}{|B|}$)
C: Phase shift (horizontal)
D: Vertical shift

$Trig Transformations$

Inverse Trigonometric Functions

$Inverse Trig Functions$

Complex Numbers

Imaginary unit ($i$) is defined as:

\[i = \sqrt{-1} \quad \text{where} \quad i^2 = -1\]

Complex number: $z = a + bi$ where $a$ is the real part and $b$ is the imaginary part.

Standard Form

\[z = a + bi\]

Argand Diagram (Complex Plane)

Complex numbers are plotted on a plane where:

Horizontal axis: Real part ($a$)
Vertical axis: Imaginary part ($b$)

The modulus (distance from origin):

\[|z| = \sqrt{a^2 + b^2}\]

The argument (angle from positive real axis):

\[\theta = \arctan\left(\frac{b}{a}\right)\]

Quick Reference Table

Function	General Form	Domain	Range
Linear	\(y = mx + b\)	\((-\infty, \infty)\)	\((-\infty, \infty)\)
Quadratic	\(y = ax^2 + bx + c\)	\((-\infty, \infty)\)	Depends on \(a\)
Polynomial	\(a_nx^n + \cdots + a_0\)	\((-\infty, \infty)\)	Depends on degree
Exponential	\(y = ab^x\)	\((-\infty, \infty)\)	\((0, \infty)\)
Logarithmic	\(y = \log_b(x)\)	\((0, \infty)\)	\((-\infty, \infty)\)
Rational	\(y = \frac{p(x)}{q(x)}\)	\(q(x) \neq 0\)	Varies
Square Root	\(y = \sqrt{x}\)	\([0, \infty)\)	\([0, \infty)\)
Absolute Value	\(y = \|x\|\)	\((-\infty, \infty)\)	\([0, \infty)\)
Step (Heaviside)	\(u(x) = \begin{cases} 0 & x < 0 \\ 1 & x \geq 0 \end{cases}\)	\((-\infty, \infty)\)	\(\{0, 1\}\)
Floor	\(y = \lfloor x \rfloor\)	\((-\infty, \infty)\)	\(\mathbb{Z}\)
Sine	\(y = \sin(x)\)	\((-\infty, \infty)\)	\([-1, 1]\)
Cosine	\(y = \cos(x)\)	\((-\infty, \infty)\)	\([-1, 1]\)
Tangent	\(y = \tan(x)\)	\(x \neq \frac{\pi}{2} + n\pi\)	\((-\infty, \infty)\)

Function

General Form

Domain

Range

Linear

$y = mx + b$

$(-\infty, \infty)$

Quadratic

$y = ax^2 + bx + c$

$(-\infty, \infty)$

Depends on $a$

Polynomial

$a_nx^n + \cdots + a_0$

$(-\infty, \infty)$

Depends on degree

Exponential

$y = ab^x$

$(-\infty, \infty)$

$(0, \infty)$

Logarithmic

$y = \log_b(x)$

$(0, \infty)$

$(-\infty, \infty)$

Rational

$y = \frac{p(x)}{q(x)}$

$q(x) \neq 0$

Varies

Square Root

$y = \sqrt{x}$

$[0, \infty)$

Absolute Value

$y = |x|$

$(-\infty, \infty)$

$[0, \infty)$

Step (Heaviside)

$u(x) = \begin{cases} 0 & x < 0 \\ 1 & x \geq 0 \end{cases}$

$(-\infty, \infty)$

$\{0, 1\}$

Floor

$y = \lfloor x \rfloor$

$(-\infty, \infty)$

$\mathbb{Z}$

Sine

$y = \sin(x)$

$(-\infty, \infty)$

$[-1, 1]$

Cosine

$y = \cos(x)$

$(-\infty, \infty)$

$[-1, 1]$

Tangent

$y = \tan(x)$

$x \neq \frac{\pi}{2} + n\pi$

$(-\infty, \infty)$