# Economics 101 Homework #1 Fall 2014 Due 09/18/2014 in lecture

Save this PDF as:

Size: px
Start display at page:

## Transcription

2 Hence the equation of Line B in slope-intercept form is y = 4x -5. c. What is the equation of Line A (from problem 1a) when its y-intercept is reduced by 4 units? What is the equation of Line B (from problem 1b) when its x-intercept is reduced by 4 units? Equation of Line A: y = -x New equation of line A, with the y-intercept reduced by 4 units is y = - x + (10 4) or, y = -x + 6. Equation of Line B: y = 4x 5. To reduce the x intercept by 4 units and get the equation of the new line, we first have to write the equation in terms of x, so that we can get the x-intercept in the equation. y = 4x 5 or, (1/4)y = x (5/4) [diving by 4 on both sides] or, x = (1/4)y + (5/4) [Note this is an equation in terms of x, and (5/4) is the x- intercept] New equation of line B, with the x-intercept reduced by 4 units is x = (1/4)y + (5/4-4) or, x = (1/4)y + (-11/4) This new equation can be written in y-intercept form as (1/4)y = x + (11/4) y = 4x Playing with two given equations. Consider the following equations of two straight lines i. y = x/5 + 4 and 0.2y = 4 - x ii. y = 7x + 1 and y = 7x + 2 Answer the following questions for each of the above 2 sets of equations. Start with the two equations given in (i) and answer the next three questions; then, repeat for the two equations given in (ii). a. What are the slopes, x-intercept and y-intercept of both lines? a. i. Calculating the slopes. y = x/ Here the slope is (1/5) since the equation is given to you in slope-intercept form. 0.2y = 4 x. Let us rewrite this equation so that it is in slope-intercept form (that is, so that y stands alone). or, (2/10)y = 4 x or, (1/5)y = 4 x or, y = 20 5x Here the slope is (-5). Calculating the x-intercept and the y-intercept y = x/ Putting y=0, we have, 0 = x/5 +4, or, x =- 20. The x-intercept is (-20) for the first equation. Putting x=0, we have, y = 0 +4, or, y = 4. The y-intercept is 4 for the first equation. 0.2y = 4 x. Putting y=0, we have 0= 4 -x or, x = 4. The x-intercept is 4 for the second equation. Putting x=0, we have 0.2y = 4-0, or, y = 20. The y-intercept is 20 for the second equation. a. ii. Slopes: 2

3 Slope of y = 7x +1 is 7. Slope of y = 7x +2 is 7. x-intercept and y-intercept: The y-intercept of y = 7x +1 is y = (7)(0) +1 or, y =1 The x-intercept of y = 7x +1 is 0 = 7x +1 or, x = -(1/7) The y-intercept of y = 7x +2 is y = (7)(0) + 2 or, y = 2 The x-intercept of y = 7x +2 is 0 = 7x + 2 or, x = (-2/7) b. Graph the two lines in a single graph measuring y on the vertical axis and x on the horizontal axis. b. i. Graphs of the two straight lines 3

4 b. ii. Graph of the two straight lines c. At which co-ordinate point (x, y) do the two straight lines intersect? [Hint: Be careful on this step when you are working with the equations given in (ii.)] c. i. Points of intersection. The point at which the two lines intersect will satisfy the equation of both the lines. Putting y = y, or, x/5 + 4 = 20 5x [note that we got y = 20 5x, by rewriting the given equation in terms of y, in the solution to part a.] or, x + 20 = x or, x = 80/26 or, x = 40/13 Plugging this value of x, in any of the two equations we can get the value of y, of the point of intersection. Plugging it in y = x/5 + 4, we have y = (1/5)(40/13) + 4 or, y = (8/13) + 4 or, y = 60/13 So the point of intersection of y = x/5 + 4 and 0.2y = 4 x, is (40/13, 60/13) c. ii. Clearly from the graph it can be seen that the two lines never intersect. This is because the two lines have the same slope and are hence parallel. Parallel straight lines never intersect at any point. So, the lines y = 7x +1 and y = 7x +2 being parallel, do not intersect at any point. 3. Adding equations Consider the following two equations 2y = 6 -x and 3y = 18 2x a. What are the intercepts on the X-axis and the Y- axis for these two equations? a. x-intercept and y-intercept. 4

5 The two given lines are 2y = 6-x and 3y = 18-2x The x-intercept of 2y = 6 - x is (2)(0) = 6 -x, or, x = 6. The y-intercept of 2y = 6 - x is 2y = 6 0, or, y = 3. The x-intercept of 3y = 18-2x is (3)(0) = 18 2x, or, 2x = 18, or, x = 9. The y-intercept of 3y = 18-2x is 3y = 18 (2)(0), or, 3y =18, or, y =6. b. Draw the graph of the two straight lines, and show graphically the graph of what you get when you add these two straight lines horizontally. Use three graphs: in the first graph draw the first line, in the second graph draw the second line, and in the third graph draw the graph that results when you add the two lines together horizontally. In your three graphs consider only the values found in the first quadrant of your coordinate graph (that is, only consider the positive values of x and y when graphing). b. Graph of the two straight lines and the horizontally summed line c. What is the equation for the line(s) you drew in the third graph? (Hint: If you have more than one line you will need to provide a range of values for each equation.) c. EXPLANATION: Note that, for all values of y in between 3 and 6, the first equation (or, graph of it) does not exist (considering only positive values of x and y). Hence in the combined graph, until the point (4.5, 3) the equation of the combined graph will be the same as that of 3y = 18-2x. (4.5 is the value of x, 5

6 when y takes the value of 3, on the line 3y = 18-2x. If you look at the x-axis, for all values of x in between 0 and 4.5, the equation of the combined graph is 3y = 18-2x.) Now for all values of y below 3 and above 0, you will need to sum together horizontally both equations. So for these values of y, the combined equation will be the horizontal summation of the two given equations. Since we are summing horizontally, we have to rewrite the equations in terms of x. So, 2y = 6 x can be rewritten as x = 6 2y. and, 3y = 18 2x, can be rewritten as x = 9 (3/2)y. So, for all values of y below 3and above 0, the equation of the combined graph is x = (6 2y ) + ( 9 (3/2)y ) or, x = 15 (7/2)y Hence, the equation of the combined graph is written as follows, - For values of y such that, 3 y 6, the equation of the combined graph is x = 9 (3/2)y And for values of y such that, 0 y 3, the equation of the combined graph is x = 15 (7/2)y Part II: Opportunity Costs, Absolute Advantage, Comparative Advantage, Production Possibility Frontier (PPF) 4. Schmidt and Alex are employed as Librarians in College Library. At the end of each day, they have to deal with a number of returning books, B, and laptops, L. Their work consists of scanning the code bar stuck to either book or laptop, so that each item is discharged. How quickly they can scan each item is given as below: # of books scanned in per minute # of laptops scanned in per minute Schmidt Alex According to the information above, answer the following questions (a)-(e): a. For Schmidt, what is the opportunity cost of scanning 1 book (remember that opportunity cost is measured in terms of the good that is given up)? What is opportunity cost of scanning 1 laptop for him? For both answers provide the units of measurement. Now, answer the same two questions for Alex. Schmidt in one minute can scan twice as many laptops as books: the opportunity cost of scanning 1 book is therefore the 2 scanned laptops he must give up. The opportunity cost for Schmidt of scanning 1 laptop is the ½ scanned books he must give up. Alex in one minute can scan either 20 books or 30 laptops: therefore the opportunity cost of scanning 1 book equals 1.5 laptops, and the opportunity cost of scanning 1 laptop is 2/3 books. b. Who has the comparative advantage in scanning books? Who has the comparative advantage 6

8 For Alex, the PPF equation is B = 600 (2/3)L: e. Draw a graph of the joint PPF between Schmidt and Alex and provide the coordinate values (L, B) for the kink point. Measure scanned books (B) on the vertical axis and scanned laptops (L) on the horizontal axis. The joint PPF equation is: B = L, if L is within the domain [0 scanned laptops, 600 scanned laptops]; and B = 1000 (2/3)L, if L is within the domain [600 scanned laptops, 1500 scanned laptops]. And the kink point is (600 scanned laptops, 600 scanned books). 5. Suppose there are two counties in a southern state, East and West. Both counties could produce sweaters (S) and woolly hats (H) by weavers. Each county employs 10 weavers per year, and each weaver works for 2000 hours per year. Both East and West have linear production possibility frontiers in the production of sweaters and hats. By using up all the labour hours in a whole year, 8

10 From the graphs and your answer to (c), it is easy to see that if East only produces sweaters, it would be able to produce 2500 sweaters in all; and if it only produces woolly hats, it would be able to produce 5000 hats in all. And the numbers for West are 1500 sweaters and 6000 hats. Comparing these numbers we can conclude that East has the absolute advantage in producing sweaters and West has the absolute advantage in producing woolly hats. e. In each county, how many labor hours are needed to produce a woolly hat? And how many labor hours are needed to produce a sweater? Fill out the table below. County East West Labor hours Needed to Produce a Woolly Hat Labor hours Needed to Produce a Sweater From the given information we know that the total labor hours each county has is 2000*10=20,000 hours of labor. Then, if we also use our answer from (d), we get: County East Labour hours Needed to Produce a Piece of Labour hours Needed to Produce a Piece Woolly Hat of Sweater 20,000 hours of labor/5000 hats = 4 hours 20,000 hours of labor/2500 sweaters = 8 of labor per hat hours of labor per sweater 10

### 5 \$75 6 \$90 7 \$105. Name Hour. Review Slope & Equations of Lines. STANDARD FORM: Ax + By = C. 1. What is the slope of a vertical line?

Review Slope & Equations of Lines Name Hour STANDARD FORM: Ax + By = C 1. What is the slope of a vertical line? 2. What is the slope of a horizontal line? 3. Is y = 4 the equation of a horizontal or vertical

### Section 1.1 Linear Equations: Slope and Equations of Lines

Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

### GRAPHING LINEAR EQUATIONS IN TWO VARIABLES

GRAPHING LINEAR EQUATIONS IN TWO VARIABLES The graphs of linear equations in two variables are straight lines. Linear equations may be written in several forms: Slope-Intercept Form: y = mx+ b In an equation

### Linear Equations Review

Linear Equations Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The y-intercept of the line y = 4x 7 is a. 7 c. 4 b. 4 d. 7 2. What is the y-intercept

### What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b.

PRIMARY CONTENT MODULE Algebra - Linear Equations & Inequalities T-37/H-37 What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of

### Section 1.4 Graphs of Linear Inequalities

Section 1.4 Graphs of Linear Inequalities A Linear Inequality and its Graph A linear inequality has the same form as a linear equation, except that the equal symbol is replaced with any one of,,

### Graphing Linear Equations

Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope

### MATH 105: Finite Mathematics 1-1: Rectangular Coordinates, Lines

MATH 105: Finite Mathematics 1-1: Rectangular Coordinates, Lines Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006 Outline 1 Rectangular Coordinate System 2 Graphing Lines 3 The Equation of

### In this section, we ll review plotting points, slope of a line and different forms of an equation of a line.

Math 1313 Section 1.2: Straight Lines In this section, we ll review plotting points, slope of a line and different forms of an equation of a line. Graphing Points and Regions Here s the coordinate plane:

### Section 3.2. Graphing linear equations

Section 3.2 Graphing linear equations Learning objectives Graph a linear equation by finding and plotting ordered pair solutions Graph a linear equation and use the equation to make predictions Vocabulary:

### Linear Equations and Graphs

2.1-2.4 Linear Equations and Graphs Coordinate Plane Quadrants - The x-axis and y-axis form 4 "areas" known as quadrants. 1. I - The first quadrant has positive x and positive y points. 2. II - The second

### Linear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)}

Linear Equations Domain and Range Domain refers to the set of possible values of the x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in

### Section 1.4 Notes Page Linear Equations in Two Variables and Linear Functions., x

Section. Notes Page. Linear Equations in Two Variables and Linear Functions Slope Formula The slope formula is used to find the slope between two points ( x, y ) and ( ) x, y. x, y ) The slope is the vertical

### Slope-Intercept Form of a Linear Equation Examples

Slope-Intercept Form of a Linear Equation Examples. In the figure at the right, AB passes through points A(0, b) and B(x, y). Notice that b is the y-intercept of AB. Suppose you want to find an equation

### Practice Problems for Exam 1 Math 140A, Summer 2014, July 2

Practice Problems for Exam 1 Math 140A, Summer 2014, July 2 Name: INSTRUCTIONS: These problems are for PRACTICE. For the practice exam, you may use your book, consult your classmates, and use any other

### Ordered Pairs. Graphing Lines and Linear Inequalities, Solving System of Linear Equations. Cartesian Coordinates System.

Ordered Pairs Graphing Lines and Linear Inequalities, Solving System of Linear Equations Peter Lo All equations in two variables, such as y = mx + c, is satisfied only if we find a value of x and a value

### WARM UP EXERCSE. 1-3 Linear Functions & Straight lines

WARM UP EXERCSE A company makes and sells inline skates. The price-demand function is p (x) = 190 0.013(x 10) 2. Describe how the graph of function p can be obtained from one of the library functions.

### Math 152 Rodriguez Blitzer 2.4 Linear Functions and Slope

Math 152 Rodriguez Blitzer 2.4 Linear Functions and Slope I. Linear Functions 1. A linear equation is an equation whose graph is a straight line. 2. A linear equation in standard form: Ax +By=C ex: 4x

### 1 Functions, Graphs and Limits

1 Functions, Graphs and Limits 1.1 The Cartesian Plane In this course we will be dealing a lot with the Cartesian plane (also called the xy-plane), so this section should serve as a review of it and its

### Sect The Slope-Intercept Form

Concepts # and # Sect. - The Slope-Intercept Form Slope-Intercept Form of a line Recall the following definition from the beginning of the chapter: Let a, b, and c be real numbers where a and b are not

### The Point-Slope Form

7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope

### Section 3.4 The Slope Intercept Form: y = mx + b

Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept Reminding! m = y x = y 2 y 1 x 2 x 1 Slope of a horizontal line is 0 Slope of a vertical line is Undefined Graph a linear

### Plot the following two points on a graph and draw the line that passes through those two points. Find the rise, run and slope of that line.

Objective # 6 Finding the slope of a line Material: page 117 to 121 Homework: worksheet NOTE: When we say line... we mean straight line! Slope of a line: It is a number that represents the slant of a line

### Chapter 2 Section 4: Equations of Lines. 4.* Find the equation of the line with slope 4 3, and passing through the point (0,2).

Chapter Section : Equations of Lines Answers to Problems For problems -, put our answers into slope intercept form..* Find the equation of the line with slope, and passing through the point (,0).. Find

### Math 113 Review for Exam I

Math 113 Review for Exam I Section 1.1 Cartesian Coordinate System, Slope, & Equation of a Line (1.) Rectangular or Cartesian Coordinate System You should be able to label the quadrants in the rectangular

### Slope-Intercept Equation. Example

1.4 Equations of Lines and Modeling Find the slope and the y intercept of a line given the equation y = mx + b, or f(x) = mx + b. Graph a linear equation using the slope and the y-intercept. Determine

### COMPARING LINEAR AND NONLINEAR FUNCTIONS

1 COMPARING LINEAR AND NONLINEAR FUNCTIONS LEARNING MAP INFORMATION STANDARDS 8.F.2 Compare two s, each in a way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example,

### Writing the Equation of a Line in Slope-Intercept Form

Writing the Equation of a Line in Slope-Intercept Form Slope-Intercept Form y = mx + b Example 1: Give the equation of the line in slope-intercept form a. With y-intercept (0, 2) and slope -9 b. Passing

### Graphing Linear Equations in Two Variables

Math 123 Section 3.2 - Graphing Linear Equations Using Intercepts - Page 1 Graphing Linear Equations in Two Variables I. Graphing Lines A. The graph of a line is just the set of solution points of the

### 5.1: Rate of Change and Slope

5.1: Rate of Change and Slope Rate of Change shows relationship between changing quantities. On a graph, when we compare rise and run, we are talking about steepness of a line (slope). You can use and

1.3 LINEAR EQUATIONS IN TWO VARIABLES Copyright Cengage Learning. All rights reserved. What You Should Learn Use slope to graph linear equations in two variables. Find the slope of a line given two points

### Exam 2 Review. 3. How to tell if an equation is linear? An equation is linear if it can be written, through simplification, in the form.

Exam 2 Review Chapter 1 Section1 Do You Know: 1. What does it mean to solve an equation? To solve an equation is to find the solution set, that is, to find the set of all elements in the domain of the

### What students need to know for... ALGEBRA II

What students need to know for... ALGEBRA II 2015-2016 NAME Students expecting to take Algebra II at Cambridge Rindge & Latin School should demonstrate the ability to... General: o Keep an organized notebook

### EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

### Many different kinds of animals can change their form to help them avoid or

Slopes, Forms, Graphs, and Intercepts Connecting the Standard Form with the Slope-Intercept Form of Linear Functions Learning Goals In this lesson, you will: Graph linear functions in standard form. Transform

### Economics 101 Fall 2013 Answers to Homework 5 Due Tuesday, November 19, 2013

Economics 101 Fall 2013 Answers to Homework 5 Due Tuesday, November 19, 2013 Directions: The homework will be collected in a box before the lecture. Please place your name, TA name and section number on

### Solving Equations Involving Parallel and Perpendicular Lines Examples

Solving Equations Involving Parallel and Perpendicular Lines Examples. The graphs of y = x, y = x, and y = x + are lines that have the same slope. They are parallel lines. Definition of Parallel Lines

### Number of Solutions to Simultaneous Equations

Worksheet 3.5 Simultaneous Equations Section 1 Number of Solutions to Simultaneous Equations In maths we are sometimes confronted with two equations in two variables and we want to find out which values

### 2 Unit Bridging Course Day 2 Linear functions II: Finding equations

1 / 38 2 Unit Bridging Course Day 2 Linear functions II: Finding equations Clinton Boys 2 / 38 Finding equations of lines If we have the information of (i) the gradient of a line (ii) the coordinates of

### Topic 15 Solving System of Linear Equations

Topic 15 Solving System of Linear Equations Introduction: A Linear Equation is an equation that contains two variables, typically x and y that when they are graphed on a cartesian grid make a straight

### A synonym is a word that has the same or almost the same definition of

Slope-Intercept Form Determining the Rate of Change and y-intercept Learning Goals In this lesson, you will: Graph lines using the slope and y-intercept. Calculate the y-intercept of a line when given

1.6 A LIBRARY OF PARENT FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Identify and graph linear and squaring functions. Identify and graph cubic, square root, and reciprocal

### Warm Up. Write an equation given the slope and y-intercept. Write an equation of the line shown.

Warm Up Write an equation given the slope and y-intercept Write an equation of the line shown. EXAMPLE 1 Write an equation given the slope and y-intercept From the graph, you can see that the slope is

### 4.1 & Linear Equations in Slope-Intercept Form

4.1 & 4.2 - Linear Equations in Slope-Intercept Form Slope-Intercept Form: y = mx + b Ex 1: Write the equation of a line with a slope of -2 and a y-intercept of 5. Ex 2:Write an equation of the line shown

### Section summaries. d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. 1 + y 2. x1 + x 2

Chapter 2 Graphs Section summaries Section 2.1 The Distance and Midpoint Formulas You need to know the distance formula d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 and the midpoint formula ( x1 + x 2, y ) 1 + y 2

### Helpsheet. Giblin Eunson Library LINEAR EQUATIONS. library.unimelb.edu.au/libraries/bee. Use this sheet to help you:

Helpsheet Giblin Eunson Library LINEAR EQUATIONS Use this sheet to help you: Solve linear equations containing one unknown Recognize a linear function, and identify its slope and intercept parameters Recognize

### Study Guide and Review - Chapter 4

State whether each sentence is true or false. If false, replace the underlined term to make a true sentence. 1. The y-intercept is the y-coordinate of the point where the graph crosses the y-axis. The

### Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.

Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear

### Economics 101 Spring 2011 Homework #4 Due Tuesday, March 29

Economics 101 Spring 2011 Homework #4 Due Tuesday, March 29 Directions: The homework will be collected in a box before the lecture. Staple your homework before handing it in. Please place your name, TA

### Economics 102 Fall 2015 Answers to Homework #2 Due Monday, October 5, 2015

Economics 102 Fall 2015 Answers to Homework #2 Due Monday, October 5, 2015 Directions: The homework will be collected in a box before the large lecture. Please place your name, TA name and section number

### Section 2.2 Equations of Lines

Section 2.2 Equations of Lines The Slope of a Line EXAMPLE: Find the slope of the line that passes through the points P(2,1) and Q(8,5). = 5 1 8 2 = 4 6 = 2 1 EXAMPLE: Find the slope of the line that passes

### 2x - y 4 y -3x - 6 y < 2x 5x - 3y > 7

DETAILED SOLUTIONS AND CONCEPTS GRAPHICAL REPRESENTATION OF LINEAR INEQUALITIES IN TWO VARIABLES Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu.

### Math 018 Review Sheet v.3

Math 018 Review Sheet v.3 Tyrone Crisp Spring 007 1.1 - Slopes and Equations of Lines Slopes: Find slopes of lines using the slope formula m y y 1 x x 1. Positive slope the line slopes up to the right.

### The slope m of the line passes through the points (x 1,y 1 ) and (x 2,y 2 ) e) (1, 3) and (4, 6) = 1 2. f) (3, 6) and (1, 6) m= 6 6

Lines and Linear Equations Slopes Consider walking on a line from left to right. The slope of a line is a measure of its steepness. A positive slope rises and a negative slope falls. A slope of zero means

### Lines and Linear Equations. Slopes

Lines and Linear Equations Slopes Consider walking on a line from left to right. The slope of a line is a measure of its steepness. A positive slope rises and a negative slope falls. A slope of zero means

### Lines, Lines, Lines!!! Slope-Intercept Form ~ Lesson Plan

Lines, Lines, Lines!!! Slope-Intercept Form ~ Lesson Plan I. Topic: Slope-Intercept Form II. III. Goals and Objectives: A. The student will write an equation of a line given information about its graph.

### Equations of Lines Derivations

Equations of Lines Derivations If you know how slope is defined mathematically, then deriving equations of lines is relatively simple. We will start off with the equation for slope, normally designated

### Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year.

Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year. Goal The goal of the summer math program is to help students

### Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20

Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding

### ) represents the original function, will a, r 1

Transformations of Quadratic Functions Focus on Content: Download Problem Sets Problem Set #1 1. In the graph below, the quadratic function displayed on the right has been reflected over the y-axis producing

### Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities

Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.

### Section 2.1 Rectangular Coordinate Systems

P a g e 1 Section 2.1 Rectangular Coordinate Systems 1. Pythagorean Theorem In a right triangle, the lengths of the sides are related by the equation where a and b are the lengths of the legs and c is

### CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS

CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS 2.01 SECTION 2.1: QUADRATIC FUNCTIONS (AND PARABOLAS) PART A: BASICS If a, b, and c are real numbers, then the graph of f x = ax2 + bx + c is a parabola, provided

### Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4)

Chapter 2: Functions and Linear Functions 1. Know the definition of a relation. Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4) 2. Know the definition of a function. 3. What

### CHAPTER 1 Linear Equations

CHAPTER 1 Linear Equations 1.1. Lines The rectangular coordinate system is also called the Cartesian plane. It is formed by two real number lines, the horizontal axis or x-axis, and the vertical axis or

### b. Given this information, describe the government budget balance for this economy.

Economics 102 Summer 2015 Answers to Homework #5 Due Wednesday, July 15, 2015 Directions: The homework will be collected in a box before the lecture. Please place your name on top of the homework (legibly).

### Vocabulary Words and Definitions for Algebra

Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

### Section 1.10 Lines. The Slope of a Line

Section 1.10 Lines The Slope of a Line EXAMPLE: Find the slope of the line that passes through the points P(2,1) and Q(8,5). = 5 1 8 2 = 4 6 = 2 1 EXAMPLE: Find the slope of the line that passes through

### Pre-AP Algebra 2 Unit 3 Lesson 1 Quadratic Functions

Unit 3 Lesson 1 Quadratic Functions Objectives: The students will be able to Identify and sketch the quadratic parent function Identify characteristics including vertex, axis of symmetry, x-intercept,

### Part 1: Background - Graphing

Department of Physics and Geology Graphing Astronomy 1401 Equipment Needed Qty Computer with Data Studio Software 1 1.1 Graphing Part 1: Background - Graphing In science it is very important to find and

### GRAPHING (2 weeks) Main Underlying Questions: 1. How do you graph points?

GRAPHING (2 weeks) The Rectangular Coordinate System 1. Plot ordered pairs of numbers on the rectangular coordinate system 2. Graph paired data to create a scatter diagram 1. How do you graph points? 2.

### SCATTER PLOTS AND TREND LINES

Name SCATTER PLOTS AND TREND LINES VERSION 2 Lessons 1 3 1. You have collected the following data while researching the Winter Olympics. You are trying to determine if there is a relationship between the

### 2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses

College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2

### Math 115 Spring 2014 Written Homework 10-SOLUTIONS Due Friday, April 25

Math 115 Spring 014 Written Homework 10-SOLUTIONS Due Friday, April 5 1. Use the following graph of y = g(x to answer the questions below (this is NOT the graph of a rational function: (a State the domain

### Section 1.8 Coordinate Geometry

Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of

### Let s explore the content and skills assessed by Heart of Algebra questions.

Chapter 9 Heart of Algebra Heart of Algebra focuses on the mastery of linear equations, systems of linear equations, and linear functions. The ability to analyze and create linear equations, inequalities,

### Educational Transfer Plan IISME Summer Fellowship Investigating Liner Equations Using Graphing Calculator

Joumana Alsumidaie Fisher Middle School Algebra 1 Educational Transfer Plan IISME Summer Fellowship 2005 Investigating Liner Equations Using Graphing Calculator Overview: Investigating Linear Equations

### Course Name: Course Code: ALEKS Course: Instructor: Course Dates: Course Content: Textbook: Dates Objective Prerequisite Topics

Course Name: MATH 1204 Fall 2015 Course Code: N/A ALEKS Course: College Algebra Instructor: Master Templates Course Dates: Begin: 08/22/2015 End: 12/19/2015 Course Content: 271 Topics (261 goal + 10 prerequisite)

### Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties

Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties Addition: (1) (Associative law) If a, b, and c are any numbers, then ( ) ( ) (2) (Existence of an

### Procedure In each case, draw and extend the given series to the fifth generation, then complete the following tasks:

Math IV Nonlinear Algebra 1.2 Growth & Decay Investigation 1.2 B: Nonlinear Growth Introduction The previous investigation introduced you to a pattern of nonlinear growth, as found in the areas of a series

### 7.5 Systems of Linear Inequalities

7.5 Systems of Linear Inequalities In chapter 6 we discussed linear equations and then saw linear inequalities, so since here, in chapter 7, we are talking about systems of linear equations, it makes sense

### Name: Class: Date: Does the equation represent a direct variation? If so, find the constant of variation. c. yes; k = 5 3. c.

Name: Class: Date: Chapter 5 Test Multiple Choice Identify the choice that best completes the statement or answers the question. What is the slope of the line that passes through the pair of points? 1.

### x x y y Then, my slope is =. Notice, if we use the slope formula, we ll get the same thing: m =

Slope and Lines The slope of a line is a ratio that measures the incline of the line. As a result, the smaller the incline, the closer the slope is to zero and the steeper the incline, the farther the

### 55x 3 + 23, f(x) = x2 3. x x 2x + 3 = lim (1 x 4 )/x x (2x + 3)/x = lim

Slant Asymptotes If lim x [f(x) (ax + b)] = 0 or lim x [f(x) (ax + b)] = 0, then the line y = ax + b is a slant asymptote to the graph y = f(x). If lim x f(x) (ax + b) = 0, this means that the graph of

### LINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0

LINEAR FUNCTIONS As previousl described, a linear equation can be defined as an equation in which the highest eponent of the equation variable is one. A linear function is a function of the form f ( )

### Section 7.1 Solving Linear Systems by Graphing. System of Linear Equations: Two or more equations in the same variables, also called a.

Algebra 1 Chapter 7 Notes Name Section 7.1 Solving Linear Systems by Graphing System of Linear Equations: Two or more equations in the same variables, also called a. Solution of a System of Linear Equations:

### Write the Equation of the Line Review

Connecting Algebra 1 to Advanced Placement* Mathematics A Resource and Strategy Guide Objective: Students will be assessed on their ability to write the equation of a line in multiple methods. Connections

### 2.7. The straight line. Introduction. Prerequisites. Learning Outcomes. Learning Style

The straight line 2.7 Introduction Probably the most important function and graph that you will use are those associated with the straight line. A large number of relationships between engineering variables

### with "a", "b" and "c" representing real numbers, and "a" is not equal to zero.

3.1 SOLVING QUADRATIC EQUATIONS: * A QUADRATIC is a polynomial whose highest exponent is. * The "standard form" of a quadratic equation is: ax + bx + c = 0 with "a", "b" and "c" representing real numbers,

### Linear Equations. 5- Day Lesson Plan Unit: Linear Equations Grade Level: Grade 9 Time Span: 50 minute class periods By: Richard Weber

Linear Equations 5- Day Lesson Plan Unit: Linear Equations Grade Level: Grade 9 Time Span: 50 minute class periods By: Richard Weber Tools: Geometer s Sketchpad Software Overhead projector with TI- 83

### The table below shows the prices of the only three commodities traded in Shire.

Economics 101 Fall 2012 Homework #4 Due 11/20/2012 Directions: The homework will be collected in a box before the lecture. Please place your name, TA name and section number on top of the homework (legibly).

### Techniques of Differentiation Selected Problems. Matthew Staley

Techniques of Differentiation Selected Problems Matthew Staley September 10, 011 Techniques of Differentiation: Selected Problems 1. Find /dx: (a) y =4x 7 dx = d dx (4x7 ) = (7)4x 6 = 8x 6 (b) y = 1 (x4

### Principles of Math 12 - Transformations Practice Exam 1

Principles of Math 2 - Transformations Practice Exam www.math2.com Transformations Practice Exam Use this sheet to record your answers. NR. 2. 3. NR 2. 4. 5. 6. 7. 8. 9. 0.. 2. NR 3. 3. 4. 5. 6. 7. NR

### Beginning of the Semester To-Do List

Beginning of the Semester To-Do List Set up your account at https://casa.uh.edu/ Read the Math 13xx Departmental Course Policies Take Course Policies Quiz until your score is 100%. You can find it on the

### N.CN.7, A.CED.1, 2, 3, N.Q.2, A.SSE.1,

Learning Targets: I can solve interpret key features of quadratic functions from different form. I can choose a method to solve, and then, solve a quadratic equation and explain my reasoning. #1 4 For

### Title: Graphing Quadratic Equations in Standard Form Class: Math 100 or 107 Author: Sharareh Masooman Instructions to tutor: Read instructions under

Title: Graphing Quadratic Equations in Standard Form Class: Math 100 or 107 Author: Sharareh Masooman Instructions to tutor: Read instructions under Activity and follow all steps for each problem exactly

### Creating Equations. Set 3: Writing Linear Equations Instruction. Student Activities Overview and Answer Key

Creating Equations Instruction Goal: To provide opportunities for students to develop concepts and skills related to writing linear equations in slope-intercept and standard form given two points and a

### Answer Key Building Polynomial Functions

Answer Key Building Polynomial Functions 1. What is the equation of the linear function shown to the right? 2. How did you find it? y = ( 2/3)x + 2 or an equivalent form. Answers will vary. For example,