The goal of this section is to be able to go from a formula of a function to an accurate graph of that function. We'll use the first and second derivatives to help find exact points on the graph and to determine the overall shape(s) of the graph.
There are three steps to drawing a graph.
Make sure you're happy with each step by itself. The following examples put all the steps together.
Let f (x) = xex. Sketch a graph of f (x). Label all intercepts, critical points, and inflection points.
We'll go through the three steps.
• Find dots (intercepts, critical points, inflection points).
To find the y-intercept we plug in 0 for x and see what we find:
f (0) = 0e0 = 0.
The y-intercept is (0, 0). This is also the only x-intercept since the only time xex can be zero is if x is zero.
Now we find the critical points. From this example we know that there's a critical point at x = -1. In order to graph this point we need the full coordinates:
f (-1) = -1e-1 ≈ -0.37.
There's a critical point at approximately (-1, -0.37).
Now for inflection points. We know from this exercise that there's an inflection point at x = -2. Since
f (-2) = -2e-2 ≈ -0.27
we have an inflection point at approximately (-2, -0.27).
Step 1 is done. We have points:
• Find shapes.
Set up a numberline, marking all the important points we found:
Now figure out the signs of f, f ' and f " in the intervals between the important points. We know f is negative when x is negative, and positive when x is positive:
We know f ' is negative when x < -1 and positive when x > -1:
Finally, we know f " is negative when x < -2 and positive when x > -2:
Using this information, we can figure out the shape of f over each interval:
• Play connect-the-dots with the shapes.
Since f is negative for all negative values of x, we know the concave down, decreasing shape to the left of x = -2 must stay below the x-axis.
We play connect-the-dots, and find this:
There are three types of points to find on the function, and the great thing is that you already know how to find all of them.
In summary, we're finding where f, f ', and f " are zero or undefined. These will mostly be dots, but there may be asymptotes or holes where f is undefined.
Here's the only thing you need to do that we didn't do earlier: after finding the x-value of a CP or IP, plug that x-value back into the original function f to find the corresponding y-value. In order to graph a point, we need to know both coordinates.
We talked earlier about how we can tell from the derivative whether the original function is increasing or decreasing. We also talked about concavity, and how if we toss the second derivative into the mix, we can also tell what shape the original function should have.
We need to go over the shapes again here, because the more fluent we are at translating between signs of derivatives and shapes of graphs, the easier this whole business will be.
There are four possible shapes a piece of graph can have. Actually, this is a lie, there are seven possible shapes, but three of them don't need calculus.
Here are the ones that do need calculus:
• increasing and concave up
If f is increasing and looks like part of a right-side up bowl, it looks like this:
• increasing and concave down
If f is increasing and looks like part of a upside-down bowl, it looks like this:
• decreasing and concave up
If f is decreasing and looks like part of a right-side up bowl, it looks like this:
• decreasing and concave down
If f is decreasing and looks like part of a upside-down bowl, it looks like this:
The shapes that don't need calculus are the ones where f has no concavity, but is a straight line. If f is a line, it can have three possible shapes:
This part is as fun as it sounds. It's like those connect-the-dots games on restaurant placemats. Maybe not exactly like them, no sweet bear or zebra shapes (most likely), but we know the dots go in order from left to right, and we're supposed to connect the dots with particular shapes.
The function f hits the points (0,0) and (1,4) and is increasing & concave up from x = 0 to x = 1. We have two dots:
We have a shape:
We connect the dots with the shape:
That's all there is to it.