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Section Graphing

Prior to an understanding of derivatives, graphing was largely a matter of:

  1. plotting points,

  2. memorizing the basic shapes of certain types of functions, and

  3. trusting the graphs generated by technology.

With algebra and differentiation at our disposal, we can make a precise science out of graphing that will give more information than even our graphing software yields.

Example 3.24.

I will work a few of these examples to provide intuition for the upcoming definitions.

  1. \(p(x) = x^4-13x^2+36\)

  2. \(\dsp q(x)=\frac{3}{x^2+1}\)

  3. \(f(x) = 2x^{5/3}-5x^{4/3}\)

  4. \(\dsp r(x)=\frac{x^2-1}{2x+1}\)

  5. \(\dsp F(t)=t^{2/3}-3\)

  6. \(\dsp h(x)=\frac{3x^2}{x^{2}+1}\)

We list several questions that you should ask when graphing a function.

Question 1. What are the intercepts and asymptotes of the function?

Recall that an \(x\)-intercept is a point where the function crosses the \(x\)-axis (i.e. \(y=0\)) and a \(y\)-intercept is a point where the function crosses the \(y\)-axis (i.e. \(x=0\)).

Recall from Definition 3.8 that asymptotes are functions (usually lines) that \(f\) approaches. Limits are helpful in understanding and locating asymptotes. Review Problem 3.22 and Problem 3.23 when studying asymptotes.

Problem 3.25.

List any intercepts and asymptotes for:

  1. \(\dsp f(x) = \frac{x}{(x-2)(x+3)}\)

  2. \(E(x) = 3-e^{-x}\)

  3. \(\dsp g(x) = \frac{3x}{\sqrt{x^2+3}}\)

Question 2. What are the local maxima and minima of the function?

The peaks and the valleys are called local maxima and local minima, respectively.

Definition 3.26.

If f is a function, then we say that the point \((c,f(c))\) is a local maximum for f if there is some interval, (a,b), containing c so that \(f(x) \lt f(c)\) for all \(x \in (a,b)\) except \(x=c.\) Local minimum is defined similarly.

We have seen examples of local minima and maxima. Problem 2.14 had a minimum at a point where the derivative did not exist. In problem 2.35 we found the local maxima and minima by finding the points where the tangent lines were horizontal. Hence points on the graph where the derivative is either zero or undefined are important points.

Definition 3.27.

If \(f\) is a function, then any value \(x\) in the domain of \(f\) where \(f'(x)\) is zero or undefined is called a critical value of \(f\text{.}\) We refer to \((x,f(x))\) as a critical point of \(f.\)

Critical points give us candidates for local maximum or minimum, but they do not guarantee a maximum of minimum. The function \(f(x) = x^3 + 2\) has \((0,2)\) as a critical point, but this point is neither a minimum nor a maximum.

Problem 3.28.

Find the critical values for these functions.

  1. \(g(x) = \frac{1}{3}x^3 + \frac{1}{2}x^2 - 12x\)

  2. \(f(x) = \sin(2x)\) on \([0,2\pi]\)

  3. \(s(x) = (x-3)^{1/3}\)

Question 3. Where is the function increasing? Decreasing?

Determining where a function is increasing or decreasing is perhaps the most useful graphing tool, so we'll call it the Fundamental Theorem of Graphing.

Definition 3.29.

We say that a function \(f\) is increasing if, for every pair of numbers \(x\) and \(y\) in the domain of \(f\) with \(x\lt y\text{,}\) we have \(f(x) \lt f(y).\) Decreasing, non-increasing, and non-decreasing are defined similarly.

If the slopes of all tangent lines to the function \(f\) are positive over an interval \((a,b)\text{,}\) then \(f\) is increasing on the interval \((a,b)\) and if the slopes of all tangent lines to the function \(f\) are negative over an interval \((a,b)\text{,}\) then \(f\) is decreasing on the interval \((a,b)\text{.}\)

Problem 3.31.

These are the functions you found the critical values for in the last problem. List the intervals on which these functions are increasing by finding where \(f'>0\text{.}\)

  1. \(g(x) = \frac{1}{3}x^3 + \frac{1}{2}x^2 - 12x\)

  2. \(f(x) = \sin(2x)\) on \([0,2\pi]\)

Question 4. Where is the function concave up? Concave down?

Sketch \(f(x) = x^3-x.\) The graph of \(f\) looks somewhat like a parabola opening downward on the interval \((-\infty, 0)\) and somewhat like a parabola opening upward on the interval \((0,\infty)\text{.}\) We say that \(f\) is concave down on \((-\infty, 0)\) and concave up on \((0,\infty)\text{.}\) The defining characteristic for concavity is that \(f\) is concave down when the slopes of the tangent lines are decreasing and \(f\) is concave up when the slopes are increasing.

Definition 3.32.

We say that \(f\) is concave up on \((a,b)\) if \(f'\) is increasing on \((a,b)\) and concave down on \((a,b)\) if \(f'\) is decreasing on \((a,b)\text{.}\)

Applying the Fundamental Theorem of Graphing, \(f\) will be concave up when \(f'\) is increasing which will be when \(f''\) is positive.

Question 5. What are the inflection points for the function?

Inflection points are points on the curve in which the function switches either from concave up to concave down, or from concave down to concave up.

Definition 3.33.

We say that \((x,f(x))\) is an inflection point for \(f\) if the concavity of \(f\) changes sign at \(x.\)

Problem 3.34.

Let \(\dsp f(x) = \frac{6}{x^2+3}\text{.}\) Find the values where \(f''=0\text{.}\) Find the intervals where \(f\) is concave up. List the inflection points.

Question 6. What is the least upper bound of the function? The greatest lower bound?

From the graph of \(\dsp f(x) = \frac{6}{x^2+1}\) in the last problem we see that \((0,6)\) is the highest point on the graph, but there is no lowest point since the graph looks like an infinite bell — it tends towards zero as \(x\) takes on large positive or large negative values. We say that 6 is the least upper bound of \(f\) and that \(f\) attains this value since \(f(0)=6.\) We say that \(0\) is the greatest lower bound and that \(f\) does not attain this value since there is no number \(x\) so that \(f(x) = 0.\)

Definition 3.35.

If there is a number \(M\) satisfying \(f(x) \leq M\) for all \(x\) in the domain of \(f\text{,}\) then we call \(M\) an upper bound of \(f.\) If there is no number less than \(M\) that satisfies this property, then we call \(M\) the least upper bound.

Problem 3.36.

For each function (1) find the least upper bound or state why it does not exist, and (2) find the greatest lower bound or state why it does not exist.

  1. \(E(x) = 3-e^{-x}\)

  2. \(\dsp f(x) = x^4-x^2\)

Problem 3.37.

Graph two functions \(f\) and \(g,\) each having a least upper bound of \(1\text{,}\) so that \(f\) attains its least upper bound, but \(g\) does not. Can you find equations for two functions with these two properties?

Problem 3.38.

Choose one of the functions below and attempt to list as much information as possible. Use this information to help you sketch a very precise graph of the function. Do not expect to be able to list all of the information for each function. You won't be able to answer every question for every function. Graphing is the art of deciding which questions will best help you graph which functions.

  • \(x\)-intercepts and \(y\)-intercepts (where \(x=0\) or \(y=0\))

  • horizonal asymptotes and vertical asymptotes (lines the function approaches)

  • critical points (places where \(f'\) is either zero or undefined)

  • intervals over which each function is increasing (intervals where \(f'\) is positive)

  • intervals over which each function is decreasing (intervals where \(f'\) is negative)

  • intervals over which each function is concave up (intervals where \(f''\) is positive)

  • intervals over which each function is concave down (intervals where \(f''\) is negative)

  • least upper bound and greatest lower bound for each function (bounds for the range)

  1. \(f(x)=2x^{3}-3x^{2}\)

  2. \(p(x) =3x^4 + 4x^3 -12x^2\)

  3. \(g(x)=(1-x)^{3}\)

  4. \(r(x) = 3-x^{1/3}\)

  5. \(\dsp F(t)=\frac{2t}{t+1}\)

  6. \(\dsp h(x)=\frac{5}{x^{2}-x-6}\)

  7. \(\dsp g(x) = \frac{x^3+1}{x}\)

  8. \(s(x) = x-\cos(x)\)

  9. \(T(x) = \tan(2x)\)

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