Precalculus by Richard Wright

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So the last will be first, and the first will be last. Matthew‬ ‭20‬:‭16‬ ‭NIV‬‬‬‬‬‬‬

2-08 Graphs of Rational Functions

Mr. Wright teaches the lesson.

Summary: In this section, you will:

SDA NAD Content Standards (2018): PC.5.3

tank
Figure 1: credit (pixabay/visavietnam)

A large mixing tank is used to prepare water for a salt-water aquarium. It currently contains 100 gallons of water into which 5 pounds of salt have been mixed. A tap will open pouring 10 gallons per minute of water into the tank at the same time salt is poured into the tank at a rate of 1 pound per minute. What is the concentration, in pounds per gallon, of salt in the tank after a given amount of time? This problem can be solved using a rational function and will be analyzed later in this lesson.

Intercepts of Rational Functions

Intercepts of Rational Functions

Start by factoring and canceling factors common to the numerator and denominator.

y-intercepts occur when x = 0.

x-intercepts occur when y = 0. A fraction only equals zero when the numerator equals zero.

Example 1: Find the Intercepts of a Rational Function

Find the intercepts of \(f(x) = \frac{(x + 1)(x - 3)}{3x-2}\).

Solution

To find the y-intercept, substitute 0 for x and simplify.

$$ f(0) = \frac{(0 + 1)(0 - 3)}{3(0)-2} $$

$$ f(0) = \frac{3}{2} $$

The y-intercept is \(\left(0, \frac{3}{2}\right)\).

To find the x-intercepts, substitute 0 for y, or f(x), and solve for x. Because a fraction can only equal zero if the numerator is zero, set the numerator equal to zero and solve for x.

(x + 1)(x − 3) = 0

x + 1 = 0 or x − 3 = 0

x = −1 or x = 3

The x-intercepts are (−1, 0) and (3, 0).

f(x)=(x+1)(x-3)/(3x-2)
Figure 2: Graph of \(f(x) = \frac{(x + 1)(x - 3)}{3x-2}\) showing the intercepts.
Try It 1

Find the intercepts of \(t(x) = \frac{3x - 4}{x+5}\).

Answer

y-intercept: \(\left(0, -\frac{4}{5}\right)\); x-intercept: \(\left(\frac{4}{3}, 0\right)\)

Graph Rational Functions

The numerator of a rational function gives the x-intercepts of the graph. When the x-intercept has an even multiplicity because the corresponding factor is present an even number of times in the numerator, then graph touches the x-axis without crossing it at that point. When the x-intercept has an odd multiplicity, then the graph crosses the x-intercept at that point. For example, then graph of \(f(x) = \frac{(x-2)^2(x+3)}{(x-3)^2(x+1)}\) is shown in figure 3 below. The x-intercepts are (2, 0) with multiplicity 2 (from the factor (x − 2)2) and (−3, 0) (from the factor (x + 3)). Notice that the graph does not cross the x-intercept at (2, 0) which has even multiplicity, but does cross it at (−3, 0) which has odd multiplicity.

f(x) = \frac{(x-2)^2(x+3)}{(x-3)^2(x+1)}
Figure 3: The x-intercept at (−3, 0) has odd multiplicity and (2, 0) has even multiplicity. The vertical asymptote x = −2 has odd multiplicity and x = 3 has even multiplicity.

The denominator gives the vertical asymptotes of the graph. When the factor giving a vertical asymptote has even multiplicity because it is present an even number of times, then both sides of the graph by the asymptote go the same direction. When the factor giving a vertical asymptote has an odd multiplicity, one side of the graph goes up and the other side goes down near the asymptote. This can be see in figure 3 above were the vertical asymptote x = 3 has an even multiplicity ((x − 3) is present twice) and the vertical asymptote x = −2 has odd multiplicity ((x + 2) is present once).

Graph a Rational Function
  1. Find the y-intercept.
  2. Factor the numerator and denominator.
  3. Cancel factors common to the numerator and denominator. Keep note of these factors for identifying removable discontinuities.
  4. Find the x-intercepts. Note the multiplicities of the x-intercepts to determine the behavior of the graph at those points.
  5. Find the vertical asymptotes. Note the multiplicities of the vertical asymptotes to determine the behavior of the graph around those lines.
  6. Find the removable discontinuities from the canceled factors in step 3.
  7. Find the horizontal or slant asymptote.
  8. Make a table of values to find some points on the graph.
  9. Sketch the graph. Start near an asymptote and end near another asymptote. Do not cross vertical asymptotes, but you may cross a horizontal or slant asymptote.

Example 2: Graph a Rational Function

Sketch a graph of \(k(x) = \frac{x^2 - 9}{x^3 - 3x + 2}\).

Solution
  1. Start by finding the y-intercept. Substitute x = 0 and simplify.

    $$ k(0) = \frac{0^2 - 9}{0^3 - 3(0) + 2} = -\frac{9}{2} $$

    The y-intercept is \(\left(0, -\frac{9}{2}\right)\).

  2. Next, factor the numerator and denominator. The numerator, x2 − 9, is a difference of squares and becomes (x − 3)(x + 3). The denominator, x3 − 3x + 2, is cubic and cannot be factored easily. The Rational Zero Theorem will need to be used.

    The p's are factors of the constant term, 2.

    p = ±1, ±2

    The q's are factors of the leading coefficient, 1. So, the \(\frac{p}{q}\) are the same as p.

    $$ \frac{p}{q} = ±1, ±2 $$

    Pick one such as −2 and use synthetic division to check to see if it is a factor.

    $$ \begin{array}{rrrrr} \underline{-2}| & 1 & 0 & -3 & 2 \\ & & -2 & 4 & -2 \\ \hline & 1 & -2 & 1 & |\underline{\phantom{0}0} \end{array} $$

    Since the remainder is zero, (x + 2) is a factor. The depressed polynomial is x2 − 2x + 1 which factors into (x − 1)2. Thus, the factored form of the function is

    $$ k(x) = \frac{(x - 3)(x + 3)}{(x - 1)^2(x + 2)} $$

  3. There are no factors common to the numerator and denominator to cancel, so there are also no removable discontinuities.

  4. The x-intercepts are found by setting the numerator equal to zero and solving.

    (x − 3)(x + 3) = 0

    x = 3 or x = −3

    The x-intercepts are (3, 0) and (−3, 0).

  5. Then, find the vertical asymptotes by setting the denominator equal to zero and solving.

    (x − 1)2(x + 2) = 0

    x = 1 (multiplicity 2) or x = −2

    The vertical asymptotes are x = 1 with a multiplicity 2 and x = −2.

  6. There no removable discontinuities because no factors canceled in step 3.

  7. Find the horizontal or slant asymptote by comparing the degrees of the numerator and denominator. The degree of the numerator, N, is 2. The degree of the denominator, D, is 3. N < D, so the horizontal asymptote is y = 0.

  8. Make a table of values so that accurate points can be plotted. This will make the sketch more accurate. Notice a few extra points were found between the two vertical asymptotes so that there are several points to make an accurate graph.

    x −5 −4 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 2 3 4 5
    y −0.15 −0.14 0 0.45 UND −2.16 −2 −2.59 −4.5 −14 UND −1.25 0 0.13 0.14
  9. Start the graph by plotting the intercepts and points from the table. Then add the asymptotes drawn as dashed lines.

    points and intercepts
    Figure 4: Start by plotting the intercepts and points from the table. Then add the asymptotes as dashed lines.

    Then draw the graph by starting near an asymptote and drawing a curve through points ending near another asymptote. Do not cross a vertical asymptote, but you may cross the horizontal asymptote. Repeat for each section of the graph. Finally, make sure the graph matches the predicted behavior. Both sides of the vertical asymptote x = 1 go the same direction because it has even multiplicity. Near the vertical asymptote x = −2, one side goes up and the other side goes down due to odd multiplicity. The graph crosses the x-axis at both x-intercepts because they each have odd multiplicity (1).

    k(x) = \frac{x^2 - 9}{x^3 - 3x + 2}
    Figure 5: \(k(x) = \frac{x^2 - 9}{x^3 - 3x + 2}\)
Try It 2

Graph the function \(f(x) = \frac{x - 2}{x^2 - 3x + 2}\).

Answer

y-intercept at (0, −1); no x-intercepts; V.A. at x = 1; Removable discontinuity at (2, 1); H.A. at y = 0

Graph of f(x)=(x-2)/(x^2-3x+2)

Write Rational Functions

Sometimes an equation is needed to match a graph. The process is somewhat like graphing in reverse. The x-intercepts give the factors of the numerator. The vertical asymptotes give the factors of the denominator. The behavior of the graph near the x-intercepts and vertical asymptotes gives the multiplicity of the factors. A stretch factor, a, can be used to scale the graph vertically for the y-intercept.

Write Rational Functions from Intercepts and Asymptotes
  1. Use the x-intercepts to find the factors of the numerator in the form (xk) where k is a x-intercept. Any factor from where the graph touches the x-axis without crossing it should be squared to give it even multiplicity.
  2. Use the vertical asymptotes to find the factors of the denominator in the form (xv) where x = v is a vertical asymptote. Any factor from where the graph on both sides of a vertical asymptote go the same direction should be squared to give it even multiplicity.
  3. Write the function in the form \(f(x) = a\frac{\text{factors from x-intercepts}}{\text{factors from vertical asymptotes}}\) where a is the stretch factor.
  4. Substitute any clear point to find a.
  5. Write the function by substituting for a.

Example 3: Write a Rational Function from Intercepts and Asymptotes

Write an equation for the rational function shown below in figure 6.

Graph of a rational function.
Figure 6
Solution
  1. The x-intercepts are −4 and 3. Write those as factors.

    (x + 4)(x − 3)

    Notice that the graph does not cross the x-axis at 3, so make the factor (x − 3) squared.

    (x + 4)(x − 3)2

  2. The vertical asymptotes are x = −2 and x = 1. Write those as factors.

    (x + 2)(x − 1)

    Notice that the graph goes up on both sides of the asymptotes x = 1, so make that factor squared.

    (x + 2)(x − 1)2

  3. Write the function in the form \(f(x) = a\frac{\text{factors from x-intercepts}}{\text{factors from vertical asymptotes}}\).

    $$ f(x) = a\frac{(x+4)(x-3)^2}{(x+2)(x-1)^2} $$

  4. Substitute any clear point to find a. (−1, 3) works in this case.

    $$ 3 = a\frac{(-1+4)(-1-3)^2}{(-1+2)(-1-1)^2} $$

    3 = a(12)

    $$ a = \frac{1}{4} $$

  5. Write the final function by substituting for a.

    $$ f(x) = \frac{(x + 4)(x - 3)^2}{4(x + 2)(x - 1)^2} $$

Solve Applied Problems Involving Rational Functions

Example 4: Solve an Applied Problem Involving a Rational Function

A large mixing tank is used to prepare water for a salt-water aquarium. It currently contains 100 gallons of water into which 5 pounds of salt have been mixed. A tap will open pouring 10 gallons per minute of water into the tank at the same time salt is poured into the tank at a rate of 1 pound per minute. Find the concentration (pounds per gallon) of salt in the tank after 12 minutes. Is that a greater concentration than at the beginning?

Solution

Let t be the number of minutes since the tap opened. Since the water starts at 100 gallons and increases at 10 gallons per minute, its volume is given by

W(t) = 100 + 10t = 10t + 100.

The salt starts at 5 pounds and increases at 1 pounds per minute, so its quantity is given by

S(t) = 5 + 1t = t + 5.

The concentration in pounds per gallon can be found by taking the pounds of salt and dividing by the gallons of water.

$$ C(t) = \frac{t + 5}{10t+100} $$

The problem asks for the concentration at t = 12 minutes. Plug in a 12 and evaluate the function.

$$ C(12) = \frac{12 + 5}{10(12) + 100} = 0.077 \text{ lbs/gal} $$

The last part asks if this concentration is higher than the beginning concentration. Answer that by finding the concentration when t = 0.

$$ C(0) = \frac{0 + 5}{10(0) + 100} = \frac{1}{20} = 0.05 \text{ lbs/gal} $$

C(12) is higher than C(0).

Analysis

Find the horizontal asymptote. Since the degree of the numerator and denominator are the same, the horizontal asymptote is the ratio of the leading coefficients.

$$ y = \frac{1}{10} = 0.1 $$

This means the concentration of salt to water, will approach 0.1 lbs/gal in the long term.

Try It 3

There are 150 mosquitoes and 25 ants at picnic lunch at noon. After 12 p.m., 10 mosquitoes arrive at the picnic every five minutes while only 2 ants come. Find the ratio of mosquitoes to ants at 1 p.m.

Answer

\(\frac{270}{49}\)

Lesson Summary

Intercepts of Rational Functions

Start by factoring and canceling factors common to the numerator and denominator.

y-intercepts occur when x = 0.

x-intercepts occur when y = 0. A fraction only equals zero when the numerator equals zero.


Graph a Rational Function
  1. Find the y-intercept.
  2. Factor the numerator and denominator.
  3. Cancel factors common to the numerator and denominator. Keep note of these factors for identifying removable discontinuities.
  4. Find the x-intercepts. Note the multiplicities of the x-intercepts to determine the behavior of the graph at those points.
  5. Find the vertical asymptotes. Note the multiplicities of the vertical asymptotes to determine the behavior of the graph around those lines.
  6. Find the removable discontinuities from the canceled factors in step 3.
  7. Find the horizontal or slant asymptote.
  8. Make a table of values to find some points on the graph.
  9. Sketch the graph. Start near an asymptote and end near another asymptote. Do not cross a vertical asymptote, but you can cross a horizontal or slant asymptote.

Write Rational Functions from Intercepts and Asymptotes
  1. Use the x-intercepts to find the factors of the numerator in the form (xk) where k is a x-intercept. Any factor from where the graph touches the x-axis without crossing it should be squared to give it even multiplicity.
  2. Use the vertical asymptotes to find the factors of the denominator in the form (xv) where x = v is a vertical asymptote. Any factor from where the graph on both sides of a vertical asymptote go the same direction should be squared to give it even multiplicity.
  3. Write the function in the form \(f(x) = a\frac{\text{factors from x-intercepts}}{\text{factors from vertical asymptotes}}\) where a is the stretch factor.
  4. Substitute any clear point to find a.
  5. Write the function by substituting for a.

Helpful videos about this lesson.

Practice Exercises

  1. Can a graph of a rational function have no x-intercepts? If so, how?
  2. Find the x- and y-intercepts for the functions.

  3. \(f(x) = \frac{x}{x^2+2x}\)
  4. \(f(x) = \frac{x^2-3x+2}{x^2-x-6}\)
  5. Find the (a) x-intercepts, (b) the y-intercept, (c) the vertical asymptotes, and (d) the horizontal or slant asymptote of the functions. (e) Use that information to sketch a graph.

  6. \(m(x) = \frac{2x-3}{x+1}\)
  7. \(r(x) = \frac{1}{(x-2)^2}\)
  8. \(s(x) = \frac{2x^2+5x-3}{2x^2-2x-4}\)
  9. \(t(x) = \frac{x^2+3x-4}{x^2-3x+2}\)
  10. \(w(x) = \frac{x^2-2x-3}{x-1}\)
  11. \(g(x) = \frac{(x-4)(x-1)(x+2)}{(x-2)^2(x+1)}\)
  12. Write an equation for a rational function with the given characteristics.

  13. Vertical asymptotes at x = 3 and x = −4, x-intercept at (1, 0), y-intercept at \(\left(0, \frac{1}{6}\right)\)
  14. Use the graphs to write an equation for the function.

  15. Graph of a rational function with vertical asymptotes at x=2 and x=−1.
  16. Graph of a rational function with vertical asymptote at x=-3.
  17. Graph of a rational function with vertical asymptote at x=−3.
  18. Graph of a rational function with vertical asymptotes at x=−4, x=3, and x=1.
  19. Problem Solving: Write a rational function that describes the situation.

  20. A large mixing tank at a frosting factory currently contains 500 gallons of water, into which 12 pounds of sugar have been mixed. A tap will open, pouring 15 gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of 2 pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after t minutes.
  21. Mixed Review

  22. (2-07) Find the slant asymptote of \(f(x) = \frac{x^2+4}{x-3}\).
  23. (2-06) Use Descartes’ Rule of Signs to determine the possible number of positive and negative solutions: g(x) = x4 − 6x3 + 10x2 + 2x − 15.
  24. (2-06) Find all the complex zeros (real and non-real): g(x) = x4 − 6x3 + 10x2 + 2x − 15.
  25. (2-01) Simplify (2 − i)(3 + 3i).
  26. (1-07) Identify the parent function and describe the transformations: \(h(x) = \frac{3}{x-2} + 1\).

Answers

  1. Yes. The numerator of the functions would have no zeros, only complex zeros, and/or factors common to both the numerator and denominator.
  2. none
  3. x-int: (2, 0) and (1, 0); y-int: \(\left(0, -\frac{1}{3}\right)\)
  4. x-int: \(\left(\frac{3}{2}, 0\right)\); yint: (0, −3); V.A. x = −1; H.A. y = 2;
  5. x-int: none; yint: \(\left(0, \frac{1}{4}\right)\); V.A. x = 2; H.A. y = 0;
  6. x-int: (−3, 0) and \(\left(\frac{1}{2}, 0\right)\); yint: \(\left(0, \frac{3}{4}\right)\); V.A. x = −1 and x = 2; H.A. y = 1;
  7. x-int: (−4, 0); yint: (0, −2); V.A. x = 2; H.A. y = 1;
  8. x-int: (−1, 0) and (3, 0); yint: (0, 3); V.A. x = 1; S.A. y = x − 1;
  9. x-int: (−2, 0), (1, 0), and (4, 0); yint: (0, 2); V.A. x = −1, x = 2; H.A. y = 1;
  10. \(y = \frac{2(x-1)}{(x-3)(x+4)}\)
  11. \(y = \frac{x-4}{(x-2)(x+1)}\)
  12. \(y = \frac{4}{x+3}\)
  13. \(y = \frac{(x-2)(x+1)}{4(x+3)}\)
  14. \(y = \frac{2(x+1)^2(x-2)}{(x+4)(x-1)(x-3)}\)
  15. \(C(t) = \frac{2t+12}{15t+500}\)
  16. y = x + 3
  17. 3 or 1 positive zeros, 1 negative zero, 0 or 2 complex zeros
  18. −1, 3, 2 + i, 2 − i
  19. 9 + 3i
  20. Reciprocal function; vertical stretch by factor of 3, shifted right 2 and up 1