Exercise 48 Page 513

Practice makes perfect

a Consider the general equation of a reciprocal function.

Reciprocal Function y = \frac{a}{x - h} + k Constant of Variation : a Horizontal Asymptote : k Vertical Asymptote : h

Let's rewrite the function so that it matches the general equation of a reciprocal function.

y = \frac{2}{3 x - 6}

FactorOut

Factor out $3$

y = \frac{2}{3 ( x - 2 )}

ReduceFrac

$\frac{a}{b} = \frac{a / 3}{b / 3}$

y = \frac{2 / 3}{x - 2}

$a = a + 0$

y = \frac{2 / 3}{x - 2} + 0

Now we can identify the constant of variation, as well as the horizontal and vertical asymptote.

y = \frac{2 / 3}{x - 2} + 0 Constant of Variation : 2 / 3 Horizontal Asymptote : 0 Vertical Asymptote : 2

To draw the function we will start by drawing the asymptotes.

To plot the graph, we have to calculate some points that fall on it.

$x$	$\frac{2}{3 x - 6}$	$y$
$- 1$	$\frac{2}{3 ( - 1 ) - 6}$	$- 0.22$
$0$	$\frac{2}{3 ( 0 ) - 6}$	$- 0.33$
$1$	$\frac{2}{3 ( 1 ) - 6}$	$- 0.67$
$1.5$	$\frac{2}{3 ( 1 . 5 ) - 6}$	$- 1.33$
$2.5$	$\frac{2}{3 ( 2 . 5 ) - 6}$	$1.33$
$3$	$\frac{2}{3 ( 3 ) - 6}$	$0.67$
$4$	$\frac{2}{3 ( 4 ) - 6}$	$0.33$
$5$	$\frac{2}{3 ( 5 ) - 6}$	$0.22$

By marking the points in our coordinate plane, we can draw the function.

b Like in Part A, we need to start by rewriting the function to match the general equation.

y = \frac{1}{2 - 4 x}

CommutativePropAdd

Commutative Property of Addition

y = \frac{1}{- 4 x + 2}

FactorOut

Factor out $- 4$

y = \frac{1}{- 4 ( x - 1 / 2 )}

MoveNegDenomToNum

Put minus sign in numerator

y \frac{- 1}{4 ( x - 1 / 2 )}

ReduceFrac

$\frac{a}{b} = \frac{a / 4}{b / 4}$

y = \frac{- 1 / 4}{x - 1 / 2}

$a = a + 0$

y = \frac{- 1 / 4}{x - 1 / 2} + 0

Now we can identify the constant of variation, as well as the horizontal and vertical asymptote.

y = \frac{- 1 / 4}{x - 1 / 2} + 0 Constant of Variation : - 1 / 4 Horizontal Asymptote : 0 Vertical Asymptote : 1 / 2

To draw the function, we will start by drawing the asymptotes.

To plot the graph, we have to calculate some points that fall on the graph.

$x$	$\frac{1}{2 - 4 x}$	$y$
$- 2$	$\frac{1}{2 - 4 ( - 2 )}$	$0.1$
$- 1$	$\frac{1}{2 - 4 ( - 1 )}$	$0.17$
$0$	$\frac{1}{2 - 4 ( 0 )}$	$0.5$
$0.25$	$\frac{1}{2 - 4 ( 0 . 2 5 )}$	$1$
$0.75$	$\frac{1}{2 - 4 ( 0 . 7 5 )}$	$- 1$
$1$	$\frac{1}{2 - 4 ( 1 )}$	$- 0.5$
$2$	$\frac{1}{2 - 4 ( 2 )}$	$- 0.17$
$3$	$\frac{1}{2 - 4 ( 3 )}$	$- 0.1$

By marking the points in our coordinate plane, we can draw the function.

c Like in previous parts, we want to rewrite the function so that it matches the general equation of a reciprocal function.

y = \frac{3 - x}{x + 2}

CommutativePropAdd

Commutative Property of Addition

y = \frac{- x + 3}{x + 2}

FactorOutNegSign

Factor out a minus sign

y = \frac{- ( x - 3 )}{x + 2}

MoveNegDenomToFrac

Put minus sign in front of fraction

y = - \frac{x - 3}{x + 2}

Rewrite

Rewrite $- 3$ as $2 - 5$

y = - \frac{x + 2 - 5}{x + 2}

WriteDiffFrac

Write as a difference of fractions

y = - (\frac{x + 2}{x + 2} - \frac{5}{x + 2})

CalcQuot

Calculate quotient

y = - (1 - \frac{5}{x + 2})

Distr

Distribute $- 1$

y = - 1 + \frac{5}{x + 2}

CommutativePropAdd

Commutative Property of Addition

y = \frac{5}{x + 2} - 1

$a = - (- a)$

y = \frac{5}{x - ( - 2 )} - 1

AddPar

Add parentheses

y = \frac{5}{x - ( - 2 )} + (- 1)

Now we can identify the constant of variation, as well as the horizontal and vertical asymptote.

y = \frac{5}{x - ( - 2 )} + (- 1) Constant of Variation : 5 Horizontal Asymptote : - 1 Vertical Asymptote : - 2

To draw the function, we will start by drawing the asymptotes.

To plot the graph, we have to calculate some points that fall on the graph.

$x$	$\frac{3 - x}{x + 2}$	$y$
$- 6$	$\frac{3 - ( - 6 )}{- 6 + 2}$	$- 2.25$
$- 5$	$\frac{3 - ( - 5 )}{- 5 + 2}$	$- 2.67$
$- 4$	$\frac{3 - ( - 4 )}{- 4 + 2}$	$- 3.5$
$- 3$	$\frac{3 - ( - 3 )}{- 3 + 2}$	$- 6$
$- 2.5$	$\frac{3 - ( - 2 . 5 )}{- 2 . 5 + 2}$	$- 11$
$- 1.5$	$\frac{3 - ( - 1 . 5 )}{- 1 . 5 + 2}$	$9$
$- 1$	$\frac{3 - ( - 1 )}{- 1 + 2}$	$4$
$0$	$\frac{3 - 0}{0 + 2}$	$1.5$
$1$	$\frac{3 - 1}{1 + 2}$	$0.67$

By marking the points in our coordinate plane, we can draw the function.

d Similar to previous parts, we will have to start by factoring and isolating

y .

x y - y = 1

FactorOut

Factor out $y$

y (x - 1) = 1

DivEqn

$LHS / (x - 1) = RHS / (x - 1)$

y = \frac{1}{x - 1}

$a = a + 0$

y = \frac{1}{x - 1} + 0

Now we can identify the constant of variation, as well as the horizontal and vertical asymptote.

y = \frac{1}{x - 1} + 0 Constant of Variation : 1 Horizontal Asymptote : 0 Vertical Asymptote : 1

To draw the function, we will start by drawing the asymptotes.

To plot the graph, we have to calculate some points that fall on the graph.

$x$	$\frac{1}{x - 1}$	$y$
$- 2$	$\frac{1}{- 2 - 1}$	$- 0.33$
$- 1$	$\frac{1}{- 1 - 1}$	$- 0.5$
$0$	$\frac{1}{0 - 1}$	$- 1$
$0.5$	$\frac{1}{0 . 5 - 1}$	$- 2$
$1.5$	$\frac{1}{1 . 5 - 1}$	$2$
$2$	$\frac{1}{2 - 1}$	$1$
$3$	$\frac{1}{3 - 1}$	$0.5$
$4$	$\frac{1}{4 - 1}$	$0.33$

By marking the points in our coordinate plane, we can draw the function.

Subchapter links

Lesson Check p.512

Practice and Problem-Solving Exercises p.512-513

Standardized Test Prep p.514

Mixed Review p.514

Exercise 48 Page 513

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