FOM 11 T15 INTERSECTIONS & OPTIMIZATION PROBLEMS - 1 1 MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) INTERSECTION = a set of coordinates of the point on the grid where two or more graphed lines touch or cross each other. DETERMINING THE INTERSECTIONS USING THE GRAPHING CALCULATOR I) An INTERSECTION is a set of coordinates of the point on the grid where two or more graphed lines touch or cross each other. It is important to be able to determine the coordinates of an intersection of two or more graphed lines, especially when the intersection is not easily read from the grid coordinates. The intersection of two or more boundaries is calculated by treating the inequalities as equations then following the procedures outlined below. You will be required to draw the graph of a system of equations then use the graphing calculator to determine the coordinates of the intersection. BE SURE YOU DRAW THE GRAPH OF THE SYSTEM BEFORE YOU CALCULATE THE INTERCEPTS USING THE GRAPHING CALCULATOR THIS WILL HELP YOU CHOOSE THE APPROPRIATE WINDOW SETTINGS TO FIT THE GRAPHS IN THE CALCULATORS SCREEN/WINDOW. A) In order to graph equations using the graphing calculator you must know the location of these buttons on your calculator. See the CALCULATOR REFERENCE SHEET, page 13 of these notes, to see the location of each button. 1) The y = button is the top most left-hand button. 2) The WINDOW button is the top button, second from the left. 3) The ZOOM button is the top button, third from the left. 4) The TRACE button is the top button, fourth from the left. 5) The GRAPH button is the top most right-hand button. 6) The x variable button, X,T,Θ, n, is located to the immediate the right of the green ALPHA button. ) The negative sign button, ( ), is located immediate below the number 3 button. 8) The open and close bracket buttons, ( and ), are located immediate above the 8 and 9 buttons. B) USE THESE STEPS TO DETERMINE THE INTERSECTION USING THE GRAPHING CALCULATOR one variable write it with the variable isolated. Enter the first equation/inequality as Y 1, the second as Y 2, the third as Y 3, etc. then press GRAPH Xmin = smallest value of your x-axis Ymin = smallest value of your y-axis Xmax = largest value of your x-axis Ymax = largest value of your y-axis one decimal place.
FOM 11 T15 INTERSECTIONS & OPTIMIZATION PROBLEMS - 1 2 B) SAMPLE PROBLEMS 1: Study these examples carefully. Be sure you understand and memorize the process used to complete them. INSTRUCTIONS: Use your graphing calculator to determine the coordinates of the intersections for the system of linear inequalities found in SAMPLE PROBLEMS 2 above. {( x, y) y 2x 1, x!, y!} 1) {( x, y) y 3, x!, y!} y 2x 1 = 2x 1 y 3 = 3 = 2x 1 Press 2, X,T,Θ, n, Blue, 1, ENTER = 3 Press ( ), 3, ENTER Xmin = 10 Xmax = 10 Ymin = 10 Ymax = 10 Intersection: ( ) 1, 3 2) {( x, y) y > x, x I, y I} {( x, y) y <5, x I, y I} y > x = x y <5 = 5 = x Press X,T,Θ, n, ENTER = 5 Press 3, ENTER Xmin = 10 Xmax = 10 Ymin = 10 Ymax = 10 Intersection: ( 5, 5)
FOM 11 T15 INTERSECTIONS & OPTIMIZATION PROBLEMS - 1 3 3) {( P, M) M < 3, P W, M W} {( P, M) M 3P 4, P W, M W} M < 3 = 3 M 3P 4 = 3x 4 = 3 Press 3, ENTER = 3x 4 Press 3, X,T,Θ, n, Blue, 4, ENTER Xmin = 10 Xmax = 10 Ymin = 10 Ymax = 10 Intersection: ( 2.3, 3) 4) {( x, y) 3x y 45, x I, y I} {( x, y) 9y +5x 32, x!, y W} y 3 x + 45 = 3 x + 45 y 5 9 x + 32 9 = 5 9 x + 32 9 NOTE: The steps to convert the inequalities to slope y-intercept form are found in question 4, step 1 on page above. = 3 x + 45 Press (, 3,,, ), X,T,Θ, n, +, (, 4, 5,,, ), ENTER = 5 9 x + 32 9 Press (,, 5,, 9, ), X,T,Θ, n, +, (, 3, 2,, 9, ), ENTER Xmin = 10 Xmax = 10 Ymin = 10 Ymax = 10 Intersection: ( ) 2.9, 5.2
FOM 11 T15 INTERSECTIONS & OPTIMIZATION PROBLEMS - 1 4 5) {( x, y) y 6 x +800, x!, y!} {( x, y) y 5 } x + 600, x!, y! y 6 x +800 = 6 x +800 y 5 x + 600 = 5 x + 600 NOTE: The steps to convert the inequalities to slope y-intercept form are found in question 5, step 1 on page 9 above. = 6 x +800 Press (,,,, 6, ), X,T,Θ, n, +, 8, 0, 0, ENTER = 5 x + 600 Press (,, 5,,, ), X,T,Θ, n, +, 6, 0, 0, ENTER Xmin = 100 Xmax = 1000 Ymin = 100 Ymax = 1000 Intersection: ( 442.1, 284.2) NOTE: SEE YOUR TEACHER TO LEARN HOW TO CALCULATE AN INTERSECTION INVOLVING A VERTICAL LINE. C) REQUIRED PRACTICE 1: Instructions: Use you graphing calculator to determine the intersections for questions 3, 4 questions. {Answers are on page 10 of these notes.} 1) Page 318: Questions 3 & 4. 2) Page 323: Question 5a. MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) OPTIMIZATION PROBLEM = a real life situation analyzed using the concepts of linear equations and systems of inequalities in order to determine the maximum or minimum quantity necessary quantity such as revenue, costs or the number of wing-nuts produced. 2) CONSTRAINT = a linear inequality that represents a limitation to the variables in the real life situation. 3) OBJECTIVE FUNCTION = a linear equation used to answer the question found in the optimization problem. OPTIMIZATION PROBLEMS I) An OPTIMIZATION PROBLEM is a real life situation analyzed using the concepts of linear equations and systems of inequalities in order to determine the maximum or minimum quantity necessary quantity such as revenue, costs or the number of wing-nuts produced. A) To solve an optimization problem you are required to create a system of linear inequalities and list their restrictions. 1) The inequalities are called constraints because they describe the limits, constraints, to the real life situation.
FOM 11 T15 INTERSECTIONS & OPTIMIZATION PROBLEMS - 1 5 2) The restrictions are the limits to the numbers (!, W, I, " ) that can be considered as possible solutions to the problem. These DOMAIN and RANGE restrictions are determined by considering the kinds of things produced or consumed in the problem. i.e. restrictions x! REMEMBER: Constraints and restrictions are different. Constraints are linear inequalities describing the real life situation while DOMAIN and RANGE restrictions are mathematical limits to the numbers that can be considered as possible solutions to the problem. i.e. constraint: y 5x + 3 ; restrictions x!, y! II) SOLVING OPTIMIZATION A) An optimization problem is solved by creating a mathematical model, which includes variables, a linear equation, restrictions, a system of constraints (linear inequalities), and a graph. The model is then used to answer the question given in the problem. 1) The graph will have an overlapping shaded area that is confined by the boundaries of the constraints and or the horizontal and vertical axes. The coordinates of the corner points of the overlapping shaded area are the possible solutions to the optimization problem. B) USE THESE STEPS TO SOLVE OPTIMIZATION PROBLEMS 1: Define the VARIABLES for the objects described in the problem. 2: Write the OBJECTIVE FUNCTION (linear equation) needed to answer the question. 3: Write the DOMAIN and RANGE restrictions. 4: Write the CONSTRAINTS (system of linear inequalities). 5: Graph the CONSTRAINTS (system of linear inequalities). 6: List the coordinates of each INTERSECTION (corner points). You may have to calculate them using the graphing calculator. : Substitute the coordinates of each INTERSECTION (corner points) into the OBJECTIVE FUNCTION (linear equation) to answer the question. C) SAMPLE PROBLEMS 2: Study these examples carefully. Be sure you understand and memorize the process used to complete them. 1) Read the problem described in INVESTIGATE the Math on page 324 of your text. What is the maximum daily cost and how many of each vehicle is made? What is the minimum daily cost and how many of each vehicle is made? 1: Define the VARIABLES for the objects described in the problem. Let C = the daily cost Let x = the number of cars Let y = the number of SUVs 2: Write the OBJECTIVE FUNCTION (linear equation) needed to answer the question. C = 8x +12y 3: Write the DOMAIN and RANGE restrictions. Because the factory makes racing cars and SUVs, the only numbers that make sense to use are nonnegative entire numbers. Thus the restrictions to the domain and range are: x W, y W 4: Write an inequality that describes the relationship between the variables. Since no more than 40 racing cars can be made each day, the first constraint is: x 40 Since no more than 60 SUVs can be made each day, the second constraint is: y 60 Since a total of 0 or more vehicles can be made each day, the third constraint is: x + y 0 5: Graph the CONSTRAINTS (system of linear inequalities). 5a: Graph the first constraint. 5b: Graph the second constraint. This constraint has a vertical boundary This constraint has a horizontal boundary passing through the x-axis at x = 40 passing through the y-axis at y = 60 x 40 y 60 Continued on the next page.
FOM 11 T15 INTERSECTIONS & OPTIMIZATION PROBLEMS - 1 6 5c: Graph the third constraint. x + y 0 x x + y x + 0 y x + 0 b = 0 m = 1 1 Because the DOMAIN and RANGE restrictions are Integers x W, y W the overlapping shaded area is stippled. 6: List the coordinates of each INTERSECTION (corner points). You may have to calculate them using the graphing calculator. The coordinates of each intersection (corner points) can be read directly from the graph. P 1 ( 10,60) P 2 ( 40,60) P 3 ( 40,30) : Substitute the coordinates of each INTERSECTION (corner points) into the OBJECTIVE FUNCTION (linear equation) to answer the question. P 1 ( 10,60) P 2 ( 40,60) P 3 ( 40,30) x = 10 y = 60 C = 8x +12y C = 8( 10) +12 ( 60 ) C = 80 + 20 C = 800 x = 40 y = 60 C = 8x +12y C = 8( 40) +12 ( 60 ) C = 320 + 20 C = 1040 x = 40 y = 30 C = 8x +12y C = 8( 40) +12 ( 30 ) C = 320 + 360 C = 680 Ans: The maximum daily cost is $1040 producing 40 racing cars and 60 SUVs. The minimum daily cost is $680 producing 40 racing cars and 30 SUVs.
FOM 11 T15 INTERSECTIONS & OPTIMIZATION PROBLEMS - 1 ANSWERS TO THE REQUIRED PRACTICE Required Practice 1 from page 4 3a) ( 1, 2) 3b) ( 1.3, 0.6) 3c) ( 3, 1) 3d) ( 1, 2) 3e) ( 1, 2) 4a) ( 1, 2) 4b) ( 0, 0) 4c) 4d) no solution, graphs do not intersect 5a) ( 45, 15) ( 1.2, 1.2)
FOM 11 T15 INTERSECTIONS & OPTIMIZATION PROBLEMS - 1 8 ASSIGNMENT: PRINT THIS INFORMATION ON YOUR OWN GRID PAPER LAST then FIRST Name T15 INTERSECTIONS & OPTIMIZATION PROBLEMS - 1 Block: Show the process required to complete each problem to avoid receiving a zero grade. Neatness Counts!!! (Marks indicated in italicized brackets.) REMEMBER TO USE GRID PAPER FOR ALL ASSIGNMENTS!!! Copy the sentences numbered 1-8 then match it with the correct term listed below. (4) Domain Range Intersection Constraint Natural numbers Whole numbers Integers Real numbers Objective Function Linear inequality Coefficient Constant Term 1) A linear equation used to answer the question in an optimization problem. 2) Represents a limitation to the variables in the real life situation. 3) The set of all possible x values. 6) The set of positive entire numbers. 4) The number in front of a variable. ) The set of all possible y values. 5) The point where two or more graphs cross. 8) The set of all entire numbers. Use your graphing calculator to determine the intersection(s) for these systems of inequalities. {( x, y) x +14y < 28, x!, y!} {( x, y) 3x + y 2, x!, y!} 9) (6) 10) {( x, y) 3x 6y 18, x!, y!} {( x, y) 2y 3x 1, x!, y!} (4.5) 11) {( x, y) 2y 4 3x, x!, y!} {( x, y) y 6, x!, y!} (3) Following the instructions. (1.5) /19
FOM 11 T15 INTERSECTIONS & OPTIMIZATION PROBLEMS - 1 Be sure the plots are not highlighted. They must look like this. Press this button the change the maximum and minimum values seen on the graphs axes. enter an equation in this window. access the yellow coloured options above the buttons. 9 zoom in or out of the graph. Press the button then this button to calculate the intersection, vertex, zeros. graph the equation. The up, down, left and right buttons enter the x variable. The open and close brackets buttons. enter a negative sign.