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?
- Which part of the function controls the location of the horizontal asymptote?
Test yourself by modifying the equation of the function in the following ways.
- Change one constant in the function so that the horizontal asymptotes occur at x = -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?
- Which part of the function controls the location of the vertical asymptotes?
- Which part of the function controls the location of the horizontal asymptote?
Test yourself by modifying the equation of the function in the following ways.
- Change one constant in the function so that the vertical asymptotes occur at x = ± 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.
- Why is there no vertical asymptote?
- Which part of the function controls the location of the horizontal asymptote?
Test yourself by modifying
the equation of the function in the following ways.
- Change one constant in the function so that the horizontal asymptote occurs at y = 2.
- Change the function so that the horizontal asymptote occurs at y = -5/2.
- 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.
- Why is there no vertical asymptote at x = -1?
- Which part of the function controls the location of the vertical asymptotes?
- Which part of the function controls the location of the horizontal asymptote?
Test yourself by modifying
the equation of the function in the following ways.
- Change one constant in the function so that the vertical asymptote occurs at x = -1.
- Now change another constant in the function so
that the horizontal asymptote occurs at y = 2.
- 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.
- Why are there two vertical asymptotes?
- Which part of the function controls the location of the vertical asymptotes?
- Which part of the function controls the location of the horizontal asymptote?
Test yourself by modifying
the equation of the function in the following ways.
- Change one constant in the function so that the vertical asymptotes occur at x = -3.
- Change the function so
that there are no vertical asymptotes.
- 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?
- Change the function so that the skew asymptote is y = -x.
- 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.
- Which part of the function controls the location of the vertical asymptotes?
- Which part of the function controls the skew asymptote?
Test yourself by modifying
the equation of the function in the following ways.
- Change the function so that the skew asymptote occurs at y = 2x.
- Change the function so that the skew asymptote occurs at y = -x.
- 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.
- Which part of the function controls the slope of the skew asymptote?
- Which part of the function controls the intercspt of the skew asymptote?
Test yourself by modifying
the equation of the function in the following ways.
- Change the constants in the function so that the skew asymptote occurs at y = x + 1.
- Change the function so that the skew asymptote occurs at y = -x + 2.
- 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?
- Change the function so that the skew asymptote occurs at y = ±2x.
- 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.
- Which part of the function controls the location of the vertical asymptotes?
- Which part of the function controls the horizontal asymptotes?
Test yourself by modifying
the equation of the function in the following ways.
- Change the function so that the horizontal asymptotes occurs at x = ±2
and a vertical asymptote at y = -1.
- Change the function so that the vertical asymptote occurs at x = 1
and it has horizontal
asymptotes at x = ±1.
- 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.
- 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.
- 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.
- 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.
- Choose f(x) and g(x) so that r(x) has only one asymptote which is
horizontal and it occurs at y = 3.
- 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.
- 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.
- 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.
- 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.
- Choose g(x) so that r(x) has one horizontal at y = 2
and one vertical asymptote at x = 0.
- 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.
- 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.
- Choose g(x) so that r(x) has one horizontal at y = 0
and two vertical asymptotes at x = -3 and x = 3, respectively.
- Choose g(x) so that r(x) has one horizontal at y = 0
and one vertical asymptote at x = 0.
- Choose g(x) so that r(x) has one horizontal at y = 1
and one vertical asymptote at x = 1.
- Choose g(x) so that r(x) has one horizontal at y = 0
and one vertical asymptote at x = 1.
- 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).
- 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.
- 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.
- 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.
- 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.