Sketching Curves Using Derivative Properties



~Note: With todays technology, when the equation of a function is given, one can easily get a graph simply by coding in the equation in your calculator and setting up a friendly viewing window.

~Note: However, a problem arises when the equation is not given.

~Note: Since the equation of the curve is not given (just properties the curve must have), you will not be able to get the graph by a calculator (you have nothing to code in the y= menu). So, your success in sketching a possible curve will totally rely on the correct translations of the mathematical properties stated.

~Note: You’ll need a pencil with an eraser since your sketch will change as more information is applied.

It's really like learning a language. You see a word or phrase and then you need to translate its meaning. This new language consists of mathematical symbols and the translation gives information about the curve in question. Properly translated, you will end up getting a curve that has all the symbolic information given. There could be other curves having the same properties but they will all look very similar. As long as you give one with all the stated properties, you will be doing the problem correctly. So, you need to concentrate on the language (symbols that give certain properties) in order to sketch a curve properly.

Let's look at the popular symbolic properties given in most problems. I will limit the properties to the usual derivative properties that most problems of this type usually have stated.

Assume the curve is the function called f.

When you see symbols like f(a)=b, this indicates a point on the graph of f, namely, (a,b). This states that the functional value (y-value) is b when the x-value is a. So, look for these first, then place these points on your axes. You are guaranteed that the curve passes through these points. Keep this in mind.

When you see a prime on f, such as f ', this indicates the derivative of f which we can interpret as slopes of tangent lines at various places on the curve. For example, f '(2) will be the slope of a tangent line drawn to the curve at the point where x=2. Usually, problems of this type will state values of x where the derivative has a value of 0. For example, If the condition f '(2)=0 is stated, then you know that the tangent line has a slope of zero at the x value of 2 on the curve.

The problem is that you may not know where the curve is at x=2. All you know is that it will be somewhere on the vertical line x=2. Further information given in the problem should establish a reasonable location. You, most likely, will have to do some erasing and repositioning.

A zero slope for a tangent line indicates a horizontal tangent line and could give a Max point, Min point or a flat point on the curve. So, I usually draw a little horizontal tangent line somewhere on the vertical line x=2, but really do not know how high or low to place it.
For example, for f '(2)=0, I place a point on the vertical line x=2 and draw a little horizontal tangent line. It may have to be moved as more information is given.

Be careful, since f '(a)=0 could give a flat point and not a Max or Min point. A good example is the curve y=x3at x=0. This is what a typical flat point looks like.

When I see f '(x)>0 for a range of x's (usually intervals on the x-axis), that tells me that the slopes of the tangent lines are positive in that range. That will force the curve to rise from left to right in that range as we move on the x-axis. Likewise, f '(x)<0 tells me that the slopes are negative in that range and the curve will fall from left to right in that range. These are derivative properties that tells you where the function is increasing (rising) and decreasing (falling).

Two primes on f will give properties of the 2nd derivative and gives information on how the curve bends, namely, the concavity over a given range. When I see that f ''(x)>0 over a range of x's, the curve is bending upwards (concave up) like a U or part of a U. On the other hand, when I see the property f ''(x)<0, the curve is bending downwards (concave down) like an up-side-down U or part of one. When I see f ''(x)=0 at an x-value, this will give a possible point where the bend in the curve changes from upward to downward or from downward to upwards. At this point the concavity could change. If so, this point is know as a point of inflection.

Be careful, f '' (x)=0 does not necessarily give an inflection point (change in concavity). A good example of this is with the curve y=x4 at x=0.

When I see limit properties, like lim f(x) = ∞ or -∞ as x→ a (fixed number on the x-axis), this tells me that the function (y-values) shoot upward or downward as I get closer to that x-value. This usually indicates a vertical asymptote. A little + sign in an exponent position on the fixed number in the limit statement means I'm approaching the fixed number from the positive side (right side) and a little - sign indicates coming in to the number on the left side.

When I see the limit f(x) = L when x → +∞ or -∞, this indicates that the functional values (y-values) get closer to a fixed level (y=L) as I travel far out to the right or far out to the left on the x-axis. This is know as a horizontal asymptote.

Sometimes information about the continuity is given. For these properties, you need to incorporate continuity properties. Remember the 3 ways a curve can be discontinuous (point, jump, ∞) and apply this information to your sketch.

Putting all these together carefully and probably doing lots of erasing, you will be able to get a reasonable sketch of the curve. These are not easy problems for beginners to do, so, a lot of practice is needed.