Fundamental Graphs and Properties of Precalculus Functions

Graphs and Properties of Basic Functions

Introduction

This study guide covers the fundamental graphs and properties of functions commonly encountered in a Precalculus course. Understanding these parent functions and their graphical characteristics is essential for analyzing and transforming more complex functions.

Basic Parent Functions

• Linear Function: $f(x) = x$ - Graph: Straight line through the origin with slope 1. - Domain: $(-\infty, \infty)$ - Range: $(-\infty, \infty)$ - Example: $f(x) = 2x + 1$ is a linear function shifted and with a different slope.

• Quadratic Function: $f(x) = x^2$ - Graph: Parabola opening upwards, vertex at (0,0). - Domain: $(-\infty, \infty)$ - Range: $[0, \infty)$ - Example: $f(x) = (x-2)^2 + 3$ is a parabola shifted right and up.

• Cubic Function: $f(x) = x^3$ - Graph: S-shaped curve passing through the origin. - Domain: $(-\infty, \infty)$ - Range: $(-\infty, \infty)$ - Example: $f(x) = (x+1)^3 - 2$

• Square Root Function: $f(x) = \sqrt{x}$ - Graph: Starts at (0,0) and increases slowly to the right. - Domain: $[0, \infty)$ - Range: $[0, \infty)$ - Example: $f(x) = \sqrt{x-3}$ is shifted right by 3 units.

• Absolute Value Function: $f(x) = |x|$ - Graph: V-shaped, vertex at (0,0). - Domain: $(-\infty, \infty)$ - Range: $[0, \infty)$ - Example: $f(x) = |x-2| + 1$

• Piecewise Functions: Defined by different expressions over different intervals. - Graph: May have jumps, holes, or different shapes in different regions. - Example: $f(x) = \begin{cases} x+2 & x < 0 \\ x^2 & x \geq 0 \end{cases}$

Key Properties of Functions

• Domain: The set of all possible input values (x-values) for which the function is defined.

• Range: The set of all possible output values (y-values) the function can produce.

• Intercepts: Points where the graph crosses the axes. - x-intercept: Where $f(x) = 0$ - y-intercept: Where $x = 0$

• Symmetry:

  • Even Function: $f(-x) = f(x)$ (symmetric about the y-axis)

  • Odd Function: $f(-x) = -f(x)$ (symmetric about the origin)

• End Behavior: Describes how the function behaves as $x \to \infty$ or $x \to -\infty$.

Common Graph Types and Their Features

• Parabola: $y = x^2$ - Opens upward if coefficient is positive, downward if negative. - Vertex is the minimum or maximum point.

• Circle (not a function): $x^2 + y^2 = r^2$ - Not a function because it fails the vertical line test.

• Step Function: $f(x) = \lfloor x \rfloor$ - Graph has jumps at integer values.

• Piecewise Linear: Combination of line segments, often used to model real-world situations.

Table: Summary of Parent Functions

Function	Equation	Domain	Range	Graph Shape
Linear	$y = x$	$(-\infty, \infty)$	$(-\infty, \infty)$	Straight line
Quadratic	$y = x^2$	$(-\infty, \infty)$	$[0, \infty)$	Parabola
Cubic	$y = x^3$	$(-\infty, \infty)$	$(-\infty, \infty)$	S-curve
Square Root	$y = \sqrt{x}$	$[0, \infty)$	$[0, \infty)$	Half-curve
Absolute Value	$y = |x|$	$(-\infty, \infty)$	$[0, \infty)$	V-shape
Step Function	$y = \lfloor x \rfloor$	$(-\infty, \infty)$	Integers	Steps

Graph Transformations

• Vertical Shifts: $f(x) + k$ shifts the graph up by $k$ units if $k > 0$, down if $k < 0$.

• Horizontal Shifts: $f(x - h)$ shifts the graph right by $h$ units if $h > 0$, left if $h < 0$.

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

• Stretching/Compressing: $a f(x)$ stretches vertically if $|a| > 1$, compresses if $0 < |a| < 1$.

Piecewise and Step Functions

• Piecewise Functions: Defined by different rules for different intervals of the domain. - Example: $f(x) = \begin{cases} x+1 & x < 0 \\ x^2 & x \geq 0 \end{cases}$

• Step Functions: Functions that increase or decrease abruptly from one constant value to another. - Example: $f(x) = \lfloor x \rfloor$ (greatest integer function)

Zeros and Intercepts

• Zero of a Function: Value of $x$ where $f(x) = 0$.

• Finding Intercepts:

  • x-intercept: Set $y = 0$ and solve for $x$.

  • y-intercept: Set $x = 0$ and solve for $y$.

Additional info:

• Some graphs in the images represent non-functions (such as circles) to illustrate the vertical line test.

• Piecewise and step functions are included to demonstrate discontinuities and jumps in graphs.

• Transformations are a key Precalculus topic, though not explicitly labeled in the images, they are inferred from the variety of graph shapes and positions.