1. Modeling, Functions, and Graphs
Introduction
Modeling, Functions, and Graphs

2. Modeling
The ability to model problems or phenomena
by algebraic expressions and equations is the
ultimate goal of any algebra course
The examples are designed with interactive
investigation and to show how mathematical
techniques are used for learning
get opportunity to explore open-minded
modeling problems.
understand new situation

3. Functions Functions are useful not only in Calculus but in
nearby every field students may pursue.
We employ celebrated “ Rule of Four” all problems should be considered using Algebraic Method
Verbal
Algebraic Expression
Numerical
Graphical
Students learn to write algebraic expression from verbal description, to recognize trends in a table of data, extract & interpret information from the graph of a function

4. Graphs
No tool for conveying information about a
system is more powerful than a graph
Large number of examples are explained
plotting by hand and using graphing
calculator(TI 83/84)

5. Ch 1 - Linear Model Mathematical techniques are used to Analyze data
Identify trends
Predict the effects of change
This quantitative methods are the concepts of skills of algebra
You will use skills you learned in elementary algebra to solve problems and to study a variety of phenomena
the description of relationships between variables by using equations, graphs, and table of values. This process is called Mathematical Modeling

6. Some examples of Linear Models ( Example 1 pg – 3)20 15 10 5
Cost of
Rentals
Length of Rentals
Equation
C = ($ 5 isurance fee) + $ 3 x (number of hours)
C = t
Length of Cost of
Rental rental (dollars)
(hours) 8 11
(t, C)
(0, 5)
(1, 8)
(2, 11)
(3, 14)
C = 5 + 3(0)
C = 5 + 3(1)
C = 5 + 3(2)
C = 5 + 3(3)

7. Graphing an Equation
To graph an equation:
Press Y = and enter the equation you wish to graph
Press WINDOW and select a suitable graphing window
Press GRAPH

8. Example 4 ( pg 7) a) Solve the equation 2y – 1575 = 45x for y in terms of x b) Graph the equation on a graphing Calculator. Use the window X min = X max = Xsc1 = 5 Y min = -500 Y max = Ysc1 = 100 c) Sketch the graph on paper. Use the window settings to choose appropriate scales for the axes Solution 2y – 1575 = 45x 2y = 45x y = 45/2 x /2 y = 22.5 x Press Y and enter Y1 = 22.5 x Press Window X min = X max = Xsc1 = 5 Y min = -500 Y max = Ysc1 = 100 Press Graph
1000 50
-50
-500

9. General Form For a Linear Equation
The graph of any equation
Ax + By = C
Where A and B are not both equal to zero, is a
straight lines

10. Linear Equations
All the models for examples have equations with a similar form
Y = ( starting value ) + ( rate of change ) . X
Graph will be straight lines
So we called these are Linear Equations
For Example
C = 6 + 5t
This can be written equivalently as
5t + C = 6
So It is a linear equation
Note : General Form for a Linear Equation
The graph of any equation
Ax + By = C
Where A and B are not both equal to zero, is a straight line

11. Intercepts of a Graph Intercepts of a graph
The points where a graph crosses the axes are called the intercepts of
the graph
To find the x-intercepts, set y = 0 and solve for x
To find the y-intercepts, set x = 0 and solve for y
To Graph a Line Using the Intercept Method:
Find the intercepts of the line
To find the x-intercept, set y = 0 and solve for x
To find the y-intercept, set x = 0 and solve for y.
Plot the intercepts.
Choose a value for x and find a third point on the line.
Draw a line through the points

12. Example of Intercepts Consider the graph of the equation
3y – 4x = 12
The y-coordinate of the x-intercept is zero, so we set y = 0
in the equation to get
3(0) – 4x = 12
x = - 3
The x-intercept is the point (-3, 0).
Also, the x-coordinate of the y-intercept is zero, so we set
x = 0 in the equation to get
3y – 4(0) = 12
y = 4
The y-intercept is (0, 4)
(0, 4)
( -3, 0)

13. Ex4( pg 15)
Leon’s computer has a 20-gallon gas tank, and he gets 12 miles to the gallon.
(that is , he uses 1/12 gallon per mile. Complete the table of values for the
amount of gas, g, left in Leon’s tank after driving m miles
Write an equation that expresses the amount of gas, g, in Leon’s fuel tank in terms of the number of miles, m, he has driven.
Graph the equation
How much gas will Leon use between 8am, when his odometer reads 96 miles, and 9 a.m, when the odometer reads 144 miles ? Illustrate the graph
If Leon has less than 5 gallons of gas left, how many miles has he driven ? Illustrate on the graph. m 48 96
144
192 g

14. Exercise 1.1 ( Example 4, pg – 15)
Leon has traveled more than
180 miles if he has
less than 5 gallons of gas left g
200
180
175
150
125
100 75 50 25
Table m
The Equation
g = 20 – m 12 m g
Let g = 5,
5 = 20 – m 12
60 = 240 – m (Multiply by 12 both sides ) = - m (multiply by – 1 both sides)
m = 180
4 gallons

15. Example 17 , Pg = 16
Find the intercepts of the graph and graph the equation by the intercept method
x - y = 1
Solution, Set x = 0, y
9 – = 1
- y
4 = 1, y = - 4
The y-intercept is the point (0, - 4)
Set y = 0, x = 1
x = 1, x = 9 9
The x-intercept is the point (9, 0)
x-intercept
(9, 0)
(0, -4)
y-intercept

16. Find y in the given equation -4y = - 3x + 1200 ( Isolate y)
39 ( Pg 18) a) Solve the equation for y in terms of x b) Graph the equation on your calculator in the specified window c) Make a pencil and paper sketch of the graph Label the scales on your axes, and the coordinates of the intercepts
3x - 4y = 1200
Xmin = Ymin =
Xmax = Y max = 1000
XSc1 = YSc 1 = 1
Solution –
3x – 4y = 1200
Find y in the given equation
-4y = - 3x ( Isolate y)
y = -3/-4 x /-4
y = ¾ x – 300
Y1 = ¾ x - 300
Hit Y, Enter
Y1 = ¾ x - 300
Hit Window , Enter the values
Hit Graph

17. Using Graphing Calculator to solve the equation, Equation 572 – 23x = 181
Y Y2
Press Window and enter Press Y1 and Y2 and enter Press 2nd and Table
Press Graph and Trace

18. Ch 1.2 Function Notation As of 2006, the Sears Tower in Chicago is the nation’s tallest building, at 1454 feet. If an algebra book is dropped from the top of the Sears Tower, its height above the ground after t seconds is given by the equation h = 1454 –16t2
f(t) = h
Independent Variable Dependent Variable
This function t = Input variable and h is the output variable
Example h = f(t) = 1454 –16t2
When t= 1, i.e after 1 second the book’s height h= f(1)= 1454 – 16 (1)2
=1438 feet, We read as “f of 1 equals 1438”
When t = 2, h = f(2) = 1454 – 16(2)2 = 1390 feet,
We read as” f of 2 equals 1390 “

19. Function Notation Input Variable Output Variable
f(x) = y
Input Variable Output Variable
Example y = f(x) = 1454 –16x2
When x= 1, y= f(1)= 1438, We read as “f of 1 equals 1438”
When x = 2, h = f(2) =1390, We read as ” f of 2 equals 1390 “

20. Graphing Calculator
Press Y = key, Enter
Y1 = 1454 –16X2
Press 2nd WINDOW to access the Tbl Set start from 0
And the increment of one unit in the x values ,
Press 2nd and graph for table Press graph
Press WINDOW

21. Ch 1.2 (pg 19) Definition and Function
Example –To rent a plane flying lessons cost $ 800 plus $30 per hour
Suppose C = 30 t (t > 0)
When t = 0, C = 30(0) + 800= 800
When t = 4, C = 30(4) = 920
When t = 10, C = 30(10) = 1100
The variable t in Equation is called the input or independent variable, and C is the output or dependent variable, because its values are determined by the value of t
This type of relationship is called a function
Table Ordered Pair t c
800 4
920 10
1100
(t, c)
(0, 800)
(4, 920)
(10, 1100)

22. Definition of function ( Pg 19)
A function is a relationship between two variables for which a unique value of the output variable can be determined from a value of the input variable.
Function Notation f(x) = y
Input variable Output Variable

23. 1.2 Functions defined by Tables
Which of the following tables define the second variable as a function of the first variable? Explain why or why not ?
24 Pg 31
Inflation rate ( I)
Unemployment rate ( U) %
5.1% %
4.5% %
4.9% %
7.4% %
6.7% %
6.8% %
1972 and 1977
same inflation
Rate 5.6 %
But two different
Unemployment
rates 5.1% and 6.85%
Not a function: Some values of I have more than one value of U

24. No26 It is a function: Each value of M has a unique value of C
Cost of merchandise ( M)
Shipping charge (C )
$0.01 – 10.00
$2.50
10.01 – 20.00
3.75
20.01 – 35.00
4.85
30.01 – 50.00
5.95
50.01 – 75.00
6.95
75.01 –
7.95
Over
8.95
It is a function: Each value of M has a unique value of C

25. Graph of the function No35 C 15 10 5 -1 0 1 2 3 4 5 6 7 8 1990
a) When did 2000 students
consider themselves
computer literate ?
Ans- In 1991
b) How long did it take that
number to double?
Ans Value of C doubled from 2 to 4
in one year
c) How long did it take for the
number to double again?
Ans- From 4 to 8 in one year
d) How many studnets became
computer literarate
Ans- t starts from January 1990
In the beginning January 1992,
t = 2 and C = 4, so 4000 students
were computer literate. In the
beginning of June 1993, t = 3 5/12 = 3.4
and C = 11. so 11,000 students were
computer literate.
Thus 11,000 – 4000
= 7000
students became computer literate
between January 1992 and June 1993 C
No35 15 10 5
No of students
In thousands
(1993, 4000)
(1991, 2000)
1990
t measured in years

26. Evaluate each function for the given values ( pg 34 )
f(x) = 6 -2x
f(3) = 6 – 2(3)=0
f(-2)= 6 – 2(-2)= 6 + 4= 10
f(12.7)= 6 -2(12) = 6 – 25.4 = -19.4
f(2/3) = 6 – 2(2/3) = 6 -4/3 = 4 2/3 48
D( r) = – r
d(4) = = 1
d( - 3) = 5 – (-3) = =
d(-9)= 5 – (-9) = = 3.742
d(4.6)= 5 – 4.6 = = 0.632

27. Ch 1.3 Graphs of Functions (Pg 39) Reading Function Values from a Graph
2500
2400
2300
2200
2100
2000
1900
1800
f(15) = 2412
f(20) = 1726
Dow Jones Industrial Average Dependent Variable
Q (20, 1726)
October 1987
Time Independent Variable

28. Graph of a function
The point ( a, b) lies on the graph of the function f if and only f (a) = b
Functions and coordinates
Each point on the graph of the function f has coordinates ( x, f(x)) for some value of x

29. Finding Coordinates with a Graphing Calculator
Graph the equation Y = -2.6x – 5.4
X min = -5, X max = 4.4,
Y min = - 20 , Y max = 15
Press Y1 enter Press 2nd and Table Press Graph and then press, Trace and enter
“Bug” begins flashing on the display. The coordinates of the
bug appear at the bottom of the display.Use the left and right
arrows to move the bug along the graph

30. Vertical Line Test ( pg 269)
A graph represents a function if and only if every vertical line intersects the graph in at most one point
Function Not a function
Go through all example 4 ( pg 270)

31. Graphical Solution of Inequalities (Pg – 45)
Consider the inequality 285 – 15x > 150 x
285 – 15x
300
200
150
100
- 100
y = 285 – 15x
The solution is x< 9

32. 5
f(f)
Find f(-1) and f(3)
The points (-1,3) and (3,6) lie on the graph so f(-1) = 3 and f(3) = 6
b) For what value(s0 of t is f(t) = 5?
The points (0,5) and (4,5) lie on the graph so f(t) = 5 when t = 0 and t = 4
c) Find the intercepts of the graph. List the function values given by the intercepts
The t-intercept is ( -2, 0) and the f- intercept is ( 0, 5) ; f(-2) = 0 , f(0) = 5
d) Find the maximum and minimum values of f(t)
The highest point is (3, 6) and the lowest is ( -4, -1) , so f(t) has a maximum value of 6 and a
Minimum value of - 1
e)For what value(s0 of t does f take on its maximum and minimum values?
The maximum occurs for t = 3
The minimum occurs for t = -4
f) On what intervals is the function increasing ? Decreasiing ?
The function increasing on the interval ( -4, 3) and decreasing on the interval ( 3, 5)

33. 13. Make a table of values and sketch a graph( Use calculator
Enter Y Enter the values in window Hit 2nd and table
Hit Graph

34. Graph y1 = 0.5x3 – 4x Enter Y1 Enter Window Hit Graph
( -1.6, 4.352)
( 1.6, 4.352)
Turning points are approximately ( -1.6, 4.352) and ( 1.6, )
b) F ( - 1.6) = 4.352
F(1.6) =

35. 1.4 Measuring Steepness ( pg 57)
Which path is more strenuous ?
5 ft
2 ft
Steepness measures how sharply the altitude increases..
To compare the steepness of two inclined paths, we compute the ratio of change
in horizontal distance for each path

36. Change in y- Coordinate Change in x- coordinate
1.4 Slope (Pg 59)
Definition of Slope: The slope of a line is the ratio
Change in y- Coordinate
Change in x- coordinate 5 4 3 2 1
A B
Slope = Change in y-coordinate = = 1
Change in x- coordinate –

37. Notation for Slope (Pg 60)
Change in y coordinate y x
Change in x coordinate y
Slope of a line is given by
m = x
, where x is not equal to zero
The slope of line measures the rate of change of the output variable with respect to the input variable

38. Significance of the slope (Ex 6, Pg 63)
250
200
150
100 50
H (4, 200)
D = 100
Distance in miles traveled
G (2, 100)
t = 2 t
No of hours
D Change in distance = 100miles = 50 miles per hour
T Change in time hours
Slope m =
The slope represents the trucker’s average speed or velocity

39. A Formula for Slope ( Pg 64 )( two point slope formY x
y2 – y1
m = = x2
= x1
x2 – x1
The slope of the line passing through the points P1(x1, y1) and P2 ( x2, y2) is given by
Slope Formula m = y2 – y1 = 9-(-6) = - 3
x2 – x – 10 5 -5
P1 (2, 9) 10
P2 (7, -6)

40. Slope formula in Function Notation
m = y2 – y f(x2) – f(x1) , x2 = x1
x2 – x = x2 - x1

41. The y-intercept is ( 0,-9)Set y = 0 2( 0) + 6(x) = -18 6x = -18 X = -3
11 . a) Graph each line by the intercept method b) Use the intercepts to compute the slope 2y + 6x = -18
Set x = 0
2y + 6(00 = -18
2y = -18
Y = -9
The y-intercept is ( 0,-9)Set y = 0
2( 0) + 6(x) = -18
6x = -18
X = -3
The x- intercept is ( -3, 0)
B) Slope m = 0 –(-9) = 9 = - 3
-3 – -2 -4 -6 -8

42. Slope Intercept Form
If we write the equation of a linear function in the form.
F(x) = b + mx
Then m is the slope of the line, and b is the y-intercept

43. 1.5 Equations of Lines ( Pg 79)
y = 2x + 1
y = 2x
y = ½ x
y = 2x + 3
y = 2x - 2
y = -2x
These lines have the
same y- intercept but different slopes
These lines have the same slope
but different y - intercepts

44. Point- Slope Form ( pg 82 ) Point- Slope Form y = y1 + m (x – x1) 5
(1, - 4)
- 4
y = -3
(5, - 7)
- 7
x = 4

45. 3x + 4y = 6 Subtract 3x from both sides
1.5 Slope – Intercept Form y = mx + b ( m is slope of a line, and b is the y- intercept)
3x + 4y = Subtract 3x from both sides
4y = - 3x Divide both sides by 4
y = - 3x + 3
m= and b = 3
General
Slope- Intercept Method of Graphing x
New point y
b Start here
y = mx + b

46. Ch 1.6 Scatter Plots ( pg 94 ) scattered not organized
decreasing trend
increasing trend

47. 1.6 Linear Regression (Lines of Best Fit)
The datas in the scatterplot are roughly linear, we can estimate the location of imaginary”lines best fit” that passes as close as possible to the data points
We can make the predictions about the data. The process of predicting a value
of y based on a straightline that fits the data is called a linear regression, and the line itself called the regression line.
The equation of the regression line is usually used(instead of graph) to predict values

48. Example of Linear Regression
Estimate a line of best fit and find the equation of the regression line
Use the regression line to predict the heat of vaporization of potassium bromide,
whose boiling temperature is 14350C
200
100
Heat of Vaporization (kJ)
(1560, 170)
Choose two points in the regression line
(900, 100)
Boiling Point 0 C
a) Slope = m= – 100 = 0.106
1560 – 900
The equation of regression line is
y- y1 = m(x – x1)
y – 100 = 0.106(x – 900) , y = 0.106x + 4.6
b) Regression equation for
potassium bromide , x = 1435
y = 0.106(1435) + 4.6

49. Interpolation- The process of estimating between known data points Extrapolation- Making predictions beyond the range of known data
112 96 80 64 48
Height (cm)
Age (months)
The graph is not linear because her rate of growth is not constant; her growth slows down
as she approaches her adult height. The short time of interval the graph is close to a line,
and that line can be used to approximate the coordinates of points on the curve.

50. Using Graphing Calculator for Linear Regression Pg99
Step 1 Press Stat , choose 1 press Enter
Step 2 Enter Y = 1.95x – 7.86
Step 3 Stat, right arrow go to 4
And enter
Step 4Press Vars 5 ,Right,Right , Enter
Step 5 Press 2nd, Stat Plot and enter
Step 6 the graph ( Pg 60)