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BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot Mathematics
§J Graph§J GraphRational Rational
FcnsFcns
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt2
Bruce Mayer, PE Chabot College Mathematics
Review §Review §
Any QUESTIONS About• §5.7 → PolyNomical Eqn Applications
Any QUESTIONS About HomeWork• §5.7 → HW-21
5.7 MTH 55
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt3
Bruce Mayer, PE Chabot College Mathematics
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
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt4
Bruce Mayer, PE Chabot College Mathematics
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
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt5
Bruce Mayer, PE Chabot College Mathematics
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
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt6
Bruce Mayer, PE Chabot College Mathematics
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).
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt7
Bruce Mayer, PE Chabot College Mathematics
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.
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt8
Bruce Mayer, PE Chabot College Mathematics
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
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt9
Bruce Mayer, PE Chabot College Mathematics
Example Example Domain & Range Domain & Range
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)
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt10
Bruce Mayer, PE Chabot College Mathematics
Example Example Domain & Range Domain & Range
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
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt11
Bruce Mayer, PE Chabot College Mathematics
Example Example Domain & Range Domain & Range
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
M55_§JBerland_Graphs_0806.xls
x
y
This Projection Pattern Reveals a Domain of
number real a isxx
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt12
Bruce Mayer, PE Chabot College Mathematics
Example Example Domain & Range Domain & Range
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
M55_§JBerland_Graphs_0806.xls
x
y
0 yyRANGE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt13
Bruce Mayer, PE Chabot College Mathematics
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
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt14
Bruce Mayer, PE Chabot College Mathematics
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
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt15
Bruce Mayer, PE Chabot College Mathematics
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
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt16
Bruce Mayer, PE Chabot College Mathematics
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
M55_§JBerland_Graphs_0806.xls
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
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt17
Bruce Mayer, PE Chabot College Mathematics
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.
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt18
Bruce Mayer, PE Chabot College Mathematics
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
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt19
Bruce Mayer, PE Chabot College Mathematics
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.
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt20
Bruce Mayer, PE Chabot College Mathematics
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
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt21
Bruce Mayer, PE Chabot College Mathematics
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.
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt22
Bruce Mayer, PE Chabot College Mathematics
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
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt23
Bruce Mayer, PE Chabot College Mathematics
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
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt24
Bruce Mayer, PE Chabot College Mathematics
Example Example Graph Graph Construct T-Table
2( )
3
xh x
x
4/5200428455
3.57h(x)x
Plot Points, “Dash In” Asymptotes
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt25
Bruce Mayer, PE Chabot College Mathematics
WhiteBoard WorkWhiteBoard Work
Problems From §J1 Exercise Set• J2, J4, J6
Watch the DENOMINATORPolyNomial; it canProduce Div-by-Zero
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt26
Bruce Mayer, PE Chabot College Mathematics
All Done for TodayAll Done for Today
Asymptote Architecture
wins competition for WBCB Tower,
to be tallest building in Asia
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt27
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot Mathematics
AppendiAppendixx
–
srsrsr 22