This is the process of sketching a curve, based on information about the function. We look at different parts of the function such as its derivative and second derivative.
Step 1. Consider any information given about the type of function. For example if the graph is polynomial or rational. Whatever type of function it is will determine its domain. The domain is very important when sketching the graph.
Step 2. Mark any asymptotes, which are straight lines approached by a given curve as one of the variables in the equation of the curve approaches infinity.
- If the limit of f(x) as x approaches negative infinity or positive infinity is equal to k, then there is a horizontal asymptote of y=k
- If the limit of f(x) as x approaches c is negative infinity or positive infinity, then there is a vertical asymptote of x=c.
Step 3. Plot all given points
- These are any known points on the graph such as x-intercept and y-intercept.
Step 4. Consider first derivative sign chart. The first derivative sign chart tells you where the graph is increasing or decreasing at a certain point. The points where the graph changes from increasing to decreasing is considered a critical value point on the graph.The critical values are where the second derivative is equal to zero. If the first derivative is positive at a point then it is increasing at that point and if the first derivative is negative at a point it is decreasing at that point.
- Mark horizontal tangent lines whenever f'(x)=0
- Mark f'(x) is not defined at a point, determine whether it has a discontinuity, vertical tangent line, or sharp bend and mark according
- Mark all local minimums and maximums. (remember to pay attention to whether it’s a curvy max/min or sharp bend.
Step 5. Consider the second derivative sign chart. The second derivative tells you if the graph is concave up or concave down at a certain point. The second derivative can tell us the inflection points of the graph which are points where the graph changes concavity. If the second derivative is negative at a point it is concave down at that certain point and if the second derivative is positive at a point it is concave up at that certain point. Sketch in the curve one piece at a time taking the intervals from the second derivative sign chart, paying attention to the concavity and also pay attention to the notes you have already marked on the graph.
Step 6. You should be able to sketch the graph.
When the derivative does not exist:
There are 3 reasons the derivative may not exist:
1. Different types of discontinuities
- Point Discontinuities
- Jump Discontinuities
- Vertical Asymptotes
2. Vertical Tangent Lines
3. Sharp Bend
Once you make the sign charts for the first derivative and the second derivative, sketching the graph is easy. You have to sketch the graph one piece at a time. It is important to focus on key points of the graph and not try to drawl the whole graph at once. This will lead to errors and simple mistakes. Sketching graphs is not a speedy process. It takes time, patience, and diligence.