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The graph of a function f is the set of all points in the plane of the form (x, f(x)). We could also define the graph of f to be the graph of the equation y = f(x). So, the graph of a function if a special case of the graph of an equation.
Example 1.
Let f(x) = x2 - 3.
Recall that when we introduced graphs of equations we noted that if we can solve the equation for y, then it is easy to find points that are on the graph. We simply choose a number for x, then compute the corresponding value of y. Graphs of functions are graphs of equations that have been solved for y!
The graph of f(x) in this example is the graph of y = x2 - 3. It is easy to generate points on the graph. Choose a value for the first coordinate, then evaluate f at that number to find the second coordinate. The following table shows several values for x and the function f evaluated at those numbers.
Each column of numbers in the table holds the coordinates of a point on the graph of f.
Exercise 1:
(a) Plot the five points on the graph of f from the table above, and based on these points, sketch the graph of f.
(b) Verify that your sketch is correct by using the Java Grapher to graph f. Simply enter the formula x^2 - 3 in the f text box and click graph.
Example 2.
Let f be the piecewise-defined function
To express f in a single formula for the Java Grapher or Java Calculator we write
(5 - x^2)*(xLE2) + (x - 1)*(2Lx).
The factor (xLE2) has the value 1 for x <= 2 and 0 for x > 2. Similarly, (2Lx) is 1 for 2 < x and 0 otherwise. If we evaluate the sum above at x = 3, the first product is 0 because (xLE2) is 0 and the second product is (3 - 1)*1=2. In other words, for x > 2, the formula evaluates to x - 1. If x <= 2, then the formula above is equal to 5 - x^2, which is exactly what we want!
The graph of f is shown below.
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