Relations, Graphs, and Functions

In this section we will study functions in more depth and begin by defining a relation as any set of ordered pairs.

An example of a relation might look like { (2,3), (2,5), (0,1) }. Here we have a relation consisting of three ordered pairs or points on the Cartesian coordinate system. We are familiar with ordered pairs and usually see them denoted as (x, y).  Typically the x-value (the first component) will be the independent variable or input and the y-value (the second component) is the dependent variable or output.

The example { (2,3), (2,5), (0,1) } is NOT a function because the x-value 2 is assigned more than one y-value, namely 3 and 5. For every x-value there can be only one y-value. Next we define the domain as the set of x-values and the range as the set of y-values for which the relation is defined.

Example:

Tip: When looking at a list of ordered pairs, if there are repeating x-values then the relation is not a function.  This usually indicates that there is an input with multiple outputs. (This does not apply to the y-values)

Determine if the relations are functions.  If so state the domain and range.

 

 

Relations can consist of an infinite number of ordered pairs, in which case, making a big list would be impossible. A graph can represent a relation by considering it as a big list of ordered pairs.  Each dot on the graph represents an ordered pair (x, y).

 

Notice that if we can draw a vertical line that intersects the graph twice we will be able to identify one x-value with two corresponding y-values.  Therefore, it can not be a function.

  

Alternatively, if any vertical line crosses the graph only once then it does represent a function.

Use the vertical line test to determine whether or not the graph represents a function.

 

Be prepared to state the domain and range given the graph.  Remember to think of the graph as an infinite set of ordered pairs (x, y).  From the graph, determine the x-values and y-values. First shade in the domain and range then convert to interval notation.

 

It should be clear, at this point, that the circle shown is not a function.  Nonetheless, we still can determine the domain and range of the relation that it represents.

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Graphing Functions using Transformations

One way to graph functions is to simply plot points.  In this section, we cover a method used to quickly sketch graphs related to some basic functions. Here we focus on rigid transformations, that is, transformations that do not change the shape of the graph.

Vertical Translations: [ Interactive Graph ]
If k is any positive real number then,

The graph of the basic function f(x) = sqrt(x) follows:

Using this basic graph and the vertical translations described above we can sketch f(x) = sqrt(x) + 2 by shifting all of the points up 2 units. Similarly, graph g(x) = sqrt(x) − 3 by shifting all points down 3 units.

     

Horizontal Translations:  [ Interactive Graph ]

If h is any positive real number then,

Using the graph of f(x) = sqrt(x) and the horizontal translations described above we can sketch f(x) = sqrt(x + 4) by shifting all of the points left 4 units. Similarly, graph g(x) = sqrt(x − 3) by shifting all points right 3 units.

     

Reflections: [ Interactive Graph ]

Given any function f(x),

Using the graph of f(x) = sqrt(x) sketch the graph of f(x) = −sqrt(x)  by reflecting all of the points about the x-axis. Similarly, graph f(x) = sqrt(−x) by reflecting all of the points about the y-axis.

     

For the first function f(x) = −sqrt(x) all of the y-values are negative which results in a reflection about the x-axis. For the second function f(x) = sqrt(−x) all of the x-values must be negative thus resulting in a reflection about the y-axis.

Sketch the graph.

The -1 indicates a reflection of the graph of the squaring function  f(x) = x^2 about the x-axis. Be sure to graph the squaring function using a dashed curve because it will be used as a guide and is not the answer. Next, reflect all points about the x-axis and draw in the final graph with a solid curve.

General Steps for Graphing Functions using Transformations:

   1. Identify and graph the basic function using a dashed curve.

   2. Identify any reflections first and sketch them using the basic function as a guide.

   3. Identify any translations.

   4. Use this information to sketch the final graph using a solid curve.

Graph the function and determine the domain and range.

 

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Use the transformations to determine the equation that represents the given function.

Example:

From the general shape of the graph we can determine the basic function and transformations. The graph has the shape of the square root function,

y = sqrt(x)

Next, notice the reflection about the y-axis,

y = sqrt(−x)

And finally, we see a shift up 1 unit.

y = sqrt(−x) + 1

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Example:

The given function has the general shape of the squaring function (parabola),

y = x^2

Next, notice the shift right 3 units,

y = (x − 3)^2 

And finally, we see a shift down 2 units.

y = (x − 3)^2 − 2

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Example:

The given function has the general shape of the absolute value function,

y = abs(x)

Next, notice the reflection about the x-axis,

y = −abs(x) 

Finally we see a shift left 1 unit and down 2 units

y = −abs(x+1) − 2

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