Chapter 1 & 2 1-1 Relations & Functions 1-2 Compostion of Functions 1-3 Graphs Linear Eqns 1-4 Writing Linear Functions 1-5 Parallel & Perpendicular Lines 1-7 Piecewise Functions 1-8 Linear Inequalities 2-1 Systems 2 Variables 2-2 Systems 3 Variables 2-6 Systems Linear Inequaliti 2-7 Linear Programming Homework Homework Ch 1 & 2 1-1 page 9-11: #5, 16-29, 38-50, 54 1-2 page 17-18: #10, 11-13, 15-16, 22-27 1-3 page 24: #12-17, 24-29 1-4 page 30: #11-24 1-5 page 36-37: #12-29 1-6 page 42-43: #6ab, 8ab 1-7 page 48-50: #1, 2, 11-20, 28 1-8 page 54-55: #1, 8-20, 23 2-1 page 71: #11-33 2-2 page 76: #8-19, 21 2-6 page 110: #9-22 2-7 page 115: #7, 9-15 1
1-1 Relations and Functions Pull Example 1: State the domain and range of each relation. D: D: R: R: 2
Ways to determine if a relation is a function Function Notation f(x) "function f of x" The easy way to think about it is y = f(x) Other letters can be used too. g(x), h(x), P(x),... Find f(-1). 3
Example 2: State the domain for the function. clicker questions 1-2 Composition of Functions Functions can be added, subtracted, multiplied (FOIL), or divided. g(x)! 0 4
1 Find IQ 2 Find IQ 5
Composition of Functions Insert the second function into the first function. Example 1: Find the two compositions for the f(x) = x 2-1 and g(x) = 3x. The domain of includes all of the elements x in the domain of g for g(x) is in the domain of f. Example 2: State the domain of 6
Example 3: Find the first three iterates, x 1, x 2, and x 3 of the function f(x) = 3x +2 for an initial value of x 0 = 4. 1-7 Piecewise Functions Piecewise Functions: different equations are used for different intervals of the domain. Example 1: if x! -5 if -5 < x! 4 if x > 4 7
Step Functions: Look like a staircase Greatest Integer Function: y = x "The greatest Integer not greater than x" 3.9 = 3 3 is not bigger than 3.9-2.5 = -3 Remember -3 is NOT bigger than -2.5 but -2 is bigger. Example 2: Graph f(x) = x x f(x) 8
Example 3: Graph f(x) = 3 x x f(x) Absolute Value Function: also a piecewise function; important reminder the graph is a V. y = x What is the piecewise if x < 0 function? Click image to reveal. if x! 0 9
Example 4: Graph y = -2 x + 3 x y Pull 1-8 Linear Inequalities Example 1: Graph each inequality. a. x! 2 b. x - 2y < 8 c. y! x + 1 10
Example 2: Graph the inequality. 7 < x + y! 9 Example 3: Arctic explorers need endurance and strength. An endurance diet has a balance between fat, carbohydrates and protein. Fat is a concentrated engery source that supplies nine calories per gram. Carbohydrates and protein provide four calories per gram. If a daily diet for the endurance diet is between 4000 and 6000 calories, graph the possible combinations of fat, carbs and protein. 11
2-1 Systems with 2 Variables Four Methods to Solve 2 Variable Systems 1. Graphing (where the lines intersect) 2. Substitution 3. Elimination 4. Cramer's Rule (uses Matrices- Finite Math) 1 Solve the system using substitution. (Answer as an ordered pair.) IQ 2x + y = 22 y = 3x - 8 12
2 Solve the system by elimination. 5x + 2y = 340 3x - 4y = 360 IQ doc camera p.68 13
2-2 Systems with 3 Variables Most 3 variable systems will have an ordered triple as the answer. Write the ordered triple in alphabetical order. (x, y, z) or (a, b, c) The graphs would be 3-dimensional. (Pictures on p. 73) Example 1: 4x = 8z 3x - 2y + z = 0-2x + y - z = -1 14
Example 2: 5x - 2y + z = 11 2x + y + 3z = 0 6x - 2y -2z = 16 1 Solve the system. 4x + 2y + z = 7 2x +2y - 4z= -4 x + 3y - 2z = -8 IQ 15
Cynthia Cooper played for the WNBA Houston Comets. In 1998, she scored 680 points by hitting 413 of her 1-point, 2-point, and 3-point attempts. She made 40% of her 160 3-point attempts. How many 1-point, 2-point, and 3-point baskets did she complete? 2-6 Systems of Linear Inequalities Green Region is where both inequalities are true. This region represents all the ordered pairs that make the system true. y > -x + 4 Gray y > x Yellow 16
Example 1: Graph x! 0 y! 0 2x + y " 4 Polygonal Convex Set: A bounded set of all points on or inside a graphed convex polygon. Vertex Theorem: The maximum or minimum value of f(x, y) = ax + by + c on a polygonal convex set occurs at a vertex of the polygonal boundary. f(x, y) = 5x - 3y 17
Example 2: Find the min and max for f(x,y)= y- 2x + 5 for the polygonal convex set determined by the system of inequalities. x! 1 2 " y " 8 x + y! 5 2x + y " 14 2-6 practice Grade: «grade» Subject: «subject» Date: «date» 18
1 For the function and system of inequalities, determine the maximum value. (NOA) 4y! x + 8 y! 2x - 5 x + y " 2 2 For the same function and graph, find the minimum value. (NOA) 4y! x + 8 y! 2x - 5 x + y " 2 19
3 For the same graph, what is the vertex point in the 4th quadrant? (Ordered pair answer) 4y! x + 8 y! 2x - 5 x + y " 2 4 Regarding this lesson... A B C I am ready to practice what I know. I want a few more examples. I have a couple questions. 20
2-7 Linear Programming s, write nction. with Pull Pull ded Pull ction. Pull /or max. Pull 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 1200 widgets and 1400 gadgets per day. The combined number of widgets and gadgets that the packaging department can handle is 2300 per day. If the company sells both widgets and gadgets for $1.59 each, how many of each item should be produced in order to maximize profit? Squares each 10 on both axis 21
Infeasible: inequalities do not overlap. Unbounded: Region continues into infinity. It may not have an optimal solution. Alternate Optimal Solutions: Two or more optimal solutions 2-7 Practice Grade: «grade» Subject: «subject» Date: «date» 22
A package delivery service has a truck that can hold 4200 pounds of cargo and has a capacity of 480 cubic feet. The service handles two types of packages: small, which weigh up to 25 pounds each and no more than 3 cubic feet each; and large, which are 25 to 50 pounds each and are 3 to 5 cubic feet each. The delivery service charges $5 for each small package and $8 for each large package. Let x be the number of small packages and y be the number of large packages in the truck. 1 What is the inequality that represents the weight of the packages in pounds the truck can carry? 23
2 What is the inequality that represents the volume, in cubic feet, of packages the truck can carry? 3 Write a function that represents the amount of money the delivery service will make with each truckload. (P (x,y)=...) 24
4 Find the number of each type of packages that should be placed on a truck to maximize revenue. (Ordered pair) 5 What is the maximum revenue per truck? (NOA) 25
6 Regarding linear programming... A B C I am ready to practice what I know. I want a few more examples. I have a couple questions. 26
1-1 Grade: «grade» Subject: «subject» Date: «date» 1 In a relation, the domain is which set of values? A B C x-values y-values neither 27
2 State the domain for the given relation. Answer: {#,#,...} Braces with numbers separated by commas. x y -3 4 0 0 3-4 6-8 3 What is the range for the given relation? Same format as #2. 28
4 Is the relation a function? Yes No 5 Which relation or relations are functions (Multiple Answers Possible-MAP) A B C D 29
6 Find f(3). (Number Only Answer-NOA) 7 Find h(a-2). 30
8 What is the domain for the function? 9 Regarding functions and relations... A B C I am ready to practice what I know. I want a few more examples. I have a couple questions. 31
Linear Equations Grade: Subject: 1-3, 1-4, 1-5 Date: This is an open book assignment, but it is a solo assignment. It covers information from Algebra 1 and Algebra 2 about Linear Equations. Draw the graphs on a piece of graph paper. Do your best and when everyone is done (or should be done) we will discuss the results. No Decimal answers, use fractions. 32
1 What is the x-intercept for the line? Write the answer as an ordered pair. 2 What is the y-intercept for the line? Write the answer as an ordered pair. Graph the line. 33
3 Calculate the slope of the line between (-3, 7) and (4, -3). Graph the line. 4 What is the equation of the line through (0,4) with slope -1/2? Graph the line. 34
5 Horizontal lines have a slope of? (Number Only Answer- NOA) 6 What is the general equation of a horizontal line? 35
7 Find the zero of the equation. If no zero exists, type none. 8 Write the equation of a line passing through (5,2) and (7,9). 36
9 Convert the previous line into standard form. 10 Are the pair of equations parallel, perpendicular, coinciding, or neither. (Text answer) 5x-3y=12 10x-6y=24 37
11 Are the pair of equations parallel, perpendicular, coinciding, or neither. (Text answer) y=-6x-2 y= 1/6 x -8 12 Write the slope-intercept form of the line that passes through (-10,-5) and is perpendicular to the graph of 6x - 5y = 24. 38
13 Regarding linear equations... A B C I am ready to practice what I know. I want a few more examples. I have a couple questions. 39