Transformation of Functions: Tutorial

Types of Functions

While it might be possible to see all functions as transformations of other functions, it's more helpful to group them by type and analyze transformations within those groups. You've probably noticed that the graphs of functions vary in their shapes and lines of symmetry, in their positions on the x- and y-coordinates, and in the width and length of the shapes they form. Graphs of functions that share common characteristics are grouped into families. Each family of functions has a prototypical function called the parent function. In this lesson, we'll study six of the eight parent functions.

1. The identity function

has the form f(x) = x. The domain and the range of this function are the set of all real numbers, or (-∞, ∞). The graph of this parent function is a straight line passing through the origin with a slope of 1, so for each point on the graph, the y-coordinate is equal to the x-coordinate. The graph is symmetrical across the origin and across the line f(x) = -x.

3. A quadratic function

has the form f(x) = x^2. The domain, or the input values, for this function is the set of all real numbers, or (-∞, ∞). But the range includes only positive real numbers and 0, or [0, ∞), because the square of both positive and negative numbers is positive. The graph of the parent quadratic function is on or above the x-axis and is symmetric about the y-axis. This function's graph is called a parabola.

2. The absolute value function

has the form f(x) = |x|, and its domain is the set of all real numbers, or (-∞, ∞). Because the output value is an absolute value, the range of this function includes only positive real numbers and 0, or [0, ∞). The graph of this parent function is always on or above the x-axis because the y-coordinate is never negative, resulting in the V-shaped graph seen here. The graph is symmetrical about the y-axis.

6. Cube root functions

have the general form .The domain and the range of these functions include all real numbers (-∞, ∞). The graph of this parent function is symmetric with respect to the origin.

5. Cubic functions

have the general form f(x) = x^3. Like quadratic functions, the domain of this type of function includes all real numbers (-∞, ∞). The cube of negative numbers is negative and the cube of positive numbers is positive, so the range of this function as well as its domain is (-∞, ∞). The graph of this parent function is symmetric about the origin, that is, it is reflected across both the x- and the y- axes.

4. Square root functions

have the general form f(x) = √x. Because we're dealing only with real-valued functions, the domain of this family of functions includes only positive real numbers and 0, or [0, ∞). The square root of a positive number is also positive, and the square root of 0 is 0, so the range of a square root function is also the positive real numbers and 0, or [0, ∞). Because both the domain and the range exclude negative numbers, the graph of this function lies completely in the first quadrant.

Reflections

Another type of transformation, called a reflection, occurs when you change the sign of a parent function. This type of transformation is said to reflect the original function around a certain line of symmetry, such as the x-axis or the y-axis, or around the origin. For example, let's take the opposite of the whole function f(x) = x^2. The result is g(x) = -f(x) = -(x^2). Here is a table of values for f(x) and g(x). The output values for f(x) and g(x) will differ only in sign. As you can see from the graph, g(x) is a reflection of f(x) across the x-axis. Each output is equidistant from the x-axis but on opposite sides; that is, for each point (x, y) on f(x) there is a point (x, -y) on g(x).

Horizontal and Vertical Shifts

Besides categorizing functions by types, we also categorize transformations based on how the graph of the resulting function differs from the graph of the parent function. For example, in the example you just saw, the graph of the transformed function shifted up two units. This type of transformation is called a vertical shift. For any parent function, a vertical shift means that the transformed graph is shifted either upward or downward. What would happen if we transformed the parent function f(x) = x2 to g(x) = x2 − 3? As you can see from the graph, the transformation causes a vertical shift downward of three units. Generalizing from these examples, we can conclude that when we transform a basic function by adding or subtracting a positive real number, a, the graph of the function undergoes a vertical shift: -If g(x) = f(x) + a, the vertical shift is a units up. -If g(x) = f(x) - a, the vertical shift is a units down.

Introduction

Bill Sands, the head accountant at Fritz and Cobb (F & C) Electronics, has for several years used this function, y = 100x^2, where x is the number of items produced per year, to project the company's approximate annual profit. Last year, the company began to lease out a piece of its land for $10,000 per year. Since the lease runs for 20 years, the profit function of the company for the next 20 years has changed. In his latest presentation to the F & C shareholders, Mr. Sands shared a new profit function that accounts for the additional income, y = 100x^2 + 10,000. The graph illustrates how the two profit functions work. What do you see in the graphs of the two functions? Specifically, how are they similar? How do the similarities and differences in the two graphs reflect the two functions for the company's profits? Observe that if you move the graph of the first function, y = 100x^2, up the y-axis by 10,000 units, you get the graph of y = 100x^2 + 10,000. The graphs have the same shape, but one has shifted up because of the additional $10,000.

Types of Transformations

If we alter any of these parent functions by performing a mathematical operation on them, the graph of the resulting function will retain the basic shape and features of the parent function's graph. The graph of the new function is called a transformation of the parent graph. Let's look first at transformations of the most basic type of function, the identity function. The form of the parent function is f(x) = x. We'll alter this function to g(x) = x + 2. As with the profit function we saw earlier in this lesson, g(x) is a transformation of f(x) in which the only change is that the graph of g(x) has shifted two units up. The shape and the slope of the two lines are the same.

Stretching and Shrinking Transformations

The transformations of functions that we've seen so far, shifts and reflections, have involved changes only in the position or orientation of the graph. The shape of the graphs remained the same. Those types of transformations are called rigid transformations. However, some transformations distort a graph, causing it to be stretched or squeezed. Such transformations, called nonrigid transformations, occur when we multiply a function or the variable of a function by a real number.