Asymptotes



Problem 1

Load Example 1 into the graphing applet above y = 2*x^2/(x^2+1).

Note that it has a horizontal asymptote at y = 2. Why?

Test yourself by modifying the equation of the function in the following ways.

  1. Change one constant in the function so that the horizontal asymptotes occur at x = -2.
  2. Now change another constant in the function so that the horizontal asymptote occurs at y = 4.

Problem 2

Load Example 2 in the graphing applet above: (x^2+1)/(x^2-1).

Note that it has vertical asymptotes at x = ± 1 and horizontal asymptote at y = 1. Why?

Test yourself by modifying the equation of the function in the following ways.

  1. Change one constant in the function so that the vertical asymptotes occur at x = ± 2.
  2. Now change another constant in the function so that the horizontal asymptote occurs at y = -1.

Problem 3

Load Example 3: (3x^2-2*x-4)/(2*x^2+1).

Note that it has no vertical asymptotes but does have a horizontal asymptote at y = 3/2.

Test yourself by modifying the equation of the function in the following ways.

  1. Change one constant in the function so that the horizontal asymptote occurs at y = 2.
  2. Change the function so that the horizontal asymptote occurs at y = -5/2.
  3. Change one constant in the function so that there are now vertical asymptotes.

Problem 4

Load Example 4: (3*x^3+x)/(x^3-1).

Note that it has a vertical asymptote at x = 1 and horizontal asymptote at y = 3.

Test yourself by modifying the equation of the function in the following ways.

  1. Change one constant in the function so that the vertical asymptote occurs at x = -1.
  2. Now change another constant in the function so that the horizontal asymptote occurs at y = 2.
  3. Change the function so that the horizontal asymptote occurs at y = -2.

Problem 5

Load Example 5: 1/(x^2-1).

Note that it has vertical asymptotes at x = ±1 and horizontal asymptote at y = 0.

Test yourself by modifying the equation of the function in the following ways.

  1. Change one constant in the function so that the vertical asymptotes occur at x = -3.
  2. Change the function so that there are no vertical asymptotes.
  3. Can you change the function so that he vertical asymptotes occur at x = 3 and x = -4?

Problem 6

Load Example 6: (x^3+9*x)/(x^2+1).

Note that it has no vertical and horizontal asymptotes. But the line y = x is a skew asymptote. In other words, as x goes to positive (or negative) infinity, the graph of the function x^3/(x^2+1) approaches the graph of the line y = x. Why does this happen?

  1. Change the function so that the skew asymptote is y = -x.
  2. Change the function so that the skew asymptote is y = 2x.

Problem 7

Load Example 7: x^3/(2*x^2-2).

Note that it has vertical asymptotes at x = ± 1 and a skew asymptote at y = x/2.

Test yourself by modifying the equation of the function in the following ways.

  1. Change the function so that the skew asymptote occurs at y = 2x.
  2. Change the function so that the skew asymptote occurs at y = -x.
  3. Change the function so that the vertical asymptotes occur at x = ±2.

Problem 8

Load Example 8: (4*x^3+2*x^2)/(2*x^2+6). Note that it has a skew asymptote at y = 2x + 1. Test yourself by modifying the equation of the function in the following ways.

  1. Change the constants in the function so that the skew asymptote occurs at y = x + 1.
  2. Change the function so that the skew asymptote occurs at y = -x + 2.
  3. Change the function so that the skew asymptote occurs at y = -3x + 1.

Problem 9

Asymptotes do not just occur in rational functions. Load Example 9: sqrt(x^2+1).

Note that it has no vertical or horizontal asymptotes but does have two skew asymptotes at y = ±x? Why?

  1. Change the function so that the skew asymptote occurs at y = ±2x.
  2. Change the function so that the skew asymptote occurs at y = ±3x.

Problem 10

It is possible to have two horizontal asymptotes. Load Example 10: abs(x^3)/(x^3+1). Here abs(x^3) = |x^3| is just the absolute value function.

Note that it has horizontal asymptotes at x = ±1 and a vertical asymptote at y = -1.

Test yourself by modifying the equation of the function in the following ways.

  1. Change the function so that the horizontal asymptotes occurs at x = ±2 and a vertical asymptote at y = -1.
  2. Change the function so that the vertical asymptote occurs at x = 1 and it has horizontal asymptotes at x = ±1.
  3. Change the only the denominator of the function (but it should still be a cubic equation) so that it has two vertical asymptotes at x = 0 and x = 2 and it has horizontal asymptotes at x = ±1 or explain why this is impossible.
  4. Change the only the denominator of the function (but it should still be a cubic equation) so that it has thre vertical asymptotes at x = 0, x = 1 and x = 1 and it has horizontal asymptotes at x = ±1 or explain why this is impossible.
  5. Change the function so that the horizontal asymptotes occurs at x = 2 and x = 4 and a vertical asymptote occurs at y = -1 or explain why this is impossible.
  6. Change the function so that there are three horizontal asymptotes occurs at x = ±2 and x =02 and a vertical asymptote occurs at y = -1 or explain why this is impossible.

Problem 11

Let r(x) = f(x)/g(x) be a rational function where both f(x) and g(x) are quadratic functions.
  1. Choose f(x) and g(x) so that r(x) has only one asymptote which is horizontal and it occurs at y = 3.
  2. Choose f(x) and g(x) so that r(x) has one horizontal at y = 4 and one vertical asymptote at x = 0 or explain why it is impossible to do so.
  3. Choose f(x) and g(x) so that r(x) has one horizontal at y = 2 and two vertical asymptotes at x = -2 and x = 2, respectively.
  4. Choose f(x) and g(x) so that r(x) has one horizontal at y = -3 and two vertical asymptotes at x = -1 and x = 3, respectively.
  5. Choose f(x) and g(x) so that r(x) has two horizontal asymptotes at y = 4 and y = 2 or explain why it is impossible to do so.

Problem 12

Let r(x) = f(x)/g(x) be a rational function where f(x) = 8x + 3 and g(x) is either linear or quadratic.
  1. Choose g(x) so that r(x) has one horizontal at y = 2 and one vertical asymptote at x = 0.
  2. Choose g(x) so that r(x) has only one asymptote which is horizontal and it occurs at y = 4 or explain why it is impossible.
  3. Choose g(x) so that r(x) has only one asymptote which is horizontal and it occurs at y = 0 or explain why it is impossible.
  4. Choose g(x) so that r(x) has one horizontal at y = 0 and two vertical asymptotes at x = -3 and x = 3, respectively.
  5. Choose g(x) so that r(x) has one horizontal at y = 0 and one vertical asymptote at x = 0.
  6. Choose g(x) so that r(x) has one horizontal at y = 1 and one vertical asymptote at x = 1.
  7. Choose g(x) so that r(x) has one horizontal at y = 0 and one vertical asymptote at x = 1.
  8. Choose g(x) so that r(x) has two horizontal asymptotes at y = 4 and y = 2 or explain why it is impossible to do so.

Problem 13

Let s(x) = f(x)/g(x).
  1. Choose f(x) and g(x) so that s(x) has two horizontal asymptotes at y = ±4 and a vertical asymptote at x = 0. Hint: You may wish to make use of the absolute value function.
  2. Choose f(x) and g(x) so that s(x) has two horizontal asymptotes at y = ±2 and two vertical asymptote at x = ±2 or explain why it is impossible to do so.
  3. Choose f(x) and g(x) so that s(x) has two horizontal asymptotes at y = ±2 and three vertical asymptote at x = ±2 and x = 0 or explain why it is impossible to do so.
  4. Choose f(x) and g(x) so that s(x) has three horizontal asymptotes at y = ±2 and y = 0 and three vertical asymptote at x = ±2 and x = 0 or explain why it is impossible to do so.