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2.12 Graphical and analytical solutions to rational equations

Lesson

Graphing calculators and rational functions

We can use technology such as graphing calculators to draw rational functions and identify certain properties of the function. This might include where the horizontal or vertical asymptotes lie, points of intersection, turning points and the domain and range of the function.

Exploration

Remember that a rational function is one that has the form

$f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$f(x)=p(x)q(x),

where $p\left(x\right)$p(x) and $q\left(x\right)$q(x) are polynomials and $q\left(x\right)\ne0$q(x)0.

Consider the rational function given by the equation $f\left(x\right)=\frac{x^2+2x-1}{\left(x+3\right)\left(x+5\right)}$f(x)=x2+2x1(x+3)(x+5).

To draw the function using your graphing calculator, first locate the graphing feature of the calculator and enter the equation. From there, draw and view the graph as below.

The graph of $y=f\left(x\right)$y=f(x) and its asymptotes.

 

Immediately we can observe two vertical asymptotes with equations $x=-5$x=5 and $x=-3$x=3 and a horizontal asymptote at $y=1$y=1.

We can also identify the existence of a maximum and minimum turning point as indicated below.

Maximum and minimum turning points of the graph.

 

A graphing calculator can be used to approximate each turning point. The maximum and minimum turning points are $\left(-3.55,-5.65\right)$(3.55,5.65) and $\left(-1.79,-0.35\right)$(1.79,0.35) respectively.

Similarly we can identify points of intersection such as the $x$x-intercepts and $y$y-intercept. This function appears to pass through the origin, however on close inspection and using the graphing calculator, we can determine the $x$x-intercepts as $\left(-2.41,0\right),\left(0.41,0\right)$(2.41,0),(0.41,0) and the $y$y-intercept as $\left(0,0.07\right)$(0,0.07) to two decimal places. Alternatively we can find the exact coordinates by substituting $x=0$x=0 and $y=0$y=0.

$x$x-intercepts marked above

 

$y$y-intercept marked above

 

Since the value of the function is undefined at $x=-5$x=5 and $x=-3$x=3 (the location of each vertical asymptote) we can state the domain of the function as $\left(-\infty,-5\right)\cup\left(-5,-3\right)\cup\left(-3,\infty\right)$(,5)(5,3)(3,) which translates to all real numbers except for $x=\left\{-5,-3\right\}$x={5,3}.

Even though there's a horizontal asymptote, notice that the range is all real values, or $\left(-\infty,\infty\right)$(,), since the function passes through the dotted line $y=1$y=1.

In general we can identify the points of intersection between any two functions, and not necessarily just the vertical and horizontal axes.

Consider the function $g\left(x\right)=x+5$g(x)=x+5 which we've drawn below.

The solution to the equation $\frac{x^2+2x-1}{\left(x+3\right)\left(x+5\right)}=x+5$x2+2x1(x+3)(x+5)=x+5 can be found by determining the $x$x-value of the point of intersection of the two graphs. In this case, the solution is $x=-2.77$x=2.77 to two decimal places.

The graph of $f\left(x\right)=\frac{x^2+2x-1}{\left(x+3\right)\left(x+5\right)}$f(x)=x2+2x1(x+3)(x+5) and $g\left(x\right)=x+5$g(x)=x+5.

 

Practice questions

Question 1

Draw $f\left(x\right)=\frac{x-5}{x-3}$f(x)=x5x3 with a graphing calculator and answer the following.

  1. How many $x$x-intercepts does the graph have?

  2. How many $y$y-intercepts does the graph have?

  3. How many horizontal asymptotes does the graph have?

  4. How many vertical asymptotes does the graph have?

Question 2

Draw $f\left(x\right)=-\frac{20}{3x^2-12x+17}$f(x)=203x212x+17 with a graphing calculator and answer the following.

  1. Find the equation of the horizontal asymptote.

  2. Find the coordinates of the minimum turning point.

  3. What is the domain of the function?

    $\left(-\infty,-2\right)\cup\left(-2,\infty\right)$(,2)(2,)

    A

    $\left(-\infty,0\right)\cup\left(0,\infty\right)$(,0)(0,)

    B

    $\left(-\infty,\infty\right)$(,)

    C

    $\left(-\infty,2\right)\cup\left(2,\infty\right)$(,2)(2,)

    D

    $\left(-\infty,-2\right)\cup\left(-2,\infty\right)$(,2)(2,)

    A

    $\left(-\infty,0\right)\cup\left(0,\infty\right)$(,0)(0,)

    B

    $\left(-\infty,\infty\right)$(,)

    C

    $\left(-\infty,2\right)\cup\left(2,\infty\right)$(,2)(2,)

    D
  4. What is the range of the function?

    $\left[-4,0\right)$[4,0)

    A

    $\left(-\infty,-4\right)\cup\left(-4,\infty\right)$(,4)(4,)

    B

    $\left(-\infty,2\right)\cup\left(2,\infty\right)$(,2)(2,)

    C

    $\left(-\infty,\infty\right)$(,)

    D

    $\left[-4,0\right)$[4,0)

    A

    $\left(-\infty,-4\right)\cup\left(-4,\infty\right)$(,4)(4,)

    B

    $\left(-\infty,2\right)\cup\left(2,\infty\right)$(,2)(2,)

    C

    $\left(-\infty,\infty\right)$(,)

    D

Question 3

Use a graphing calculator to solve $\frac{8x-22}{x-3}=\frac{3x-25}{x-9}$8x22x3=3x25x9.

  1. Round your answer to two decimal places and write each answer on the same line separated by a comma.

 

Solve rational functions analytically

A variety of equations arise from time to time that involves the sums and differences of rational expressions. The solution of these almost always begins by multiplying both sides of the equation by well-chosen factors that will 'clear' the denominators.

Worked examples

Question 4

Solve $\frac{x}{x-5}+\frac{10}{x+5}=\frac{4}{x^2-25}$xx5+10x+5=4x225

Observe that $x^2-25=\left(x-5\right)\left(x+5\right)$x225=(x5)(x+5), and so the strategy becomes clear. We need to multiply both sides of the equation by $x^2-25$x225 to clear the denominators.

$\frac{x}{x-5}+\frac{10}{x+5}$xx5+10x+5 $=$= $\frac{4}{x^2-25}$4x225
$\left(x-5\right)\left(x+5\right)\frac{x}{\left(x-5\right)}+\left(x-5\right)\left(x+5\right)\frac{10}{\left(x+5\right)}$(x5)(x+5)x(x5)+(x5)(x+5)10(x+5) $=$= $\left(x-5\right)\left(x+5\right)\frac{4}{x^2-25}$(x5)(x+5)4x225
$x\left(x+5\right)+10\left(x-5\right)$x(x+5)+10(x5) $=$= $4$4
$x^2+15x-54$x2+15x54 $=$= $0$0
$\left(x+18\right)\left(x-3\right)$(x+18)(x3) $=$= $0$0
$\therefore$     $x$x $=$= $3,-18$3,18

 

question 5

Solve $\frac{48}{x^2-8x}-\frac{80}{x^2-64}=0$48x28x80x264=0

Here we need to notice that the factor $\left(x-8\right)$(x8) is common to both denominator expressions. Thus multiplying both sides of the equation by $x\left(x-8\right)\left(x+8\right)$x(x8)(x+8) will clear the denominators. 

$\frac{48}{x^2-8x}-\frac{80}{x^2-64}$48x28x80x264 $=$= $0$0
$\frac{48}{x\left(x-8\right)}-\frac{80}{\left(x-8\right)\left(x+8\right)}$48x(x8)80(x8)(x+8) $=$= $0$0
$x\left(x-8\right)\left(x+8\right)\frac{48}{x\left(x-8\right)}-x\left(x-8\right)\left(x+8\right)\frac{80}{\left(x-8\right)\left(x+8\right)}$x(x8)(x+8)48x(x8)x(x8)(x+8)80(x8)(x+8) $=$= $0$0
$48\left(x+8\right)-80x$48(x+8)80x $=$= $0$0
$32x$32x $=$= $384$384
$\therefore$    $x$x $=$= $12$12

  

question 6

Find all real solutions to the equation  $x^{-4}+7x^{-2}-144=0$x4+7x2144=0. Are there any complex solutions?

To find the real solutions we could proceed as follows:

Re-write the equation as $\frac{1}{x^4}+\frac{7}{x^2}-144=0$1x4+7x2144=0, and then multiply both sides by $x^4$x4. This would lead to the new equation $1+7x^2-144x^4=0$1+7x2144x4=0

Then by setting $u=x^2$u=x2, and rearranging, we establish the quadratic equation given by $144u^2-7u-1=0$144u27u1=0. We solve this as follows:

$144u^2-7u-1$144u27u1 $=$= $0$0
$\left(16u+1\right)\left(9u-1\right)$(16u+1)(9u1) $=$= $0$0
$\therefore$     $u$u $=$= $-\frac{1}{16},\frac{1}{9}$116,19
     

For $u=x^2=-\frac{1}{16}$u=x2=116, there is no real solution because, for all $x$x, $x^2\ge0$x20

For $u=x^2=\frac{1}{9}$u=x2=19, we have that $x=\pm\frac{1}{3}$x=±13.

NOTE:

A slightly simpler alternative approach to the problem is to immediately set $u=x^{-2}$u=x2 so that the equation $x^{-4}+7x^{-2}-144=0$x4+7x2144=0 changes to $u^2+7u-144=0$u2+7u144=0.

Factoring, we have $\left(u+16\right)\left(u-9\right)=0$(u+16)(u9)=0 and therefore $u=-16,9$u=16,9.

Setting $u=x^{-2}=-16$u=x2=16, and $u=x^{-2}=9$u=x2=9, we again arrive at the same conclusion that the real solutions are given by $x=\pm\frac{1}{3}$x=±13

If we admit complex solutions, then we can solve the equation $x^2=-\frac{1}{16}$x2=116 as follows:

$x^2$x2 $=$= $-\frac{1}{16}$116
$16x^2+1$16x2+1 $=$= $0$0
$16x^2-i^2$16x2i2 $=$= $0$0
$\left(4x+i\right)\left(4x-i\right)$(4x+i)(4xi) $=$= $0$0
$\therefore$     $x$x $=$= $\pm\frac{i}{4}$±i4
     

 

Practice questions

Question 7

Find all solutions to $\frac{x}{x-2}+\frac{1}{x+2}=\frac{8}{x^2-4}$xx2+1x+2=8x24.

Question 8

Solve $\frac{3}{x^2-2x}-\frac{5}{x^2-4}=0$3x22x5x24=0.

Question 9

Find all complex solutions to $x^{-4}+8x^{-2}-9=0$x4+8x29=0.

Outcomes

A.REI.D.11

Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) =g(x); find the solutions approximately.

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