1.3 Functions


Contents


1.3.1 Terminology

$17+5=22$ sum
$17-5=12$ difference
$17·5=22$ product
$\frac{17}{5}=3+\frac{2}{5}$ $17$: dividend
$5$: divisor
$17/5$: simplified fraction
$3$: quotient
$2$: remainder
$3+2/5$: proper fraction
$\frac{x}{x_∘}+\frac{y}{y_∘}=1$ $x$: abscissa
$x_∘$: horizontal axis intercept
$+$: operator
$y$: ordinate
$y$: vertical axis intercept
$1$: value
$y=a·x^2+b·x+c$ $y$: dependent variable
$a$: leading coefficient
$x$: independent variable
$2$: order
$b$: coefficient
$c$: constant
$(a+b·i)(a-b·i)=a^2+b^2$ ( expression = expression ) ← equation
factored form = expanded form
$a$: real component
$b·i$: imaginary component
$(a±b·i)$: roots
$(a+b·i)(conjugate)$
$\sqrt[n]{x}$ $n$: $n$th root
$x$: radicand
$x/x_∘$
$y/y_∘$
$a·x^2$
$b·x$
$(a±b·i)$
$a^2$
$b^2$
$\sqrt[n]{x}$
terms
Jump to Completing the Square
Jump to Matrix Notation

1.3.2 The Term "General"

General means that all possible situations are represented or that something applies to multiple scenarios.
Jump to Quadratic Equations Definition
Jump to Completing the Square
Jump to Quadratic Formula
Example 1
Using the quadratic formula to solve for:
Zeros of $x$ only (specific) $$x=\frac{-b \pm \sqrt{b^2-4·a·y_∘}}{2·a}$$ All values of $x$ (general) $$x=\frac{-b \pm \sqrt{b^2+4·a·(y-y_∘)}}{2·a}$$
Example 2
The conic sections general formula describes ellipses, circles, parabolas, hyperbolas, and lines. $$A·x^2+B·x·y+C·y^2+D·x+E·y+F=0$$

1.3.3 Definitions

Even - symmetric about $y$-axis
$$f(-x)=f(x)$$
Odd - symmetric about origin
$$f(-x)=-f(x)$$
Inverse - reflected about $y=x$
$$f^{-1}(f(x))=x$$
Absolute value - magnitude; non-negative
$$ |x| = \begin{cases} \phantom{-}x &x ≥ 0 \\ -x &x \lt 0 \end{cases} $$
Periodic - repeated segment ($P$)
$$f(x+P)=f(x)$$
All images in this section were taken from Wolfram Alpha and edited
Learn: Paul's Online Notes
Jump to Integer Powers of Imaginary Units

1.3.4 Transformations

$$a·f(b·x+c)+d$$
Learn: Yoshiwara Books