On-line Math 21

4.5 Curve Sketching

Example 1 f(x) = x²-2x+3 .

Solution

Although it was only mentioned in passing, it is useful to first find the intercepts, the places where the graph crosses an axis. This give a place to start drawing the graph.

Intercepts

The y -axis intercept is always easy to find. Just evaluate f(0) . In this case, f(0) = 3 , so (0,3) is the y -intercept.

The x -intercepts, where f(x) = 0 , (there may be more than one) are harder to find, since it requires solving an equation. Only try to solve these by hand if it is straightforward. If the solution looks difficult, wait until the rest of the graph is drawn to see where to look for solutions - or use Maple. In this case, we just have to solve the equation

0

=

x²-2x+3.

Wait a minute. Is that so easy to solve? Well, if you can't factor it with the standard game of trying to write

x²-2x+3 = (x-?)(x-??),

which I can't get to work out, you then try the quadratic formula. What this does is find the numbers r₁ and r₂ so that

x²-2x+3 = (x-r₁)(x-r₂),

or, equivalently (and what we want), find the roots of the equation x²-2x+3 = 0 , by a direct formula, in general, ax²+bx+c = 0 has roots r determined by

r =
-bą
Ö

b²-4ac

2a
.

In this case,

r =
+2ą
Ö

4-4·3

2
.

But this is impossible. There are no (real number) solutions, since that square root is of a negative number. So, there are no x -intercepts.

Increasing/decreasing, and critical points

If we find the derivative of the function,

f˘(x) = 2x-2,

which clearly means that f˘(x) = 0 only when x = 1 , and that the number line for f˘(x) in this case is

Thus x = 1 is a local minimum (decreasing down to it, increasing away). It is helpful to find the values at any critical point. Here, f(1) = 1²-2·1+3 = 2 , so (1,2) is the critical point.

Draw the graph

Then use this information to draw a fair representation of the graph.

The first information you plot are the intercepts; [Link to ex1-1.gif]
Then plot the critical point (and value). I always mark it with a horizontal line to remind myself that it is a critical point. [Link to ex1-2.gif]
Then fill in the graph, connecting the dots and making sure that the graph you draw is increasing/decreasing where the number line indicates it should be. [Link to ex1-3.gif]

[Each of these items should trigger the appearance of a new drawing with that information added.]

File translated from T_EX by T_TH, version 2.61.
On 21 Dec 2000, 00:24.