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Plotting the Graph of a Function Help

By — McGraw-Hill Professional
Updated on Aug 31, 2011

Introduction to Plotting the Graph of a Function

Until we learn some more sophisticated techniques, the basic method that we shall use for graphing functions is to plot points and then to connect them in a plausible manner.

Examples

Example 1

Sketch the graph of f (x) = x3x .

Solution 1

We complete a table of values of the function f .

  x       y = x 3x

  −3    −24

  −2    −6

  −1      0

  0        0

  1        0

  2        6

  3        24

We plot these points on a pair of axes and connect them in a reasonable way ( Fig. 1.41 ). Notice that the domain of f is all of Basics 1.8 Sets and Functions , so we extend the graph to the edges of the picture.

Basics 1.8 Sets and Functions

Fig. 1.41

Example 2

Sketch the graph of

Basics 1.8 Sets and Functions

Solution 2

We again start with a table of values.

x

y = f ( x )

 −3  

 −1

 −2  

 −1 

 −1  

 −1 

 0  

 −1 

 1  

 −1 

 2  

 −1 

 3  

 3 

 4  

 4 

 5  

 5 

We plot these on a pair of axes ( Fig. 1.42 ).

Basics 1.8 Sets and Functions

Fig. 1.42

Since the definition of the function changes at x = 2, we would be mistaken to connect these dots blindly. First notice that, for x ≤ 2, the function is identically constant. Its graph is a horizontal line. For x > 2, the function is a line of slope 1. Now we can sketch the graph accurately ( Fig. 1.43 ).

Basics 1.8 Sets and Functions

Fig. 1.43

You Try It: Sketch the graph of Basics 1.8 Sets and Functions

Examples

Example 3

Sketch the graph of Basics 1.8 Sets and Functions .

Solution 3

We begin by noticing that the domain of f , that is the values of x for which the function makes sense, is { x : x ≥ −1}. The square root is understood to be the positive square root. Now we compute a table of values and plot some points.

 

x

Basics 1.8 Sets and Functions

−1

0

0

1

1

Basics 1.8 Sets and Functions

2

Basics 1.8 Sets and Functions

3

2

4

Basics 1.8 Sets and Functions

5

Basics 1.8 Sets and Functions

6

Basics 1.8 Sets and Functions

Connecting the points in a plausible way gives a sketch for the graph of f ( Fig. 1.44 ).

Basics 1.8 Sets and Functions

Fig. 1.44

Example 4

Sketch the graph of x = y 2

Solution 4

The sketch in Fig. 1.45 is obtained by plotting points. This curve is not the graph of a function.

Basics 1.8 Sets and Functions

Fig. 1.45

A curve that is the plot of an equation but which is not necessarily the graph of a function is sometimes called the locus of the equation. When the curve is the graph of a function we usually emphasize this fact by writing the equation in the form y = f (x).

You Try It: Sketch the locus x = y2 + y.

Practice problems for this concept can be found at: Calculus Basics Practice Test.

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