Pre-Calculus Calendar Chapter 1: Linear Relations and Functions Chapter 2: Systems of Linear Equations and Inequalities

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1 Pre-Calculus Calendar Chapter : Linear Relations and Functions Chapter : Systems of Linear Equations and Inequalities /9 Monday Tuesday Wednesday Thursday Friday /0 / / /. p. 0: 9-5,, 5, 7-,, 5, p. 7: 0-0 even, -,.-. Review.-. Quiz *All.-. homework must be handed in BEFORE the quiz to receive full credit. p. 7: 9-5 odd, 8, /5 / (Sandtveit out) /7. p. 7: 5-9 odd, 8,, 5, 8 /8 /9.-. Review.-. Quiz *All.-. homework must be handed in BEFORE the quiz to receive full credit. p. 8: 5-9 odd, 5-5 odd, 50, 5. p. 9: - odd,, 0,.-. Review.-. Quiz *All.-. homework must be handed in BEFORE the quiz to receive full credit.5 p. 0: 5-9 odd, 5, 8, 5 / / / /5 /. p. 0: 5-9 odd,.5-. Review.5-. Quiz *All.5-. homework must be handed in BEFORE the quiz to receive full credit.7 Worksheet Start Chapter Test Review ½ Day.7 p. : -5 /9 /0 / / / Mid-Winter Break Mid-Winter Break Chapter Test Review *All Chapter Homework must be handed in BEFORE the end of class to qualify for a test retake Chapter Test

2 Success Criteria I can determine whether or not a relation is a function. Page(s) I can use interval notation to write the domain of a function Page(s) I can add, subtract, multiply, divide, and compose (composition) functions. Page(s) I can solve systems of equations by graphing. Page(s) I can solve systems of equations by substitution. Page(s) I can solve systems of equations by elimination. Page(s) I can solve a system of variables. Page(s) I can add and subtract matrices. Page(s) I can multiply matrices. Page(s)

3 I can use matrices to perform reflections over the x and y axes Page(s) I can use matrices to rotations of 90, 80, and 70 degrees Page(s) I can use the Matrix Transformation Theorem to generate transformation matrices. Page(s) I find the determinant of x and x matrices Page(s) I can find the inverse of x matrices by hand; x and x with a calculator Page(s) I can use a matrix equation to solve a system of equation with any number of variables Page(s) I can solve a system of linear inequalities Page(s) I can use linear programming to maximize and minimize variables Page(s)

4 Name: Pre-Calculus Notes: Chapter Relations, Functions, and Graphs Section Relations and Functions Relation: Domain: Range: Function: a set of ordered pairs x-value, input, independent variable y-value, output, dependent variable a relation in which each x-value is paired with exactly one y-value Example State the relation of the wind chill data as a set of ordered pairs. Also state the domain and range of the relation. Windchill Factors at 0 o F Wind Speed (mph) Windchill Temperature ( o F) Example The domain of a relation is all consecutive integers between - and. The range y of the relation is less than twice x, where x is a member of the domain. Write the relation as a table of values and as an equation. Then graph the relation. x y

5 Example State the domain and range of each relation. Then state whether the relation is a function. a. b. c. {(-, 0), (, ), (, 5)} d. {(-, -), (-, -), (0, -), (0, )} Vertical Line Test If every vertical line drawn on the graph of a relation passes through no more than one point of the graph, then the relation is a function. Example 5 Determine whether the graph of each relation represents a function. Explain. a. b. Function Notation Example Evaluate each function for the given value. a. f(-) if f(x) = -x b. g() if g(x) = x 5

6 c. g(m) if g(x) = x 0x x + 5 d. h(a ) if h(x) = x x + Interval Notation Intervals whose graphs are segments: Closed interval from a to b Interval notation Set notation Open interval from a to b Half-open interval from a to b, including a Intervals whose graphs are rays: Closed infinite interval Open infinite interval

7 Example 7 State the domain of each function. a. f(x) = x 9 b. g x x c. h x d. k x x x x x Section Composition of Functions Operations with Functions: Sum: (f + g)(x) = Difference: (f g)(x) = f g Product: f gx = Quotient: x = Composition of Functions: Denoted The domain of the composition includes all of the elements in x in the domain of g for which f(x) is in the domain of f. Example Given f(x) = x and g(x) = x, find each function. a. (f + g)(x) b. (f g)(x) f g c. f gx d. x 7

8 Example For the Lotsa Coffee Shop, the revenue r(x) in dollars from selling x cups of coffee is r(x) =.5x. The cost c(x) for making and selling the coffee is c(x) = 0.x + 0. a. Write the profit function. b. Find the profit on 00, 00, and 500 cups of coffee sold. Example Find f g x and g f x for f(x) = x and g(x) = x. Example g. x a. State the domain of f gx for f x x and x b. State the domain of f gx for f x x and x g. x 8

9 Name: Pre-Calculus Notes: Chapter Systems of Linear Equations and Inequalities Section Solving Systems of Equations in Two Variables System of equations Solution to the system Consistent system Independent system Dependent system Inconsistent system Systems of equations can be solved using one of three different methods: Graphing Substitution Elimination Example Solve the system of equations by graphing. y = x 8 y = x 5 9

10 Example Use the substitution method to solve the system of equations. y = x 8 x + y = Example Use the elimination method to solve the system of equations. 5x + y = 0 x y = 0 Example Madison is thinking about leasing a car for two years. The dealership says that they will lease her the car she has chosen for $ per month with only $00 down. However, if she pays $00 down, the lease payment drops to $ per month. What is the break-even point when comparing these lease options? Which -year lease should she choose if the down payment is not a problem? 0

11 Section Solving Systems of Equations in Three Variables Example Solve the system of equations. y = -9z x + y z = 0 -x y + z = -

12 Example Solve the system of equations. 5x y + z = x + y + z = 0 x y z = Example In the 998 WNBA season, Sheryl Swoopes made 8% of her 8 attempted free throws. She made of her -point, -point, and -point attempts, resulting in 5 points. Find the number of - point free throws, -point field goals, and -point field goals Swoopes made in the 998 season.

13 Section Modeling Real-World Data with Matrices Matrix mn matrix Dimensions matrix matrix matrix There are special names for certain matrices: Row Matrix Column Matrix Square Matrix nth order Matrix Equal Matrices Example During the summer, Ms. Robbins received several types of grains on her farm to feed her livestock. Use a matrix to represent the data. June 5,000 bushels corn, 000 bushels soybeans, 500 bushels oats July,500 bushels corn, 500 bushels soybeans, 000 bushels oats August,000 bushels corn, 5500 bushels soybeans, 500 bushels oats

14 Exploring Matrix Operations - For each operation, determine the rule. Matrix Addition.) ) Rule: Zero Matrix Additive Inverse Matrix Subtraction.) ) Rule: Scalar Multiplication 5.) ) Rule: Matrix Multiplication 7.) ) ) ) ) Rule:

15 Example Find the values of x and y for which the matrix equation y x y x is true. Example Find A + B if 7 A 5 0 and 0 B Example Find S T if 5 S and T Example 5 5 If A 8, find A. 9 5

16 Example Use matrices A 0 0, B, C to find each product. 0 a.) BC b.) CB c.) AC d.) CA Example 7 At Ohio State University, professional students pay different tuition rates based on the programs they have chosen. For the 00-0 school year, in-state students in the school of medicine paid $5 per quarter, dental school students paid $79 per quarter, and veterinary medicine students paid $05 per quarter. The chart lists the total student enrollment in those programs for each quarter of the 00-0 school year. Use matrix multiplication to find the amount of tuition paid for each of these four quarters. Quarter Enrollment Med. Dent. Vet. Autumn Winter Spring Summer Source: The Ohio State University Registrar

17 Section Modeling Motion with Matrices Transformations translations reflections rotations dilations Triangle ABC can be represented by the following vertex matrix. Triangle A B C is congruent to and has the same orientation as ABC, but is moved from ABC s location. The coordinates of A' B' C' can by expressed as the following vertex matrix: Compare the two matrices. If you add to the first matrix you get the second matrix. This type of matrix is called a. In this transformation ABC is the and A' B' C' is the after the translation. 7

18 Example Suppose the quadrilateral RSTU with vertices R(, ), S(7, ), T(9, 8), and U(5, ) is translated units right and units down. a.) Represent the vertices of the quadrilateral as a matrix. b.) Write the translation matrix. c.) Use the translation matrix to find the vertices of R S T U, the translated image of the quadrilateral. d.) Graph the quadrilateral RSTU and its image. 8

19 Reflections over the x-axis Reflect the point (,) over the x-axis. How did the coordinate change? What matrix could you multiply by to yield your new coordinate? Reflections over the y-axis Reflect the point (,) over the y-axis. How did the coordinate change? What matrix could you multiply by to yield your new coordinate? Reflections over the line y = x Reflect the point (,) over the line y = x. How did the coordinate change? What matrix could you multiply by to yield your new coordinate? Reflection Matrices For a reflection over the: Symbolized by: Multiply the vertex matrix by: x-axis y-axis line y = x 9

20 Rotations about the origin of 90 o Rotate the point (,) about the origin How did the coordinate change? What matrix could you multiply by to yield your new coordinate? Rotations about the origin of 80 o Rotate the point (,) about the origin How did the coordinate change? What matrix could you multiply by to yield your new coordinate? Rotations about the origin of 70 o Rotate the point (,) about the origin How did the coordinate change? What matrix could you multiply by to yield your new coordinate? For a counterclockwise rotation about the origin of Rotation Matrices Symbolized by: Multiply the vertex matrix by: 90 o 80 o 70 o 0

21 Matrix Transformation Theorem Example Use a reflection matrix to find the coordinates of a reflection over the y-axis of square SQAR with vertices S(, ), Q(7, ), A(9, 0), R(, -). Then graph the pre-image and the image on the same coordinate grid. Example An animated figure rotates about the origin. The image has key points at (5, ), (, -), (, -), (-, ), and (.5,.5). Find the locations of these points at the 90 o, 80 o, and 70 o counterclockwise rotations.

22 Example A parallelogram has vertices W(-, ), X(0, 8), Y(, ), and Z(, ). Find the coordinates of the dilated parallelogram W X Y Z for a scale factor of.5. Describe the dilation. Section 5 Determinants and Multiplicative Inverses of Matrices Our goal is to solve a system of equations using matrices, but before we can do that, we need some math magic. Solve the system of equations by using matrix equations. x y = x + y = 7 Each square matrix has a. The determinant of 8 7 is a number denoted by or det 7. Second-Order Determinant Third-Order Determinant Example Find the value of 0 8.

23 Example Find the value of Identity Matrix for Multiplication Identity Matrix for Second-Order Matrix Inverse Matrix A - Inverse of a Second-Order Matrix Example Find the inverse of the matrix 8 9.

24 Example Solve the system of equations by using matrix equations. x y = x + y = 7 Section Solving Systems of Linear Inequalities Example Belan Chu is a graphic artist who makes greeting cards. Her startup cost will be $500 plus $0.0 per card. In order for her to remain competitive with large companies, she must sell her cards for no more than $.70 each. How many cards must Ms. Chu sell in order to make a profit? Polygonal Convex Set

25 Example Solve the system of inequalities by graphing and name the coordinates of the polygonal convex set. x 0 y 0 x y 5 Vertex Theorem Example Find the maximum and minimum values of f(x, y) = y x + 5 for the polygonal convex set determined by the system of inequalities. x y 8 x y y x y 5 5

26 Section 7 Linear Programming Linear. Define all variables. Programming Procedure. Write the constraints as a system of inequalities.. Graph the system and find the coordinates of the vertices of the polygon formed.. Write an expression whose value is to be minimized or maximized. 5. Substitute values from the coordinates of the vertices into the expression.. Select the greatest or least result. Example Suppose a lumber mill can turn out 00 units of product each week. To meet the needs of its regular customers, the mill must produce 50 units of lumber and 5 units of plywood. If the profit for each unit of lumber is $0 and the profit for each unit of plywood is $5, how many units of each type of wood product should the mill produce to maximize profit?

27 Example The profit on each set of cassettes that is manufactured by MusicMan, Inc., is $8. The profit on a single cassette is $. Machines A and B are used to produce both types of cassettes. Each set takes nine minutes on Machine A and three minutes on Machine B. Each single takes one minute on Machine A and one minute on Machine B. If Machine A is run for 5 minutes and Machine B is run for minutes, determine the combination of cassettes that can be manufactured during the time period that most effectively generates profit within the given constraints. Example The Woodell Carpentry Shop makes bookcases and cabinets. Each bookcase requires 5 hours of woodworking and 9 hours of finishing. The cabinets require 0 hours of woodworking and.5 hours of finishing. The profit is $0 on each bookcase and $0 on each cabinet. There are 70 hours available each week for woodworking and hours available for finishing. How many of each item should be produced in order to maximize profit? 7

28 Example A manufacturer makes widgets and gadgets. At least 500 widgets and 700 gadgets are needed to meet minimum daily demands. The machinery can produce no more than 00 widgets and 00 gadgets per day. The combined number of widgets and gadgets that the packaging department can handle is 00 per day. If the company sells both widgets and gadgets for $.59 each, how many of each item should be produced in order to maximize profit? 8