[email protected] MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical.

[email protected] • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§G §G TranslateTranslateRational Rational

PlotsPlots



Review §Review §

Any QUESTIONS About• §G → Graphing Rational Functions

Any QUESTIONS About HomeWork• §G → HW-22

G MTH 55



GRAPH BY PLOTTING POINTSGRAPH BY PLOTTING POINTS Step1. Make a representative

T-table of solutions of the equation.

Step 2. Plot the solutions as ordered pairs in the Cartesian coordinate plane.

Step 3. Connect the solutions (dots) in Step 2 by a smooth curve



Translation of GraphsTranslation of Graphs

Graph y = f(x) = x2. • Make T-Table & Connect-Dots

Select integers for x, starting with −2 and ending with +2. The T-table:

x 2xy Ordered Pair yx,

2 42 2 y 4,2

1 11 2 y 1,1

0 002 y 0,0

1 112 y 1,1

2 422 y 4,2



Translation of GraphsTranslation of Graphs

Now Plot the Five Points and connect them with a smooth Curve

-2

-1

0

1

2

3

4

5

6

-4 -3 -2 -1 0 1 2 3 4

M55_§JBerland_Graphs_0806.xls

x

y

(−2,4) (2,4)

(−1,1) (1,1)

(0,0)



Axes TranslationAxes Translation

Now Move UP the graph of y = x2 by two units as shown

y

x

-1

0

1

2

3

4

5

6

7

-4 -3 -2 -1 0 1 2 3 4


(−2,4) (2,4)

(−1,1) (1,1)

(0,0)

(−2,6) (2,6)

(−1,3) (1,3)

(0,2) What is the What is the

Equation Equation for the new for the new Curve?Curve?




Compare ordered pairs on the graph of with the corresponding ordered pairs on the new curve:

2xy

New Curve (2 units up)

4,2 6,2

1,1 3,1

0,0 2,0

1,1 3,1

4,2 6,2

y

x

-1

0

1

2

3

4

5

6

7

-4 -3 -2 -1 0 1 2 3 4


(−2,4) (2,4)

(−1,1) (1,1)

(0,0)

(−2,6) (2,6)

(−1,3) (1,3)

(0,2)



Axes TranslationAxes Translation Notice that the x-coordinates

on the new curve are the same, but the y-coordinates are 2 units greater

So every point on the new curve makes the equation y = x2+2 true, and every point off the new curve makes the equation y = x2+2 false.

An equation for the new curve is thus

y = x2+2

2xy

New Curve (2 units up)

4,2 6,2

1,1 3,1

0,0 2,0

1,1 3,1

4,2 6,2




Similarly, if every point on the graph of y = x2 were is moved 2 units down, an equation of the new curve is y = x2−2

2xy

-3

-2

-1

0

1

2

3

4

5

-4 -3 -2 -1 0 1 2 3 4


y

x

22 xy




When every point on a graph is moved up or down by a given number of units, the new graph is called a vertical translation of the original graph.

y

x

-1

0

1

2

3

4

5

6

7

-4 -3 -2 -1 0 1 2 3 4


yy

xx

-1

0

1

2

3

4

5

6

7

-4 -3 -2 -1 0 1 2 3 4

M55_§JBerland_Graphs_0806.xls-3

-2

-1

0

1

2

3

4

5

-4 -3 -2 -1 0 1 2 3 4


y

x

-3

-2

-1

0

1

2

3

4

5

-4 -3 -2 -1 0 1 2 3 4


yy

xx

2xy 22 xy

2xy

22 xy



Vertical TranslationVertical Translation

Given the Graph of y = f(x), and c > 0

1. The graph of y = f(x) + c is a vertical translation c-units UP

2. The graph of y = f(x) − c is a vertical translation c-units DOWN



Horizontal TranslationHorizontal Translation What if every point on the graph of

y = x2 were moved 5 units to the right as shown below.

What is the eqn of the new curve?What is the eqn of the new curve?

2xy ?y

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5 6 7 8 9

Xlate_ABS_Graphs_1010.xls

x

y

(−2,4) (2,4) (3,4) (7,4)



Horizontal TranslationHorizontal Translation

Compare ordered pairs on the graph of with the corresponding ordered pairs on the new curve:

2xy

New Curve

(5 units right)

4,2 4,3

1,1 1,4

0,0 0,5

1,1 1,6

4,2 4,7



Horizontal TranslationHorizontal Translation Notice that the y-coordinates on the

new curve are the same, but the x-coordinates are 5 units greater.

Does every point on the new curve make the equation y = (x+5)2 true? • No; for example if we input (5,0) we get 0 = (5+5)2,

which is false. • But if we input (5,0) into the equation y = (x−5)2 , we get

0 = (5−5)2 , which is TRUE.

In fact, every point on the new curve makes the equation y = (x−5)2 true, and every point off the new curve makes the equation y = (x−5)2 false. Thus an equation for the new curve is y = (x−5)2

2xy

New Curve

(5 units right)

4,2 4,3

1,1 1,4

0,0 0,5

1,1 1,6

4,2 4,7



Horizontal TranslationHorizontal Translation

Given the Graph of y = f(x), and c > 0

1. The graph of y = f(x−c) is a horizontal translation c-units to the RIGHT

2. The graph of y = f(x+c) is a horizontal translation c-units to the LEFT



Example Example Plot by Translation Plot by Translation

Use Translation to graph f(x) = (x−3)2−2 LET y = f(x) → y = (x−3)2−2 Notice that the graph of y = (x−3)2−2

has the same shape as y = x2, but is translated 3-unit RIGHT and 2-units DOWN.

In the y = (x−3)2−2, call −3 and −2 translators




The graphs of y=x2 and y=(x−3)2−2 are different; although they have the Same shape they have different locations

Now remove the translators by a substitution of x’ (“x-prime”) for x, and y’ (“y-prime”) for y

Remove translators for an (x’,y’) eqn

22 23 xyxy




Since the graph of y=(x−3)2−2 has the same shape as the graph of y’ =(x’)2 we can use ordered pairs of y’ =(x’)2 to determine the shape

T-tablefory’ =(x’)2

x y Ordered Pair yx ,

1 1 1,1

0 0 0,0

1 1 1,1



Example Example Plot by Translation Plot by Translation Next use the translation rules to find the

origin of the x’y’-plane. Draw the x’-axis and y’-axis through the translated origin 22 23 xyxy

• The origin of the x’y’-plane is 3 units right and 2 units down from the origin of the xy-plane.

• Through the translated origin, we use dashed lines to draw a new horizontal axis (the x’-axis) and a new vertical axis (the y’-axis).




Locate the Origin of the Translated Axes Set using the translator values

Move: 3-Right, 2-Down




Now Plot the ordered pairs of the x’y’ equation on the x’y’-plane, and use the points to draw an appropriate graph.• Remember that this graph is smooth




Perform a partial-check to determine correctness of the last graph. Pick any point on the graph and find its (x,y) CoOrds; e.g., (4, −1) is on the graph

The Ordered Pair (4, −1) should make the xy Eqn True




Sub (4, −1) into y=(x−3)2−2

23 2 xy:

11

2341 2

Thus (4, −1) does make y = (x−3)2−2 true. In fact, all the points on the translated graph make the original Eqn true, and all the points off the translated graph make the original Eqn false




What are the Domain &Range of y=(x−3)2−2?

y

x

-3

-2

-1

0

1

2

3

4

5

6

7

-1 0 1 2 3 4 5 6 7 8 9


yy

xx

-3

-2

-1

0

1

2

3

4

5

6

7

-1 0 1 2 3 4 5 6 7 8 9


23 2 xy

To find the domain & range of the xy-eqn, examine the xy-graph (not the x’y’ graph).

The xy graph showns• Domain of f is {x|x is any real number}

• Range of f is {y|y ≥ −2}



Graphing Using TranslationGraphing Using Translation

1. Let y = f(x)2. Remove the x-value & y-value

“translators” to form an x’y’ eqn.3. Find ordered pair solutions of the x’y’ eqn4. Use the translation rules to find the origin

of the x’y’-plane. Draw dashed x’ and y’ axes through the translated origin.

5. Plot the ordered pairs of the x’y’ equation on the x’y’-plane, and use the points to draw an appropriate graph.



Example Example ReCall Graph ReCall Graph yy = | = |xx||

Make T-tablex y = |x |

-6 6-5 5-4 4-3 3-2 2-1 10 01 12 23 34 45 56 6

x

y

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

file =XY_Plot_0211.xls



Example Example Graph Graph yy = | = |xx+2|+3+2|+3

Step-1 32 xxf 32 xy

3;2 :Xlators yx xy Step-2

Step-3 → T-table in x’y’

x y Ordered Pair yx ,

1 1 1,1

0 0 0,0

1 1 1,1



Example Example Graph Graph yy = | = |xx+2|+3+2|+3 Step-4: the x’y’-plane origin is 2 units

LEFT and 3 units UP from xy-plane

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6


y

x

x

Up 3

Left 2

y




Step-5: Remember that the graph of y = |x| is V-Shaped:

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6


yyy

x

x

Up 3

Left 2



Rational Function TranslationRational Function Translation

A rational function is a function f that is a quotient of two polynomials, that is,

Where• where p(x) and q(x) are polynomials and

where q(x) is not the zero polynomial.

• The domain of f consists of all inputs x for which q(x) ≠ 0.

( )( ) ,

( )

p xf x

q x



Example Example Find the DOMAIN

and GRAPH for f(x) SOLUTION

When the denom x = 0, we have x = 0, so the only input that results in a denominator of 0 is 0. Thus the domain {x|x 0} or (–, 0) (0, )

Construct T-table

x

xf1

x y = f(x)

-8 -1/8-4 -1/4-2 -1/2-1 -1

-1/2 -2-1/4 -4-1/8 -81/8 81/4 41/2 21 12 1/24 1/48 1/8

Next Plot points & connect Dots



Plot Plot x

xf1

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10


x

y Note that the Plot

approaches, but never touches, • the y-axis (as x ≠ 0)

– In other words the graph approaches the LINE x = 0

• the x-axis (as 1/ 0)– In other words the graph

approaches the LINE y = 0

A line that is approached by a graph is called an ASYMPTOTE



ReCall Asymptotic BehaviorReCall Asymptotic Behavior

The graph of a rational function never crosses a vertical asymptote

The graph of a rational function might cross a horizontal asymptote but does not necessarily do so



Recall Vertical TranslationRecall Vertical Translation Given the graph of the equation

y = f(x), and c > 0, the graph of y = f(x) + c

is the graph of y = f(x) shifted UP (vertically) c units;

the graph of y = f(x) – c is the graph of y = f(x) shifted DOWN (vertically) c units

y = 3x2

y = 3x2−3

y = 3x2+2



Recall Horizontal TranslationRecall Horizontal Translation Given the graph of the equation

y = f(x), and c > 0, the graph of y = f(x– c)

is the graph of y = f(x) shifted RIGHT (Horizontally) c units;

the graph of y = f(x + c) is the graph of y = f(x) shifted LEFT (Horizontally) c units.

y = 3x2

y = 3(x-2)2

y = 3(x+2)2



ReCall Graphing by TranslationReCall Graphing by Translation

1. Let y = f(x)

2. Remove the translators to form an x’y’ eqn

3. Find ordered pair solutions of the x’y’ eqn

4. Use the translation rules to find the origin of the x’y’-plane. Draw the dashed x’ and y’ axes through the translated origin.

5. Plot the ordered pairs of the x’y’ equation on the x’y’-plane, and use the points to draw an appropriate graph.



Example Example Graph Graph

Step-1

Step-2

Step-3 → T-table in x’y’

12

1

xxf

12

1

xxf 1

2

1

xy

12

1

xy 2

11

xy

xy

1

x ' y ' x ' y '

-4 -1/4 1/4 4-2 -1/2 1/2 2-1 -1 1 1

-1/2 -2 2 1/2-1/4 -4 4 1/4



Example Example Graph Graph Step-4: The

origin of the x’y’ -plane is 2 units left and 1 unit up from the origin of the xy-plane:

12

1

xxf

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6


x

yy

x



Example Example Graph Graph Step-5: We know

that the basic shape of this graph is Hyperbolic. Thus we can sketch the graph using Fewer Points on the translated axis using the T-Table

12

1

xxf

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6


x

yy

x



Example Example Graph Graph Examination of

the Graph reveals• Domain →

{x|x ≠ −2}

• Range →{y|y ≠ 1}

12

1

xxf

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6


x

yy

x




Step-1

Step-2

Step-3 → T-table in (x’y’) by y’ = −2/x’

13

2

xxg

3

21

xy

xy

2

13

2

xxg

13

2

xy

13

2

xy

x ' y ' x ' y '

-4 1/2 1/4 -8-2 1 1/2 -4-1 2 1 -2

-1/2 4 2 -1-1/4 8 4 -1/2




origin of the x’y’ -plane is 3 units RIGHT and 1 unit DOWN from the origin of the xy-plane

13

2

xxg

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

-3 -2 -1 0 1 2 3 4 5 6 7 8 9


x

y y

x





13

2

xxg

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

-3 -2 -1 0 1 2 3 4 5 6 7 8 9


x

y y

x



Example Example Graph Graph Notice for this

Graph that the Hyperbola is• “mirrored”, or

rotated 90°, by the leading Negative sign

• “Spread out”, or expanded, by the 2 in the numerator

13

2

xxg

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

-3 -2 -1 0 1 2 3 4 5 6 7 8 9


x

y y

x



Example Example Graph Graph Examination

of the Graph reveals• Domain →

{x|x ≠ 3}

• Range →{y|y ≠ −1}

13

2

xxg

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

-3 -2 -1 0 1 2 3 4 5 6 7 8 9


x

y y

x




Step-1

Step-2

Step-3 → T-table in x’y’ by y’ = −1/(2x’)

212

1

xxh

xy

2

1

212

1

xxh 2

12

1

xy

212

1

xy 12

12

xy

x ' y ' x ' y '

-4 1/8 1/4 -2-2 1/4 1/2 -1-1 1/2 1 -1/2

-1/2 1 2 -1/4-1/4 2 4 -1/8




origin of the x’y’ -plane is 1 unit RIGHT and 2 units UP from the origin of the xy-plane

212

1

xxh

x

y y

x

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7






212

1

xxh

x

y y

x

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7




Example Example Graph Graph Notice for this

Graph that the Hyperbola is• “mirrored”, or

rotated 90°, by the leading Negative sign

• “Pulled in”, or contracted, by the 2 in the Denominator

212

1

xxh

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7


x

y y

x



Example Example Graph Graph Examination

of the Graph reveals• Domain →

{x|x ≠ 1}

• Range →{y|y ≠ 2}

212

1

xxh

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7


x

y y

x



WhiteBoard WorkWhiteBoard Work

Problems From §G1 Exercise Set• G8, G10, G12

A Function with TWO Vertical Asymptotes

2

3( ) .

2 5 3

xf x

x x



All Done for TodayAll Done for Today

AnotherCool Design

byAsymptote

Architecture



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

–

srsrsr 22



Translate Up or DownTranslate Up or Down

Make Graphs for

Notice: 5

3

xfxh

xfxg

f (x) x , g(x) x 3 and h(x) x 5

• Of the form of VERTICAL Translations

f (x) x

g(x) x 3

h(x) x 5



Translate Left or RightTranslate Left or Right

Make Graphs for

f (x) x , g(x) x 3 and h(x) x 5

Notice: 53 xfxhxfxg

f (x) x

g(x) x 3

h(x) x 5

• Of the form of HORIZONTAL Translations




Step-4: The origin of the x’y’ -plane is 2 units LEFT and 3 units UP from the origin of the xy-plane:




Step-5: Remember that the graph of y = |x| is V-Shaped:



-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6


yy

x

x

Up 3

Over 2



-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6


yyy

x

Up 3

Over 2



-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5 6 7 8 9


x

y y

x



-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5 6 7 8 9


x

y

[email protected] MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical.

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