bruce mayer, pe licensed electrical & mechanical engineer bmayer@chabotcollege

[email protected] • MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§1.2 Graphs

Of Functions



Review §

Any QUESTIONS About• §1.1 → Introduction to Functions

Any QUESTIONS About HomeWork• §1.1 → HW-01

1.1



§1.2 Learning Goals

Review the rectangular coordinate system

Graph several functions Study intersections of graphs, the

vertical line test, and intercepts Sketch and use graphs of quadratic

functions in applications



Points and Ordered-Pairs

To graph, or plot, points we use two perpendicular number lines called axes. The point at which the axes cross is called the origin. Arrows on the axes indicate the positive directions

Consider the pair (2, 3). The numbers in such a pair are called the CoOrdinates. The first coordinate, x, in this case is 2 and the second, y, coordinate is 3.



Plot-Pt using Ordered Pair

To plot the point (2, 3) we start at the origin, move horizontally to the 2, move up vertically 3 units, and then make a “dot”• x = 2• y = 3

(2, 3)



Example Plot the point (–4,3)

Starting at the origin, we move 4 units in the negative horizontal direction. The second number, 3, is positive, so we move 3 units in the positive vertical direction (up)• x = –4; y = 3

4 units left

3 u

nit

s u

p



Example Read XY-Plot

Find the coordinates of pts A, B, C, D, E, F, G

AB

C

D

E

F

G

• Solution: Point A is 5 units to the right of the origin and 3 units above the origin. Its coordinates are (5, 3). The other coordinates are as follows: – B: (−2,4)– C: (−3,−4)– D: (3,−2)– E: (2, 3)– F: (−3,0)– G: (0, 2)



Tool For XY Graphing

Called “ Engineering Computation Pad”• Light Green

Backgound• Tremendous Help

with Graphing and Sketching

• Available in Chabot College Book Store

• I use it for ALL my Hand-Work

Graph on this side!



XY Quadrants

The horizontal and vertical axes divide the plotting plane into four regions, or quadrants• Note the

Ordinate & Abscissa

(Ordinate)

(Abscissa)



The Distance Formula

The distance between the points (x1, y1) and (x2, y1) on a horizontal line is |x2 – x1|.

Similarly, the distance between the points (x2, y1) and (x2, y2) on a vertical line is |y2 – y1|.

-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

6y

O

(-4,-2) (3,-2)

(5,4)

(5,-4)

x

XYGraph6x6HnVdistance.fig

743 Hd

844 Vd



Pythagorean Distance

Now consider any two points (x1, y1) and (x2, y2).

These points, along with (x2, y1), describe a right triangle. The lengths of the legsare |x2 – x1| and |y2 – y1|.



Pythagorean Distance

Find d, the length of the hypotenuse, by using the Pythagorean theorem:

d2 = |x2 – x1|2 + |y2 – y1|2

Since the square of a number is the same as the square of its opposite, we can replace the absolute-value signs with parentheses:

d2 = (x2 – x1)2 + (y2 – y1)2



Distance Formula Formally

The distance d between any two points (x1, y1) and (x2, y2) is given by

212

212 yyxxd



Example Find Distance Find Distance

Between Pt1 & Pt2 Use Dist Formula

Pt-1

Pt-2

212

212 yyxxd

22 2682 d

483686 22 d

10100 d



Graphing by Dot Connection

“Connecting the Dots” ALWAYS works for plotting any y = f(x) from an eqn

The procedure• Use Fcn Eqn to make

a “T-Table”• Properly Construct and

Label Graph• Plot Ordered-Pairs in T-Table• Connect Dots with Straight

or Curved Lines

x y

-6 0.74-5 1.02-4 1.40-3 1.92-2 2.65-1 3.640 5.001 6.872 9.453 12.994 17.865 24.566 33.76

T-Table for xexfy 5



Making 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



Example Graph f(x) = 2x2

Solution:Make T-Table andConnect-Dotsx y (x, y)

01

–12

–2

02288

(0, 0)(1, 2)

(–1, 2)(2, 8)

(–2, 8) x = 0 is Axis of Symm (0,0) is Vertex

x

y

(-1,2 )

(2,8)(-2,8)

-5 -4 -3 -2 -1 1 2 3 4 5

4

3

6

2

5

1

(1,2)

(0, 0)-1

-2

78



Plot PieceWise Function: f(x) = |x|

ReCalling the Absolute Value Definition can State Function in PieceWise Form

Make T-Table from Above Fcn Def

Class Question: What will be the SHAPE of the the Graph of this Function?

0

0

xx

xxxfy



Example Graph f(x) = |x|

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

Plot Points, and Connect Dots



Graph Intersections

How To Find Solutions to the Equality of Functions? • Graph Both Functions and Find

Intersections– At Intersections x & y are the SAME for both

functions, and ANY point on the graph is a “Solution” to Fcn

Thus at Intersections BOTH Fcns are Simultaneously Solved



Graph InterSection Example

Consider two Functions: Want to Find solution(s), xs, such that

Note that this Equation can NOT Solved exactly; The solutions are irrational Numbers• Such “NonAlgebraic” Eqns are Called

“Transcendental”

Find Solution by Graph Intersection(s)

1lncos7 21 xxfxxf

ssss xfxxxf 21 1lncos7



Graph InterSection Example

Plot Both Functions on Same Graph

Find Intersection(s)

Read xs from intersection points

0 1 2 3 4 5 6 7 8-8

-6

-4

-2

0

2

4

6

8

x

f 1(x)

= 7

cos(

x) •

f 2(x)

= ln

(x+

1)

f1

f2

≈1.44 ≈4.97 ≈7.54



MSExcel vs Transcendental The “Goal Seek”

Command in MicroSoft Excel to Find xs with greater Accuracy

Use Excel to Solve the Transcendental Equation

Collect Terms on One Side, and use “Goal Seek” to find x that satisfies eqn

For the Eqn Above the solutions, xs, are called the “zeros” or “roots” of the “zeroed” eqn

1lncos7 xx

01lncos7 xx



MSExcel vs Transcendental

Use The “Goal Seek” Command in MicroSoft Excel to Find xs with greater Accuracy



Goal Seek (on Data Tab)



Goal Seek Results (2 Roots)



Zeros Graphed by MATLAB

0 1 2 3 4 5 6 7 8-10

-8

-6

-4

-2

0

2

4

6

8

u

v

>> u = linspace(0, 2.5*pi, 300);>> v = cos_ln(u);>> xZ = [0,8]; yZ = [0, 0];>> plot(u,v, xZ,yZ, 'LineWidth',3), grid, xlabel('u'), ylabel('v');>> Z1 = fzero(cos_ln,2)Z1 = 1.4429>> Z2 = fzero(cos_ln,5)Z2 = 4.9705>> Z3 = fzero(cos_ln,8)Z3 = 7.5425



Power Function f(x) = Kxn

In the Power Function “n” can be ANY number, positive, negative, rational or Irrational. Some Examples

0 2 4 6 8 103

3.5

4

4.5

5

5.5

6

x

f(x)

= 3

x2/7

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x

f(x)

= 5

x-ln9

0 2 4 6 8 10-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

x

f(x)

= -

2x1.

7

M15PwrFcnGraphs_1306.m



PolyNomial Function

The General PolyNomial Function

Where• n ≡ a positive integer constant• ak ≡ any real number constant

n (the largest exponent) is called the DEGREE of the Polynomial

0122

1

0122

1

221

01221

or

axaxaxaxaxaxp

xaxaxaxaxaxaxp

nnn

nnn

nnn

nnn



PolyNomial Function

The plot of p(x) is continuous and crosses the X-axis no more than n-times

Some Examples

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-500

0

500

1000

1500

2000

2500

3000

3500

4000

x

p(x

) =

1.7

x4 - 8

x3 + 3

x2 - 8

x -

24

7

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-1200

-1000

-800

-600

-400

-200

0

200

400

600

800

x

p(x

) =

-5

x3 - 7

x2 + 4

x +

23

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-140

-120

-100

-80

-60

-40

-20

0

20

40

x

p(x

) =

-3

x2 - 7

x +

19

M15PloyNomialFcnGraphs_1306.m



Rational 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 xf x

q x



Rational Fcn Examples

Note the Asymptotic Behavior

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-25

-20

-15

-10

-5

0

5

10

15

20

25

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

-25

-20

-15

-10

-5

0

5

10

15

20

25

x-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-10

-8

-6

-4

-2

0

2

4

6

8

10

x

4 2

4

6 3 1( )

9 3 2

x xf x

x x

2

2 3( )

4

xf x

x

3

2

x

xxf



Graphing & Vertical-Line-Test

Test a Reln-Graph to see if the Relation represents a Fcn

If no VERTICAL line intersects the graph of a relation at more than one point, then the graph is the graph of a function.



Example Vertical-Line-Test

Use the Vertical Line Test to determine if the graph represents a function

SOLUTION• NOT a function as

the Graph Does not pass the vertical line test





SOLUTION• NOT a function as

the Graph Does not pass the vertical line test

TRIPLEValued





SOLUTION• IS a function as the

Graph Does pass the vertical line test

SINGLEValuedSINGLEValued





SOLUTION• IS a function as the

Graph Does pass the vertical line test

SINGLEValued



Quadratic Functions

All quadratic functions have graphs similar to y = x2. Such curves are called parabolas. They are U-shaped and symmetric with respect to a vertical line known as the parabola’s line of symmetry or axis of symmetry.

For the graph of f(x) = x2, the y-axis is the axis of symmetry. The point (0, 0) is known as the vertex of this parabola.



The Vertex of a Parabola

The FORMULA for the vertex of a parabola given by f(x) = ax2 + bx + c:

24, or , .

2 2 2 4

b b b ac bf

a a a a

• The x-coordinate of the vertex is −b/(2a). • The axis of symmetry is x = −b/(2a). • The second coordinate of the vertex is most

commonly found by computing f(−b/[2a])



Graphing f(x) = ax2 + bx + c

1. The graph is a parabola. Identify a, b, and c

2. Determine how the parabola opens• If a > 0, the parabola opens up. • If a < 0, the parabola opens down

3. Find the vertex (h, k). Use the formula

h, k b

2a, f

b

2a

.




4. Find the x-interceptsLet y = f(x) = 0. Find x by solving the equation ax2 + bx + c = 0.

• If the solutions are real numbers, they are the x-intercepts.

• If not, the parabola either lies – above the x–axis when a > 0 – below the x–axis when a < 0




5. Find the y-intercept. Let x = 0. The result f(0) = c is the y-intercept.

6. The parabola is symmetric with respect to its axis, x = −b/(2a)

• Use this symmetry to find additional points.

7. Draw a parabola through the points found in Steps 3-6.



Example Graph SOLUTION

f x 2x2 8x 5.

Step 1 a = –2, b = 8, and c = –5Step 2 a = –2, a < 0, the parabola opens down.Step 3 Find (h, k).

h b

2a

8

2 2 2

k f 2 2 2 2 8 2 5 3

h, k 2, 3 Maximum value of y = 3 at x = 2




f x 2x2 8x 5.

Step 4 Let f (x) = 0.

f 0 2 0 2 8 0 5

y-intercept is 5 .Step 5 Let x = 0.

2x2 8x 5 0

x 8 8 2 4 2 5

2 2 4 6

2

x-intercepts are 4 6

2 and

4 6

2.




f x 2x2 8x 5.

Step 6 Axis of symmetry is x = 2. Let x = 1, then the point (1, 1) is on the graph, the symmetric image of (1, 1) with respect to the axis x = 2 is (3, 1). The symmetric image of the y–intercept (0, –5) with respect to the axis x = 2 is (4, –5).

Step 7 The parabola passing through the points found in Steps 3–6 is sketched on the next slide.



Example Graph f x 2x2 8x 5.

SOLUTION cont.• Sketch Graph

Using the pointsJust Determined

f x 2x2 8x 5



WhiteBoard Work

Problems §1.2-44• Supply & Demand

http://www.google.com/url?sa=i&rct=j&q=&source=images&cd=&cad=rja&docid=I8vKurx1h27TaM&tbnid=Bnv4NmwePKSv9M:&ved=0CAUQjRw&url=http://resourcesforhistoryteachers.wikispaces.com/E.5.1+Define+aggregate+supply+and+demand,+Gross+Domestic+Product+(GDP),+economic+growth,+unemployment,+and+inflation.&ei=XUvHUZ-bOKr-iQKd1YGYCw&bvm=bv.48293060,d.cGE&psig=AFQjCNFuiXlVXrFBhDk0AIAdSa6XHZo5DQ&ust=1372101785981511



All Done for Today

AutoMobileStoppingDistance

http://www.google.com/url?sa=i&rct=j&q=&source=images&cd=&cad=rja&docid=FIqFRAhIWPbAcM&tbnid=6m6hfBAHIDh6LM:&ved=0CAUQjRw&url=http://www.cyberphysics.co.uk/topics/forces/stopping_distance.htm&ei=RKDIUYSzKcrHigKspYCQBQ&bvm=bv.48293060,d.cGE&psig=AFQjCNEOIpAqnuJhdNs62LhvbFJBOE7jzQ&ust=1372189038980398



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

Appendix

–

srsrsr 22



(120, 0)



Grap

h b

y MA

TL

AB

0 20 40 60 80 100 1200

100

200

300

400

500

600

700

800

p ($/Unit)

E (

$k/

Mo

nth

)MTH15 P1.2-44 • Bruce Mayer, PE

M15P12441306.m



MATLAB Code% Bruce Mayer, PE% MTH-15 • 23Jun13% M15P12441306.m%% The FUNCTIONp = linspace(0,120,500); E = -p.^2/5 + 24*p;% % the Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greenplot(p,E, 'LineWidth', 3),axis([0 120 0 800]),... grid, xlabel('\fontsize{14}p ($/Unit)'), ylabel('\fontsize{14}E ($k/Month)'),... title(['\fontsize{16}MTH15 P1.2-44 • Bruce Mayer, PE',]),... annotation('textbox',[.55 .055 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'M15P12441306.m','FontSize',9)

bruce mayer, pe licensed electrical & mechanical engineer bmayer@chabotcollege

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