Calculus Graphs Study Guide
A function can be fully described by showing what happens at each number in its domain (for example, 4→ 2) or by giving its formula (for example, f(x) = √x). However, neither of these provides a clear overall picture of the function.
Luckily for us, René Descartes came up with the idea of a graph, a visual picture of a function. Rather than say 4→ 2 or f( 4) = 2, we plot (4,2) on the Cartesian plane, which would look like Figure 2.1.
If we plotted all the points in the domain of f(x) = √x (not just the whole numbers, but all the fractions and decimals, too), then the points would be so close together that they would form a continuous curve as in Figure 2.2.
The graph shows us several interesting characteristics of the function f(x) = √x. Because the graph starts at x = 0 and runs to the right, this means that the domain is x ≥ 0.
We can see that the function f(x) = √x is increasing (going up from left to right) and not decreasing (going down from left to right).
The function f(x) = √x is concave down because it curves downward (see Figure 2.3) like a frown and not concave up like a smile (see Figure 2.4).
Note on Finding Coordinates
We put the y into the formula y = f(x) = √x to imply that the y-coordinates of our points are the numbers we get by plugging the x-coordinates into the function f.
Use the graph of the following function (see Figure 2.5) to determine the domain, where the function is increasing and decreasing, and where the function is concave up and concave down.
The domain of g consists of all real numbers because there is a point above or below every number on the x-axis.
The function g is increasing up to the point at x = 2, where it then decreases down to x = 8, and then increases ever afterward. To save space, we say that g increases on (–∞,2) and on (8,∞), and that g decreases on (2,8).
The point at (2,6) where g stops increasing and begins to decrease is the highest point in its immediate area and is called a local maximum. The point at (8,3) is similarly a local minimum, the lowest point in its neighborhood. These points tend to be the most interesting points on a graph.
The concavity of g is trickier to estimate. Clearly g is concave down in the vicinity of x = 2 and concave up around x = 7 and x = 8. The exact point where the concavity changes is called a point of inflection. On this graph, it seems to be at the point (5,4), though some people might imagine it a bit earlier or later. Thus, we say that g is concave down on (–∞,5) and concave up on (5,∞).
To be completely honest, any information obtained by looking at a graph is going to be a rough estimate. Is the local maximum at (2,6), or is it at (2.0003,5.9998)? There is no way to tell the difference. Graphs made up by people, like the ones in this lesson, tend to have everything interesting happen at whole numbers. Graphs formed using real-world data tend to be much less kind.
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