Sketch A Curve

(You need Java Runtime Environment (JRE) to run GraphFunc applet in this website)

The techniques of sketching a curve consist of

Determining the domain of a curve
Finding first and second derivatives (to find relative max, relative minima and point of reflection)
Noting any symmetry
Finding any asymptotes
Noting some special values
Considering some limits as x and y go to infinity
Construct a table

These techniques are used to determine the curve’s trend. They may not be applicable in all cases, but, depending on the particular problem, some can always be used.

Example 1, Example3, Return to Main Page

Example 2. Graph function. Use GraphFunc utility online to confirm the results.

Solution

1. The domain of f(x) is determined for all.

2. Derivatives:

Solutions are: x = -1 and x = -4.

At x = -1 => f(-1) = -1 and

3. x intercept(s): f(x) = 0 <=> 2x + 5 = 0 => x = -5/2

y intercept(s): f(0) = -5/4

4. Vertical asymptotes: Set denominator of f(x) = 0 and solve for x, namely

. We consider the following limits:

and

Horizontal asymptotes: take limit of f(x) as x approaches to infinity, namely

. Therefore, y = 0 is horizontal asymptote.

5. Construct a table:

From the above table, we see that f(x) has a relative minima at x = -4 because the sign of f’(x) is changed from negative to positive as x passes through this point, and a relative maxima at x = -1 for f’(x) is changed from positive to negative as x passes through this point.

6. The information from the constructed table in step 5 shows the prediction of the trend of the graph of f(x) as shown in Figure 2.

Figure 2 – The graph of

If you use GraphFunc applet online to plot the graph and confirm the result, the valid syntax of the function f(x) be entered at the command line is (2*x + 5)/(x^2 – 4) .

Example 1, Example 3, Return to Main Page