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) = x^{3} − x .
Solution 1
We complete a table of values of the function f .
x y = x ^{3} − x
−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 , so we extend the graph to the edges of the picture.
Example 2
Sketch the graph of
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 ).
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 ).
You Try It: Sketch the graph of
Examples
Example 3
Sketch the graph of .
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 

−1 
0 
0 
1 
1 

2 

3 
2 
4 

5 

6 
Connecting the points in a plausible way gives a sketch for the graph of f ( 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.
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 = y^{2} + y.
Practice problems for this concept can be found at: Calculus Basics Practice Test.
Ask a Question
Have questions about this article or topic? AskPopular Articles
 Kindergarten Sight Words List
 First Grade Sight Words List
 10 Fun Activities for Children with Autism
 Child Development Theories
 Social Cognitive Theory
 Signs Your Child Might Have Asperger's Syndrome
 Why is Play Important? Social and Emotional Development, Physical Development, Creative Development
 A Teacher's Guide to Differentiating Instruction
 Theories of Learning
 Definitions of Social Studies