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.
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