Modeling, Functions, and Graphs Introduction Modeling, Functions, and Graphs

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

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

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)

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

Some examples of Linear Models ( Example 1 pg – 3) 20 15 10 5 Cost of Rentals 0 1 2 3 4 5 Length of Rentals Equation C = ($ 5 isurance fee) + $ 3 x (number of hours) C = 5 + 3. t Length of Cost of Rental rental (dollars) (hours) 0 5 8 11 3 14 (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)

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

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 = - 50 X max = 50 Xsc1 = 5 Y min = -500 Y max = 1000 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 + 1575 y = 45/2 x + 1575/2 y = 22.5 x + 787.5 Press Y and enter Y1 = 22.5 x + 787.5 Press Window X min = - 50 X max = 50 Xsc1 = 5 Y min = -500 Y max = 1000 Ysc1 = 100 Press Graph 1000 50 -50 -500

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

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

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. 2. Plot the intercepts. 3. Choose a value for x and find a third point on the line. 4. Draw a line through the points

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)

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

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 0 4 8 12 16 20 24 200 180 175 150 125 100 75 50 25 Table m The Equation g = 20 – 1 m 12 m 0 48 96 144 192 g 20 16 12 8 4 Let g = 5, 5 = 20 – 1 m 12 60 = 240 – m (Multiply by 12 both sides ) - - 180 = - m (multiply by – 1 both sides) m = 180 4 gallons

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

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 = - 1000 Ymin = - 1000 Xmax = 1000 Y max = 1000 XSc1 = 1 YSc 1 = 1 Solution – 3x – 4y = 1200 Find y in the given equation -4y = - 3x + 1200 ( Isolate y) y = -3/-4 x + 1200/-4 y = ¾ x – 300 Y1 = ¾ x - 300 Hit Y, Enter Y1 = ¾ x - 300 Hit Window , Enter the values Hit Graph

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

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 “

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 “

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

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 + 800 (t > 0) When t = 0, C = 30(0) + 800= 800 When t = 4, C = 30(4) + 800 = 920 When t = 10, C = 30(10) + 800 = 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)

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

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) 1972 5.6% 5.1% 1973 6.2% 4.5% 1974 10.1% 4.9% 1975 9.2% 7.4% 1976 5.8% 6.7% 1977 5.6% 6.8% 1978 6.7% 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

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 – 100.00 7.95 Over 100.00 8.95 It is a function: Each value of M has a unique value of C

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) -1 0 1 2 3 4 5 6 7 8 1990 t measured in years

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) = 5 – r d(4) = 5 -4 = 1 d( - 3) = 5 – (-3) = 8 2 2 = 2.828 d(-9)= 5 – (-9) = 14 = 3.742 d(4.6)= 5 – 4.6 = .4 = 0.632

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) 12 13 14 15 16 19 20 21 22 23 October 1987 Time Independent Variable

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

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

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)

Graphical Solution of Inequalities (Pg – 45) Consider the inequality 285 – 15x > 150 x 0 2 4 6 8 10 12 285 – 15x 285 255 225 195 165 135 105 300 200 150 100 - 100 y = 285 – 15x 5 9 10 25 The solution is x< 9

5 f(f) - 5 5 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)

13. Make a table of values and sketch a graph( Use calculator Enter Y1 Enter the values in window Hit 2nd and table Hit Graph

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, - 4.352) b) F ( - 1.6) = 4.352 F(1.6) = -4.352

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

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 0 2 3 4 Slope = Change in y-coordinate = 5 - 4 = 1 Change in x- coordinate 4 – 2 2

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

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 1 2 3 4 5 t No of hours D Change in distance = 100miles = 50 miles per hour T Change in time 2 hours Slope m = The slope represents the trucker’s average speed or velocity

A Formula for Slope ( Pg 64 )( two point slope form Y 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) -15 = - 3 x2 – x1 7 – 2 5 10 5 -5 P1 (2, 9) 10 P2 (7, -6)

Slope formula in Function Notation m = y2 – y1 f(x2) – f(x1) , x2 = x1 x2 – x1 = x2 - x1

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 – 0 -3 -4 -2 2 -2 -4 -6 -8

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

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

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

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 = 6 Subtract 3x from both sides 4y = - 3x + 6 Divide both sides by 4 y = - 3x + 3 4 2 m= -3 and b = 3 4 2 General Slope- Intercept Method of Graphing x New point y b Start here y = mx + b

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

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

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) 1000 2000 Boiling Point 0 C a) Slope = m= 170 – 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

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) 20 40 60 80 100 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.

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)