Section 3.1 - Basic Graphs and Transformations

Section Objectives

1. Develop a familiarity with the graphs of basic functions (Toolbox Functions).
2. Apply the transformations (shifts, reflections, stretches, and compressions) to basic graphs to obtain more general graphs.
3. Determine the transformations that result in a given graph.

Summary of the Toolbox Functions

Before we study the graphs of general functions, we must build up a library, or toolbox, of basic functions and their graphs. The "toolbox functions" are the constant functions, the linear functions, the squaring and cubing functions (eventually general power functions), the square and cube root functions (eventually $n$-th roots), and the absolute value function. We must become familiar with the following characteristics of these functions and their graphs:

• Domain
• Range
• Interesting features (vertices, flat spots, opens up/down, asymptotes, intercepts, etc.)
• Symmetry
• End (long-term) behavior
• Increasing/Decreasing
• Extreme values

Examples

• Discuss the characteristics of each of the toolbox functions.

(1) Constant, (2) Linear, (3) Squaring, (4) Cubing, (5) Square Root, (6) Cube Root, (7) Absolute Value

Transformations

This table summarizes typical transformations of a given function/graph.

Transformed Function	Effect on Graph
$y=f(x)+k$	Graph of $f$ shifted $k$ units up
$y=f(x+k)$	Graph of $f$ shifted $k$ units left
$y=-f(x)$	Graph of $f$ reflected about $x$-axis
$y=f(-x)$	Graph of $f$ reflected about $y$-axis
$y=k\,f(x), \, k>1$	Graph of $f$ stretched vertically
$y=k\,f(x), \, 0<k<1$	Graph of $f$ compressed vertically
$y=f(kx), \, k>1$	Graph of $f$ compressed horizontally
$y=f(kx), \, 0<k<1$	Graph of $f$ stretched horizontally

Examples

• Describe, then sketch, the graph of $f(x) =x^2-4$. (Solution)

• Describe the graph of $g(x)=(x+1)^2-4$. (Solution)

• Describe the graph of $y=2\sqrt{x}$ . How about $y=-2\sqrt{x}$? (Solution)

Important idea: When a function is transformed, many of its special features, such as domain, range, and intervals on which it is increasing/decreasing, change in corresponding ways.

• Discuss the features of the function $g(x)=-2(x+3)^2-5$. Domain? Range? Vertex? (Solution)

For further help...

There are lots of resources available to help you with transformations.

1. Do a Google search for "Transformations of functions".
2. Here is a video posted by a colleague at UIC.
3. Use Desmos or Geogebra to graph functions, and transform by sliding the graph.