Functions and Quadratic Functions: Definitions, Properties, Graphs, and Applications

Functions and Their Graphs

Definition of a Function

A function is a relation between two non-empty sets, X (domain) and Y (co-domain), such that each element in X is assigned a unique element in Y. This is denoted as $f: X \rightarrow Y$.

• Domain: The set X, containing all possible input values.

• Co-domain: The set Y, containing all possible output values.

• Vertical Line Test: A graph represents a function if every vertical line intersects the graph at most once.

Determining Functions from Relations

• A relation is a function if no input (x-value) is paired with more than one output (y-value).

• Example: The set {(1, 1), (−1, 2), (2, 1), (3, 1), (−3, 3)} is a function because each x-value is unique.

• Example: {(1, 1), (1, 2), (2, 1), (3, 1), (−3, 3)} is not a function because x = 1 is paired with two different y-values.

Evaluating Functions

• To evaluate a function, substitute the given value into the function's formula.

• Example: For $f(x) = x^2$, $f(0) = 0^2 = 0$, $f(1) = 1^2 = 1$, $f(-1) = (-1)^2 = 1$.

Difference Quotient

The difference quotient is used to measure the average rate of change of a function:

• Formula: $\frac{f(x+h) - f(x)}{h}$

• Example: For $f(x) = x^2$, $\frac{(x+h)^2 - x^2}{h} = \frac{x^2 + 2xh + h^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h$

Operations on Functions

• Addition: $(f+g)(x) = f(x) + g(x)$

• Multiplication: $(f \cdot g)(x) = f(x) \cdot g(x)$

• Division: $(f/g)(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

• Example: If $f(x) = x^2$, $g(x) = x + 2$, then $(f+g)(x) = x^2 + x + 2$, $(f \cdot g)(x) = x^2(x + 2)$, $(f/g)(x) = \frac{x^2}{x + 2}$

Graph of a Function and the Vertical Line Test

• A graph is a function if every vertical line crosses it at most once.

• Example: The graph of $y = x^2$ passes the vertical line test.

Intercepts and Domain

• x-intercept: Where the graph crosses the x-axis ($y = 0$).

• y-intercept: Where the graph crosses the y-axis ($x = 0$).

• Domain: All possible input values for which the function is defined.

Properties of Functions

Even and Odd Functions

Functions can be classified based on their symmetry:

• Even Function: $f(-x) = f(x)$ for all $x$ in the domain. The graph is symmetric with respect to the y-axis. Example: $f(x) = x^2$.

• Odd Function: $f(-x) = -f(x)$ for all $x$ in the domain. The graph is symmetric with respect to the origin. Example: $f(x) = x^3$.

• Neither: If neither condition holds, the function is neither even nor odd.

Increasing, Decreasing, and Constant Intervals

• A function is increasing on an interval if $f(x_1) < f(x_2)$ whenever $x_1 < x_2$.

• A function is decreasing on an interval if $f(x_1) > f(x_2)$ whenever $x_1 < x_2$.

• A function is constant if $f(x_1) = f(x_2)$ for all $x_1, x_2$ in the interval.

Local and Absolute Extrema

• Local Maximum: $f(c)$ is a local maximum if $f(c) \geq f(x)$ for all $x$ near $c$.

• Local Minimum: $f(c)$ is a local minimum if $f(c) \leq f(x)$ for all $x$ near $c$.

• Absolute Maximum: $f(c)$ is the largest value on the interval.

• Absolute Minimum: $f(c)$ is the smallest value on the interval.

Library of Functions and Piece-wise Defined Functions

Common Functions and Their Properties

Function	Graph Symmetry	Domain	Range	Intercepts
$f(x) = x$	None	All real numbers	All real numbers	(0,0)
$f(x) = x^2$	y-axis	All real numbers	$[0, \infty)$	(0,0)
$f(x) = x^3$	Origin	All real numbers	All real numbers	(0,0)
$f(x) = \sqrt{x}$	None	$[0, \infty)$	$[0, \infty)$	(0,0)
$f(x) = \frac{1}{x}$	Origin	$x \neq 0$	$y \neq 0$	None
$f(x) = |x|$	y-axis	All real numbers	$[0, \infty)$	(0,0)

Piece-wise Defined Functions

A piece-wise function is defined by different expressions on different intervals of the domain.

• Example: $f(x) = \begin{cases} 2x + 3, & x < 2 \\ x + 1, & x \geq 2 \end{cases}$

• To evaluate, determine which interval x belongs to and use the corresponding formula.

• Piece-wise functions must have mutually exclusive intervals to be valid functions.

Example: Cellphone Plan as a Piece-wise Function

• $C(t) = \begin{cases} 30, & 0 \leq t \leq 120 \\ 30 + 0.30(t - 120), & t > 120 \end{cases}$

• This models a plan with a flat rate for the first 120 minutes and a per-minute charge thereafter.

Transformations of Functions

Vertical and Horizontal Shifts

• Vertical Shift: $f(x) + k$ shifts the graph up by $k$ units; $f(x) - k$ shifts it down by $k$ units.

• Horizontal Shift: $f(x + k)$ shifts the graph left by $k$ units; $f(x - k)$ shifts it right by $k$ units.

Stretching and Compressing

• Vertical Stretch: $k f(x)$, $k > 1$, stretches the graph vertically by a factor of $k$.

• Vertical Compression: $\frac{1}{k} f(x)$, $k > 1$, compresses the graph vertically by a factor of $1/k$.

• Horizontal Stretch: $f(\frac{1}{k} x)$, $k > 1$, stretches the graph horizontally by a factor of $k$.

• Horizontal Compression: $f(kx)$, $k > 1$, compresses the graph horizontally by a factor of $1/k$.

Reflections

• Across y-axis: $f(-x)$ reflects the graph horizontally.

• Across x-axis: $-f(x)$ reflects the graph vertically.

Linear and Quadratic Functions

Linear Functions

• A linear function has the form $f(x) = mx + b$.

• Slope (m): Measures the rate of change; $m = \frac{f(b) - f(a)}{b - a}$.

• y-intercept (b): The value of $f(0)$.

• The graph is a straight line.

• Increasing if $m > 0$, decreasing if $m < 0$, constant if $m = 0$.

Quadratic Functions

• A quadratic function has the form $f(x) = ax^2 + bx + c$, $a \neq 0$.

• Vertex: $\left( -\frac{b}{2a}, f\left( -\frac{b}{2a} \right) \right)$

• Axis of Symmetry: $x = -\frac{b}{2a}$

• Discriminant: $d = b^2 - 4ac$

• If $a > 0$, parabola opens upward; if $a < 0$, opens downward.

Graphing Quadratic Functions

1. Find the vertex and plot it.

2. Compute the discriminant to determine the number of x-intercepts.

3. Plot intercepts and additional points using symmetry.

Quadratic Models and Applications

• Quadratic functions are used to model maximum/minimum problems, such as area, revenue, and geometric applications.

• Example: The area of a rectangle with perimeter constraints can be expressed as a quadratic function of one side.

Quadratic Inequalities

Solving Quadratic Inequalities

• Graphically: Plot the functions and identify intervals where the inequality holds.

• Analytically: Solve $ax^2 + bx + c \leq 0$ or $ax^2 + bx + c \geq 0$ by finding roots and testing intervals.

• Example: $25x^2 + 16 \geq 40x$ can be rewritten as $25x^2 - 40x + 16 \geq 0$ and solved using the quadratic formula.

Domain of Functions Involving Square Roots

• For $f(x) = \sqrt{x^2 - 6x}$, the domain is all $x$ such that $x^2 - 6x \geq 0$.

• Solve $x^2 - 6x \geq 0$ to find the domain.

Summary Table: Types of Function Transformations

Transformation	Operation	Effect
Vertical Shift	$f(x) + k$	Up by $k$ units
Vertical Shift	$f(x) - k$	Down by $k$ units
Horizontal Shift	$f(x + k)$	Left by $k$ units
Horizontal Shift	$f(x - k)$	Right by $k$ units
Vertical Stretch	$k f(x)$, $k > 1$	Stretched vertically by $k$
Vertical Compression	$\frac{1}{k} f(x)$, $k > 1$	Compressed vertically by $1/k$
Horizontal Stretch	$f(\frac{1}{k} x)$, $k > 1$	Stretched horizontally by $k$
Horizontal Compression	$f(kx)$, $k > 1$	Compressed horizontally by $1/k$
Reflection (y-axis)	$f(-x)$	Reflects across y-axis
Reflection (x-axis)	$-f(x)$	Reflects across x-axis

Examples and Applications

• Piece-wise Function Example: $f(x) = \begin{cases} 3x + 5, & x < 0 \\ 4x + 7, & x \geq 0 \end{cases}$

• Quadratic Model Example: The area of a rectangle with perimeter $P$ and side $x$ is $A(x) = x(P/2 - x)$.

• Supply and Demand Example: $S = -1000 + 100p$, $D = 5000 - 200p$; equilibrium occurs when $S = D$.

Additional info: Some context and examples were inferred for completeness, such as the general forms of piece-wise functions and quadratic applications.