[email protected] MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 1 Bruce Mayer, PE Chabot...

[email protected] • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§J Graph§J GraphRational Rational

FcnsFcns



Review §Review §

Any QUESTIONS About• §5.7 → PolyNomical Eqn Applications

Any QUESTIONS About HomeWork• §5.7 → HW-21

5.7 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 in Step 2 by a smooth curve



Making Complete PlotsMaking Complete Plots1. Arrows in

POSITIVE Direction Only

2. Label x & y axes on POSITIVE ends

3. Mark and label at least one unit on each axis

4. Use a ruler for Axes & Straight-Lines

5. Label significant points or quantities



Rational FunctionRational Function

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 x

f xq x



Visualizing Domain and RangeVisualizing Domain and Range

Domain = the set of a function’s Inputs, as found on the horizontal axis (the x-Axis)

Range = the set of a function’s OUTputs , found on the vertical axis (the y-Axis).



Find Rational Function DomainFind Rational Function Domain

1. Write an equation that sets the DENOMINATOR of the rational function equal to 0.

2. Solve the equation.

3. Exclude the value(s) found in step 2 from the function’s domain.



Example Example Domain & Range Domain & Range

Graph y = f(x) = x2. Then State the Domain & Range of the function

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




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)




The DOMAIN of a function is the set of all first (or “x”) components of the Ordered Pairs.

Projecting on the X-axis the x-components of ALL POSSIBLE ordered pairs displays the DOMAIN of the function just plotted




Domain of y = f(x) = x2 Graphically

-2

-1

0

1

2

3

4

5

6

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


x

y

This Projection Pattern Reveals a Domain of

number real a isxx




The RANGE of a function is the set of all second (or “y”) components of the ordered pairs. The projection of the graph onto the y-axis shows the range -2

-1

0

1

2

3

4

5

6

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


x

y

0 yyRANGE



Domain RestrictionsDomain Restrictions

EVERY element, x, in a functional Domain MUST produce a VALID Range output, y

ReCall the Real-Number Operations that Produce INvalid Results• Division by Zero

• Square-Root of a Negative Number

x-values that Produce EITHER of the above can NOT be in the Function Domain



Example Example find Domain: find Domain: 3 2

6( ) .

5 4f y

y y y

SOLUTION Avoid Division by Zero3 25 4 0y y y

2 5 4 0y y y

4 1 0y y y 0 or 4 0 or 1 0y y y

4 1y y

Set the DENOMINATOR equal to 0.

Factor out the monomial GCF, y.

Use the zero-products theorem.

The function is UNdefined if y is replaced by 0, −4, or −1, so the domain is {y|y ≠ −4, −1, 0}

FOIL Factor by Guessing

Solve the MiniEquations for y



Example Example Find the DOMAIN

and GRAPH for f(x) SOLUTION

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



Vertical AsymptotesVertical Asymptotes The VERTICAL asymptotes of a

rational function f(x) = p(x)/q(x) are found by determining the ZEROS of q(x) that are NOT also ZEROS of p(x). • If p(x) and q(x) are polynomials with no

common factors other than constants, we need to determine only the zeros of the denominator q(x).

If a is a zero of the denominator, then the Line x = a is a vertical asymptote for the graph of the function.



Example Example Vertical Asymptote Vertical Asymptote Determine the

vertical asymptotes of the function

2

2 3( )

4

xf x

x

Factor to find the

zeros of the denominator:

x2 − 4 = 0 = (x + 2)(x − 2)

Thus the vertical asymptotes are the lines x = −2 & x = 2



Horizontal AsymptotesHorizontal Asymptotes

When the numerator and the denominator of a rational function have the same degree, the line y = a/b is the horizontal asymptote, • where a and b are the leading

coefficients of the numerator and the denominator, respectively.

In This case The line y = c = a/b is a horizontal asymptote.



Example Example Horiz. Asymptote Horiz. Asymptote

Find the horizontal asymptote for4 2

4

6 3 1( )

9 3 2

x xf x

x x

The numerator and denominator have the same degree. The ratio of the leading coefficients is 6/9, so the line y = 2/3 is the horizontal asymptote



Finding a Horizontal AsymptoteFinding a Horizontal Asymptote When the numerator and the denominator of a

rational function have the same degree, the line y = a/b is the horizontal asymptote, where a and b are the leading coefficients of the numerator and the denominator, respectively.

When the degree of the numerator of a rational function is less than the degree of the denominator, the x-axis, or y = 0, is the horizontal asymptote.

When the degree of the numerator of a rational function is greater than the degree of the denominator, there is no horizontal asymptote.



Asymptotic BehaviorAsymptotic 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



Example Example Graph Graph

SOLUTION Vertical asymptotes: x + 3 = 0, so x = −3 The degree of the numerator and

denominator is the same. Thus y = 2 is the horizontal asymptote

Graph Plan• Draw the asymptotes with dashed lines.

• Compute and plot some ordered pairs and connect the dots to draw the curve.

2( )

3

xh x

x



Example Example Graph Graph Construct T-Table

2( )

3

xh x

x

4/5200428455

3.57h(x)x

Plot Points, “Dash In” Asymptotes



WhiteBoard WorkWhiteBoard Work

Problems From §J1 Exercise Set• J2, J4, J6

Watch the DENOMINATORPolyNomial; it canProduce Div-by-Zero



All Done for TodayAll Done for Today

Asymptote Architecture

wins competition for WBCB Tower,

to be tallest building in Asia



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

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