Find Asymptotes, Critical and Inflection Points

This example describes how to analyze a simple function to find its asymptotes, maximum, minimum, and inflection point.

Define a Function

The function in this example is


To create the function, enter the following commands:

syms x
num = 3*x^2 + 6*x -1;
denom = x^2 + x - 3;
f = num/denom
f = 
(3*x^2 + 6*x - 1)/(x^2 + x - 3)

Plot the function f by using fplot. The fplot function automatically shows horizontal asymptotes.


Find Asymptotes

To mathematically find the horizontal asymptote of f, take the limit of f as x approaches positive infinity:

limit(f, inf)
ans = 

The limit as x approaches negative infinity is also 3. This result means the line y = 3 is a horizontal asymptote to f.

To find the vertical asymptotes of f, set the denominator equal to 0 and solve by entering the following command:

roots = solve(denom)
roots =
 - 13^(1/2)/2 - 1/2
   13^(1/2)/2 - 1/2


MATLAB® does not always return the roots to an equation in the same order.

roots indicates that the vertical asymptotes are the lines




Find Maximum and Minimum

You can see from the graph that f has a local maximum between the points x = –2 and x = 0, and might have a local minimum between x = –6 and x = –2. To find the x-coordinates of the maximum and minimum, first take the derivative of f:

f1 = diff(f)
f1 = 
(6*x + 6)/(x^2 + x - 3) - ((2*x + 1)*(3*x^2 + 6*x - 1))/(x^2 + x - 3)^2

To simplify this expression, enter

f1 = simplify(f1)
f1 =
 -(3*x^2 + 16*x + 17)/(x^2 + x - 3)^2

Next, set the derivative equal to 0 and solve for the critical points:

crit_pts = solve(f1)
crit_pts =
 - 13^(1/2)/3 - 8/3
   13^(1/2)/3 - 8/3

It is clear from the graph of f that it has a local minimum at


and a local maximum at


You can plot the maximum and minimum of f with the following commands:

hold on
plot(double(crit_pts), double(subs(f,crit_pts)),'ro')
title('Maximum and Minimum of f')
text(-4.8,5.5,'Local minimum')
text(-2,4,'Local maximum')
hold off

Find Inflection Point

To find the inflection point of f, set the second derivative equal to 0 and solve.

f2 = diff(f1);
inflec_pt = solve(f2,'MaxDegree',3);

This returns

ans =
  -5.2635 + 0.0000i
  -1.3682 - 0.8511i
  -1.3682 + 0.8511i

In this example, only the first element is a real number, so this is the only inflection point. The order of the roots can vary.

Rather than selecting the real root by indexing into inter_pt, identify the real root by determining which roots have a zero-valued imaginary part.

idx = imag(double(inflec_pt)) == 0;
inflec_pt = inflec_pt(idx);

To obtain the value of the inflection point, enter

ans =

Plot the inflection point. The extra argument, [-9 6], in fplot extends the range of x values in the plot so that you see the inflection point more clearly, as shown in the following figure.

fplot(f, [-9 6])
hold on
plot(double(inflec_pt), double(subs(f,inflec_pt)),'ro')
title('Inflection Point of f')
text(-7,1,'Inflection point')
hold off