In this class, we addressed problem 14 from Chapter 2. So first step, we expressed the problem in STANDARD FORM:

Size: px
Start display at page:

Download "In this class, we addressed problem 14 from Chapter 2. So first step, we expressed the problem in STANDARD FORM:"

Transcription

1 In this class, we addressed problem 14 from Chapter 2. So first step, we expressed the problem in STANDARD FORM: Now that we have done that, we want to plot our constraint lines, so we can find our feasible region. Since we have three constraints (for material 1, material 2 and material 3) we will plot three lines. The easiest way to plot our graphs is to find both the x- and y- intercepts on each of the three lines and connect them. For the Material 1 constraint, we first set f to zero, and solve the equation: So we have one point (0, 40). Next we set s equal to zero and solve for f:

2 Here then we have another point on the graph: (50, 0). For the material two constraint, we determined that, since f is not part of the equation, it will just be a horizontal line, where s = 25: The line happens to be horizontal on our plot, just because we are choosing to put the s amount on the y (vertical) axis

3 For the material 3 constraint, we found our two points: (0, 70) and (35, 0): So to summarize, we have enough points to draw our three constraint lines. Here are the lines plotted:

4 You can see that there are three lines, labeled M1, M2 and M3. These lines represent the constraints imposed by the limitations of the amounts of material 1, 2, and 3, respectively. The arrows next to the label indicate the side where solutions will be feasible. (I ve blocked out some things for simplicity). After all of the lines are plotted, we can shade in and label the feasible region. The feasible region is the region where any solution will be feasible, according to all of the constraints. We have done so. Now we will find a FEASIBLE (not the optimal) solution line by selecting an ARBITRARY amount, and solve the solution. For instance we chose $1200 as our arbitrary amount. In a sense, we are asking if it is feasible to contribute $1200 to the profit, given the constraints. The line is calculated in a similar way to above, by finding both the x- and y- intercept points:

5 Here, we have found two points for our feasible solution line: (0, 40) and (30, 0) This line is plotted on the graph, in red. We can see that part of the line is inside of the feasible region, so it is feasible that we can make $1200 given these constraints. Since part of the line is outside of the feasible region, not all of the solutions along that line are feasible. Now on to the optimal solution. Remember that the optimal solution is ALWAYS where a line intercepts with another line, or possibly where a line meets an axis. First, we can approximate the optimal solution using the graphical method. We do this by taking a ruler, and, while keeping it parallel to the feasible line, slide it away from the origin (0, 0). Keep in mind, this is a maximization model, so we slide it away from the origin. If this was a minimization problem, we would slide it TOWARDS the origin. Here is an example where the feasible solution line has slid to the optimal solution.

6 Optimal solution line (approximated) Feasible line 40f + 30s = $1200 Now we can approximate an optimal solution. By eyeballing the graph, we can see that the optimal solution is to produce about 25 tons of fuel additive and about 20 tons of solvent base. But there is a more accurate way to find a solution. For instance it may be too close to call, compared to another possible optimal solution. You may need to solve for both, and see what the more optimal solution is. Essentially, the optimal solution is always at an intersect of two lines. In this case, it is at the intersect of the constraint lines for materials 1 and 3. The way to solve for the optimal amounts of fuel additive (f) and solvent base (s) is the same method used for determining the intersect between two lines. This method consists of three steps, but can be done in a variety of ways (ie there is no one right way to do it, but there is only one correct answer). So take the equations for the two lines, and perform the following three steps: 1. For the first equation (for instance the material 1 constraint), isolate one of the terms (for instance f ). this will give you what f equals, in terms of s 2. In the second equation (for instance the material 3 constraint), substitute the results of the first step for one of the terms (for instance f ), to solve for the other (for instance s ). this will give you the value of s 3. Back to the first equation (for instance the material 1 constraint), substitute in the value for s and solve for f

7 So here is what we did. We solved for f in the first equation. The result was f = 50 5/4s. It has to go through a couple of calculations, and these calculations will not be the same for other ones, but this happens to be the method we used. Here is the original equation: Then we isolated the f term: So we want to turn that 2/5f into just f. So we divided everything by two: Then we multiplied by five: So now we have completed the first step, because we know what f equals (although we don t yet know s, but we still have to do some more calculation). Start of step two: we put this result into the second equation, wherever we see f. Here is the original equation: So we substitute f with the results from the first phase. It is shown in the brackets:

8 Now we got rid of the brackets by multiplying the 3/5 with each of the two terms in the brackets: This step just solved the one 150/5 terms, because it reduces to 30. This just arranges the s terms on one side, and the integers on the other: is -9, so we do this calculation: We take the ¾ and 3/10 terms, and make them like terms. The lowest common denominator of 4 and 10 is 20, so we change the terms of these fractions to 20. For the ¾, we multiply the top term 3 by 5 to get 15, because we also have to multiply 5 to turn 4 into 20. For the 3/10, we multiply the top 3 by 2 to get 6, because we multiply 10 by 2 to get 20: = 9, so we get 9/20 for the fraction: Cancel out the negatives:

9 And then we know that s = 20. This is the end of the second step. For the third step, we will sub the 20 in the s terms from the FIRST equation: Which gives us Moving the 10 to the right side gives us : Therefore: Third step is done. Now we have s = 20 and f = 25. Check out the graph that we drew. Our estimate was actually pretty accurate. 20 tons of solvent base, and 25 tons of fuel additive, is the optimal amount we can produce, while satisfying all of the constraints. One last step: now we want to find out how much profit contribution we will make when we produce the optimal amount: We go back to the optimization equation that we first created in the beginning, and substitute in 20 for s and 25 for f, and we get a profit contribution of $1600.

10

Graphing Linear Equations

Graphing Linear Equations Graphing Linear Equations Question 1: What is a rectangular coordinate system? Answer 1: The rectangular coordinate system is used to graph points and equations. To create the rectangular coordinate system,

More information

Section Graphs and Lines

Section Graphs and Lines Section 1.1 - Graphs and Lines The first chapter of this text is a review of College Algebra skills that you will need as you move through the course. This is a review, so you should have some familiarity

More information

Chapter 1. Linear Equations and Straight Lines. 2 of 71. Copyright 2014, 2010, 2007 Pearson Education, Inc.

Chapter 1. Linear Equations and Straight Lines. 2 of 71. Copyright 2014, 2010, 2007 Pearson Education, Inc. Chapter 1 Linear Equations and Straight Lines 2 of 71 Outline 1.1 Coordinate Systems and Graphs 1.4 The Slope of a Straight Line 1.3 The Intersection Point of a Pair of Lines 1.2 Linear Inequalities 1.5

More information

Review for Mastery Using Graphs and Tables to Solve Linear Systems

Review for Mastery Using Graphs and Tables to Solve Linear Systems 3-1 Using Graphs and Tables to Solve Linear Systems A linear system of equations is a set of two or more linear equations. To solve a linear system, find all the ordered pairs (x, y) that make both equations

More information

Graphing by. Points. The. Plotting Points. Line by the Plotting Points Method. So let s try this (-2, -4) (0, 2) (2, 8) many points do I.

Graphing by. Points. The. Plotting Points. Line by the Plotting Points Method. So let s try this (-2, -4) (0, 2) (2, 8) many points do I. Section 5.5 Graphing the Equation of a Line Graphing by Plotting Points Suppose I asked you to graph the equation y = x +, i.e. to draw a picture of the line that the equation represents. plotting points

More information

SNAP Centre Workshop. Graphing Lines

SNAP Centre Workshop. Graphing Lines SNAP Centre Workshop Graphing Lines 45 Graphing a Line Using Test Values A simple way to linear equation involves finding test values, plotting the points on a coordinate plane, and connecting the points.

More information

Section 1.2. Graphing Linear Equations

Section 1.2. Graphing Linear Equations Graphing Linear Equations Definition of Solution, Satisfy, and Solution Set Definition of Solution, Satisfy, and Solution Set Consider the equation y = 2x 5. Let s find y when x = 3. y = 2x 5 Original

More information

Specific Objectives Students will understand that that the family of equation corresponds with the shape of the graph. Students will be able to create a graph of an equation by plotting points. In lesson

More information

Graphs of Increasing Exponential Functions

Graphs of Increasing Exponential Functions Section 5 2A: Graphs of Increasing Exponential Functions We want to determine what the graph of an exponential function y = a x looks like for all values of a > We will select a value of a > and examine

More information

Graphs of Increasing Exponential Functions

Graphs of Increasing Exponential Functions Section 5 2A: Graphs of Increasing Exponential Functions We want to determine what the graph of an exponential function y = a x looks like for all values of a > We will select a value of a > and examine

More information

College Prep Algebra II Summer Packet

College Prep Algebra II Summer Packet Name: College Prep Algebra II Summer Packet This packet is an optional review which is highly recommended before entering CP Algebra II. It provides practice for necessary Algebra I topics. Remember: When

More information

Four Types of Slope Positive Slope Negative Slope Zero Slope Undefined Slope Slope Dude will help us understand the 4 types of slope

Four Types of Slope Positive Slope Negative Slope Zero Slope Undefined Slope Slope Dude will help us understand the 4 types of slope Four Types of Slope Positive Slope Negative Slope Zero Slope Undefined Slope Slope Dude will help us understand the 4 types of slope https://www.youtube.com/watch?v=avs6c6_kvxm Direct Variation

More information

Practice Test (page 391) 1. For each line, count squares on the grid to determine the rise and the run. Use slope = rise

Practice Test (page 391) 1. For each line, count squares on the grid to determine the rise and the run. Use slope = rise Practice Test (page 91) 1. For each line, count squares on the grid to determine the rise and the. Use slope = rise 4 Slope of AB =, or 6 Slope of CD = 6 9, or Slope of EF = 6, or 4 Slope of GH = 6 4,

More information

Question 2: How do you solve a linear programming problem with a graph?

Question 2: How do you solve a linear programming problem with a graph? Question : How do you solve a linear programming problem with a graph? Now that we have several linear programming problems, let s look at how we can solve them using the graph of the system of inequalities.

More information

Chapter 1 Section 1 Lesson: Solving Linear Equations

Chapter 1 Section 1 Lesson: Solving Linear Equations Introduction Linear equations are the simplest types of equations to solve. In a linear equation, all variables are to the first power only. All linear equations in one variable can be reduced to the form

More information

Sec 4.1 Coordinates and Scatter Plots. Coordinate Plane: Formed by two real number lines that intersect at a right angle.

Sec 4.1 Coordinates and Scatter Plots. Coordinate Plane: Formed by two real number lines that intersect at a right angle. Algebra I Chapter 4 Notes Name Sec 4.1 Coordinates and Scatter Plots Coordinate Plane: Formed by two real number lines that intersect at a right angle. X-axis: The horizontal axis Y-axis: The vertical

More information

ax + by = 0. x = c. y = d.

ax + by = 0. x = c. y = d. Review of Lines: Section.: Linear Inequalities in Two Variables The equation of a line is given by: ax + by = c. for some given numbers a, b and c. For example x + y = 6 gives the equation of a line. A

More information

Section 4.3. Graphing Exponential Functions

Section 4.3. Graphing Exponential Functions Graphing Exponential Functions Graphing Exponential Functions with b > 1 Graph f x = ( ) 2 x Graphing Exponential Functions by hand. List input output pairs (see table) Input increases by 1 and output

More information

2.1 Basics of Functions and Their Graphs

2.1 Basics of Functions and Their Graphs .1 Basics of Functions and Their Graphs Section.1 Notes Page 1 Domain: (input) all the x-values that make the equation defined Defined: There is no division by zero or square roots of negative numbers

More information

Notes Lesson 3 4. Positive. Coordinate. lines in the plane can be written in standard form. Horizontal

Notes Lesson 3 4. Positive. Coordinate. lines in the plane can be written in standard form. Horizontal A, B, C are Notes Lesson 3 4 Standard Form of an Equation: Integers Ax + By = C Sometimes it is preferred that A is Positive All lines in the plane can be written in standard form. Oblique Coordinate Horizontal

More information

slope rise run Definition of Slope

slope rise run Definition of Slope The Slope of a Line Mathematicians have developed a useful measure of the steepness of a line, called the slope of the line. Slope compares the vertical change (the rise) to the horizontal change (the

More information

2.1 Transforming Linear Functions

2.1 Transforming Linear Functions 2.1 Transforming Linear Functions Before we begin looking at transforming linear functions, let s take a moment to review how to graph linear equations using slope intercept form. This will help us because

More information

Using Linear Programming for Management Decisions

Using Linear Programming for Management Decisions Using Linear Programming for Management Decisions By Tim Wright Linear programming creates mathematical models from real-world business problems to maximize profits, reduce costs and allocate resources.

More information

Pure Math 30: Explained!

Pure Math 30: Explained! www.puremath30.com 30 part i: stretches about other lines Stretches about other lines: Stretches about lines other than the x & y axis are frequently required. Example 1: Stretch the graph horizontally

More information

Summer Packet 7 th into 8 th grade. Name. Integer Operations = 2. (-7)(6)(-4) = = = = 6.

Summer Packet 7 th into 8 th grade. Name. Integer Operations = 2. (-7)(6)(-4) = = = = 6. Integer Operations Name Adding Integers If the signs are the same, add the numbers and keep the sign. 7 + 9 = 16 - + -6 = -8 If the signs are different, find the difference between the numbers and keep

More information

Mathematics for Business and Economics - I. Chapter7 Linear Inequality Systems and Linear Programming (Lecture11)

Mathematics for Business and Economics - I. Chapter7 Linear Inequality Systems and Linear Programming (Lecture11) Mathematics for Business and Economics - I Chapter7 Linear Inequality Systems and Linear Programming (Lecture11) A linear inequality in two variables is an inequality that can be written in the form Ax

More information

This assignment is due the first day of school. Name:

This assignment is due the first day of school. Name: This assignment will help you to prepare for Geometry A by reviewing some of the topics you learned in Algebra 1. This assignment is due the first day of school. You will receive homework grades for completion

More information

Chapter 12: Quadratic and Cubic Graphs

Chapter 12: Quadratic and Cubic Graphs Chapter 12: Quadratic and Cubic Graphs Section 12.1 Quadratic Graphs x 2 + 2 a 2 + 2a - 6 r r 2 x 2 5x + 8 2y 2 + 9y + 2 All the above equations contain a squared number. They are therefore called quadratic

More information

MAT 003 Brian Killough s Instructor Notes Saint Leo University

MAT 003 Brian Killough s Instructor Notes Saint Leo University MAT 003 Brian Killough s Instructor Notes Saint Leo University Success in online courses requires self-motivation and discipline. It is anticipated that students will read the textbook and complete sample

More information

GRAPHING WORKSHOP. A graph of an equation is an illustration of a set of points whose coordinates satisfy the equation.

GRAPHING WORKSHOP. A graph of an equation is an illustration of a set of points whose coordinates satisfy the equation. GRAPHING WORKSHOP A graph of an equation is an illustration of a set of points whose coordinates satisfy the equation. The figure below shows a straight line drawn through the three points (2, 3), (-3,-2),

More information

Lesson 18: There is Only One Line Passing Through a Given Point with a Given

Lesson 18: There is Only One Line Passing Through a Given Point with a Given Lesson 18: There is Only One Line Passing Through a Given Point with a Given Student Outcomes Students graph equations in the form of using information about slope and intercept. Students know that if

More information

Name Course Days/Start Time

Name Course Days/Start Time Name Course Days/Start Time Mini-Project : The Library of Functions In your previous math class, you learned to graph equations containing two variables by finding and plotting points. In this class, we

More information

1.1 - Functions, Domain, and Range

1.1 - Functions, Domain, and Range 1.1 - Functions, Domain, and Range Lesson Outline Section 1: Difference between relations and functions Section 2: Use the vertical line test to check if it is a relation or a function Section 3: Domain

More information

3.1. 3x 4y = 12 3(0) 4y = 12. 3x 4y = 12 3x 4(0) = y = x 0 = 12. 4y = 12 y = 3. 3x = 12 x = 4. The Rectangular Coordinate System

3.1. 3x 4y = 12 3(0) 4y = 12. 3x 4y = 12 3x 4(0) = y = x 0 = 12. 4y = 12 y = 3. 3x = 12 x = 4. The Rectangular Coordinate System 3. The Rectangular Coordinate System Interpret a line graph. Objectives Interpret a line graph. Plot ordered pairs. 3 Find ordered pairs that satisfy a given equation. 4 Graph lines. 5 Find x- and y-intercepts.

More information

Lesson 20: Every Line is a Graph of a Linear Equation

Lesson 20: Every Line is a Graph of a Linear Equation Student Outcomes Students know that any non vertical line is the graph of a linear equation in the form of, where is a constant. Students write the equation that represents the graph of a line. Lesson

More information

Chapter 1 Section 1 Solving Linear Equations in One Variable

Chapter 1 Section 1 Solving Linear Equations in One Variable Chapter Section Solving Linear Equations in One Variable A linear equation in one variable is an equation which can be written in the form: ax + b = c for a, b, and c real numbers with a 0. Linear equations

More information

2-4 Graphing Rational Functions

2-4 Graphing Rational Functions 2-4 Graphing Rational Functions Factor What are the zeros? What are the end behaviors? How to identify the intercepts, asymptotes, and end behavior of a rational function. How to sketch the graph of a

More information

Rational number operations can often be simplified by converting mixed numbers to improper fractions Add EXAMPLE:

Rational number operations can often be simplified by converting mixed numbers to improper fractions Add EXAMPLE: Rational number operations can often be simplified by converting mixed numbers to improper fractions Add ( 2) EXAMPLE: 2 Multiply 1 Negative fractions can be written with the negative number in the numerator

More information

Section 18-1: Graphical Representation of Linear Equations and Functions

Section 18-1: Graphical Representation of Linear Equations and Functions Section 18-1: Graphical Representation of Linear Equations and Functions Prepare a table of solutions and locate the solutions on a coordinate system: f(x) = 2x 5 Learning Outcome 2 Write x + 3 = 5 as

More information

Section 10.1 Polar Coordinates

Section 10.1 Polar Coordinates Section 10.1 Polar Coordinates Up until now, we have always graphed using the rectangular coordinate system (also called the Cartesian coordinate system). In this section we will learn about another system,

More information

Section 1.2: Points and Lines

Section 1.2: Points and Lines Section 1.2: Points and Lines Objective: Graph points and lines using x and y coordinates. Often, to get an idea of the behavior of an equation we will make a picture that represents the solutions to the

More information

MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED

MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED FOM 11 T9 GRAPHING LINEAR EQUATIONS REVIEW - 1 MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) -INTERCEPT = the point where the graph touches or crosses the -ais. It occurs when = 0. ) -INTERCEPT = the

More information

Example 1: Give the coordinates of the points on the graph.

Example 1: Give the coordinates of the points on the graph. Ordered Pairs Often, to get an idea of the behavior of an equation, we will make a picture that represents the solutions to the equation. A graph gives us that picture. The rectangular coordinate plane,

More information

3x 4y 2. 3y 4. Math 65 Weekly Activity 1 (50 points) Name: Simplify the following expressions. Make sure to use the = symbol appropriately.

3x 4y 2. 3y 4. Math 65 Weekly Activity 1 (50 points) Name: Simplify the following expressions. Make sure to use the = symbol appropriately. Math 65 Weekl Activit 1 (50 points) Name: Simplif the following epressions. Make sure to use the = smbol appropriatel. Due (1) (a) - 4 (b) ( - ) 4 () 8 + 5 6 () 1 5 5 Evaluate the epressions when = - and

More information

Mini-Project 1: The Library of Functions and Piecewise-Defined Functions

Mini-Project 1: The Library of Functions and Piecewise-Defined Functions Name Course Days/Start Time Mini-Project 1: The Library of Functions and Piecewise-Defined Functions Part A: The Library of Functions In your previous math class, you learned to graph equations containing

More information

Algebra 2 Semester 1 (#2221)

Algebra 2 Semester 1 (#2221) Instructional Materials for WCSD Math Common Finals The Instructional Materials are for student and teacher use and are aligned to the 2016-2017 Course Guides for the following course: Algebra 2 Semester

More information

Section 2.1 Graphs. The Coordinate Plane

Section 2.1 Graphs. The Coordinate Plane Section 2.1 Graphs 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 numbers to form

More information

Writing and Graphing Linear Equations. Linear equations can be used to represent relationships.

Writing and Graphing Linear Equations. Linear equations can be used to represent relationships. Writing and Graphing Linear Equations Linear equations can be used to represent relationships. Linear equation An equation whose solutions form a straight line on a coordinate plane. Collinear Points that

More information

Math-2. Lesson 3-1. Equations of Lines

Math-2. Lesson 3-1. Equations of Lines Math-2 Lesson 3-1 Equations of Lines How can an equation make a line? y = x + 1 x -4-3 -2-1 0 1 2 3 Fill in the rest of the table rule x + 1 f(x) -4 + 1-3 -3 + 1-2 -2 + 1-1 -1 + 1 0 0 + 1 1 1 + 1 2 2 +

More information

Section 2.0: Getting Started

Section 2.0: Getting Started Solving Linear Equations: Graphically Tabular/Numerical Solution Algebraically Section 2.0: Getting Started Example #1 on page 128. Solve the equation 3x 9 = 3 graphically. Intersection X=4 Y=3 We are

More information

Test Name: Chapter 3 Review

Test Name: Chapter 3 Review Test Name: Chapter 3 Review 1. For the following equation, determine the values of the missing entries. If needed, write your answer as a fraction reduced to lowest terms. 10x - 8y = 18 Note: Each column

More information

MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED DETERMINING THE INTERSECTIONS USING THE GRAPHING CALCULATOR

MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED DETERMINING THE INTERSECTIONS USING THE GRAPHING CALCULATOR 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

More information

4.3 Rational Thinking

4.3 Rational Thinking RATIONAL EXPRESSIONS & FUNCTIONS -4.3 4.3 Rational Thinking A Solidify Understanding Task The broad category of functions that contains the function!(#) = & ' is called rational functions. A rational number

More information

Unit 3, Lesson 3.1 Creating and Graphing Equations Using Standard Form

Unit 3, Lesson 3.1 Creating and Graphing Equations Using Standard Form Unit 3, Lesson 3.1 Creating and Graphing Equations Using Standard Form Imagine the path of a basketball as it leaves a player s hand and swooshes through the net. Or, imagine the path of an Olympic diver

More information

Lesson 10 Rational Functions and Equations

Lesson 10 Rational Functions and Equations Lesson 10 Rational Functions and Equations Lesson 10 Rational Functions and Equations In this lesson, you will embark on a study of rational functions. Rational functions look different because they are

More information

Important Things to Remember on the SOL

Important Things to Remember on the SOL Notes Important Things to Remember on the SOL Evaluating Expressions *To evaluate an expression, replace all of the variables in the given problem with the replacement values and use (order of operations)

More information

Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 14

Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 14 Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 14 Scan Converting Lines, Circles and Ellipses Hello everybody, welcome again

More information

Graphing Techniques. Domain (, ) Range (, ) Squaring Function f(x) = x 2 Domain (, ) Range [, ) f( x) = x 2

Graphing Techniques. Domain (, ) Range (, ) Squaring Function f(x) = x 2 Domain (, ) Range [, ) f( x) = x 2 Graphing Techniques In this chapter, we will take our knowledge of graphs of basic functions and expand our ability to graph polynomial and rational functions using common sense, zeros, y-intercepts, stretching

More information

Unit 1 and Unit 2 Concept Overview

Unit 1 and Unit 2 Concept Overview Unit 1 and Unit 2 Concept Overview Unit 1 Do you recognize your basic parent functions? Transformations a. Inside Parameters i. Horizontal ii. Shift (do the opposite of what feels right) 1. f(x+h)=left

More information

EXERCISE SET 10.2 MATD 0390 DUE DATE: INSTRUCTOR

EXERCISE SET 10.2 MATD 0390 DUE DATE: INSTRUCTOR EXERCISE SET 10. STUDENT MATD 090 DUE DATE: INSTRUCTOR You have studied the method known as "completing the square" to solve quadratic equations. Another use for this method is in transforming the equation

More information

Rational functions, like rational numbers, will involve a fraction. We will discuss rational functions in the form:

Rational functions, like rational numbers, will involve a fraction. We will discuss rational functions in the form: Name: Date: Period: Chapter 2: Polynomial and Rational Functions Topic 6: Rational Functions & Their Graphs Rational functions, like rational numbers, will involve a fraction. We will discuss rational

More information

MA 1128: Lecture 02 1/22/2018

MA 1128: Lecture 02 1/22/2018 MA 1128: Lecture 02 1/22/2018 Exponents Scientific Notation 1 Exponents Exponents are used to indicate how many copies of a number are to be multiplied together. For example, I like to deal with the signs

More information

Meeting 1 Introduction to Functions. Part 1 Graphing Points on a Plane (REVIEW) Part 2 What is a function?

Meeting 1 Introduction to Functions. Part 1 Graphing Points on a Plane (REVIEW) Part 2 What is a function? Meeting 1 Introduction to Functions Part 1 Graphing Points on a Plane (REVIEW) A plane is a flat, two-dimensional surface. We describe particular locations, or points, on a plane relative to two number

More information

We want to determine what the graph of an exponential function y = a x looks like for all values of a such that 0 < a < 1

We want to determine what the graph of an exponential function y = a x looks like for all values of a such that 0 < a < 1 Section 5 2B: Graphs of Decreasing Eponential Functions We want to determine what the graph of an eponential function y = a looks like for all values of a such that 0 < a < We will select a value of a

More information

Math 2 Coordinate Geometry Part 3 Inequalities & Quadratics

Math 2 Coordinate Geometry Part 3 Inequalities & Quadratics Math 2 Coordinate Geometry Part 3 Inequalities & Quadratics 1 DISTANCE BETWEEN TWO POINTS - REVIEW To find the distance between two points, use the Pythagorean theorem. The difference between x 1 and x

More information

STRAIGHT LINE GRAPHS THE COORDINATES OF A POINT. The coordinates of any point are written as an ordered pair (x, y)

STRAIGHT LINE GRAPHS THE COORDINATES OF A POINT. The coordinates of any point are written as an ordered pair (x, y) THE COORDINATES OF A POINT STRAIGHT LINE GRAPHS The coordinates of any point are written as an ordered pair (x, y) Point P in the diagram has coordinates (2, 3). Its horizontal distance along the x axis

More information

Chapter 2. Polynomial and Rational Functions. 2.2 Quadratic Functions

Chapter 2. Polynomial and Rational Functions. 2.2 Quadratic Functions Chapter 2 Polynomial and Rational Functions 2.2 Quadratic Functions 1 /27 Chapter 2 Homework 2.2 p298 1, 5, 17, 31, 37, 41, 43, 45, 47, 49, 53, 55 2 /27 Chapter 2 Objectives Recognize characteristics of

More information

Section 7D Systems of Linear Equations

Section 7D Systems of Linear Equations Section 7D Systems of Linear Equations Companies often look at more than one equation of a line when analyzing how their business is doing. For example a company might look at a cost equation and a profit

More information

Domain: The domain of f is all real numbers except those values for which Q(x) =0.

Domain: The domain of f is all real numbers except those values for which Q(x) =0. Math 1330 Section.3.3: Rational Functions Definition: A rational function is a function that can be written in the form P() f(), where f and g are polynomials. Q() The domain of the rational function such

More information

Integer Operations. Summer Packet 7 th into 8 th grade 1. Name = = = = = 6.

Integer Operations. Summer Packet 7 th into 8 th grade 1. Name = = = = = 6. Summer Packet 7 th into 8 th grade 1 Integer Operations Name Adding Integers If the signs are the same, add the numbers and keep the sign. 7 + 9 = 16-2 + -6 = -8 If the signs are different, find the difference

More information

Algebra Unit 2: Linear Functions Notes. Slope Notes. 4 Types of Slope. Slope from a Formula

Algebra Unit 2: Linear Functions Notes. Slope Notes. 4 Types of Slope. Slope from a Formula Undefined Slope Notes Types of Slope Zero Slope Slope can be described in several ways: Steepness of a line Rate of change rate of increase or decrease Rise Run Change (difference) in y over change (difference)

More information

Watkins Mill High School. Algebra 2. Math Challenge

Watkins Mill High School. Algebra 2. Math Challenge Watkins Mill High School Algebra 2 Math Challenge "This packet will help you prepare for Algebra 2 next fall. It will be collected the first week of school. It will count as a grade in the first marking

More information

2-3 Graphing Rational Functions

2-3 Graphing Rational Functions 2-3 Graphing Rational Functions Factor What are the end behaviors of the Graph? Sketch a graph How to identify the intercepts, asymptotes and end behavior of a rational function. How to sketch the graph

More information

Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 2 Section 12 Variables and Expressions

Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 2 Section 12 Variables and Expressions Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 2 Please watch Section 12 of this DVD before working these problems. The DVD is located at: http://www.mathtutordvd.com/products/item67.cfm

More information

In math, the rate of change is called the slope and is often described by the ratio rise

In math, the rate of change is called the slope and is often described by the ratio rise Chapter 3 Equations of Lines Sec. Slope The idea of slope is used quite often in our lives, however outside of school, it goes by different names. People involved in home construction might talk about

More information

Chapter 4 Graphing Linear Equations and Functions

Chapter 4 Graphing Linear Equations and Functions Chapter 4 Graphing Linear Equations and Functions 4.1 Coordinates and Scatter plots on the calculator: On the graph paper below please put the following items: x and y axis, origin,quadrant numbering system,

More information

NOTES Linear Equations

NOTES Linear Equations NOTES Linear Equations Linear Parent Function Linear Parent Function the equation that all other linear equations are based upon (y = x) Horizontal and Vertical Lines (HOYY VUXX) V vertical line H horizontal

More information

WEEK 4 REVIEW. Graphing Systems of Linear Inequalities (3.1)

WEEK 4 REVIEW. Graphing Systems of Linear Inequalities (3.1) WEEK 4 REVIEW Graphing Systems of Linear Inequalities (3.1) Linear Programming Problems (3.2) Checklist for Exam 1 Review Sample Exam 1 Graphing Linear Inequalities Graph the following system of inequalities.

More information

Learning Log Title: CHAPTER 3: PORTIONS AND INTEGERS. Date: Lesson: Chapter 3: Portions and Integers

Learning Log Title: CHAPTER 3: PORTIONS AND INTEGERS. Date: Lesson: Chapter 3: Portions and Integers Chapter 3: Portions and Integers CHAPTER 3: PORTIONS AND INTEGERS Date: Lesson: Learning Log Title: Date: Lesson: Learning Log Title: Chapter 3: Portions and Integers Date: Lesson: Learning Log Title:

More information

(Refer Slide Time: 00:03:51)

(Refer Slide Time: 00:03:51) Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 17 Scan Converting Lines, Circles and Ellipses Hello and welcome everybody

More information

1.) ( ) Step 1: Factor the numerator and the denominator. Find the domain. is in lowest terms.

1.) ( ) Step 1: Factor the numerator and the denominator. Find the domain. is in lowest terms. GP3-HW11 College Algebra Sketch the graph of each rational function. 1.) Step 1: Factor the numerator and the denominator. Find the domain. { } Step 2: Rewrite in lowest terms. The rational function is

More information

Math 1330 Section : Rational Functions Definition: A rational function is a function that can be written in the form f ( x ), where

Math 1330 Section : Rational Functions Definition: A rational function is a function that can be written in the form f ( x ), where 2.3: Rational Functions P( x ) Definition: A rational function is a function that can be written in the form f ( x ), where Q( x ) and Q are polynomials, consists of all real numbers x such that You will

More information

List of Topics for Analytic Geometry Unit Test

List of Topics for Analytic Geometry Unit Test List of Topics for Analytic Geometry Unit Test 1. Finding Slope 2. Rule of 4 (4 forms of a line) Graph, Table of Values, Description, Equation 3. Find the Equations- Vertical and Horizontal Lines 4. Standard

More information

Unit 1 Integers, Fractions & Order of Operations

Unit 1 Integers, Fractions & Order of Operations Unit 1 Integers, Fractions & Order of Operations In this unit I will learn Date: I have finished this work! I can do this on the test! Operations with positive and negative numbers The order of operations

More information

Name: NOTES 5: LINEAR EQUATIONS AND THEIR GRAPHS. Date: Period: Mrs. Nguyen s Initial: LESSON 5.1 RATE OF CHANGE AND SLOPE. A. Finding rates of change

Name: NOTES 5: LINEAR EQUATIONS AND THEIR GRAPHS. Date: Period: Mrs. Nguyen s Initial: LESSON 5.1 RATE OF CHANGE AND SLOPE. A. Finding rates of change NOTES : LINEAR EQUATIONS AND THEIR GRAPHS Name: Date: Period: Mrs. Nguen s Initial: LESSON. RATE OF CHANGE AND SLOPE A. Finding rates of change vertical change Rate of change = = change in x The rate of

More information

Algebra I Notes Linear Equations and Inequalities in Two Variables Unit 04c

Algebra I Notes Linear Equations and Inequalities in Two Variables Unit 04c Big Idea: Describe the similarities and differences between equations and inequalities including solutions and graphs. Skill: graph linear equations and find possible solutions to those equations using

More information

Maths Revision Worksheet: Algebra I Week 1 Revision 5 Problems per night

Maths Revision Worksheet: Algebra I Week 1 Revision 5 Problems per night 2 nd Year Maths Revision Worksheet: Algebra I Maths Revision Worksheet: Algebra I Week 1 Revision 5 Problems per night 1. I know how to add and subtract positive and negative numbers. 2. I know how to

More information

Section 3.1 Graphing Using the Rectangular Coordinate System

Section 3.1 Graphing Using the Rectangular Coordinate System Objectives Section 3.1 Graphing Using the Rectangular Coordinate System n Construct a rectangular coordinate system n Plot ordered pairs and determine the coordinates of a point n Graph paired data n Read

More information

Chpt 1. Functions and Graphs. 1.1 Graphs and Graphing Utilities 1 /19

Chpt 1. Functions and Graphs. 1.1 Graphs and Graphing Utilities 1 /19 Chpt 1 Functions and Graphs 1.1 Graphs and Graphing Utilities 1 /19 Chpt 1 Homework 1.1 14, 18, 22, 24, 28, 42, 46, 52, 54, 56, 78, 79, 80, 82 2 /19 Objectives Functions and Graphs Plot points in the rectangular

More information

Name: Tutor s

Name: Tutor s Name: Tutor s Email: Bring a couple, just in case! Necessary Equipment: Black Pen Pencil Rubber Pencil Sharpener Scientific Calculator Ruler Protractor (Pair of) Compasses 018 AQA Exam Dates Paper 1 4

More information

PreCalculus 300. Algebra 2 Review

PreCalculus 300. Algebra 2 Review PreCalculus 00 Algebra Review Algebra Review The following topics are a review of some of what you learned last year in Algebra. I will spend some time reviewing them in class. You are responsible for

More information

This is called the vertex form of the quadratic equation. To graph the equation

This is called the vertex form of the quadratic equation. To graph the equation Name Period Date: Topic: 7-5 Graphing ( ) Essential Question: What is the vertex of a parabola, and what is its axis of symmetry? Standard: F-IF.7a Objective: Graph linear and quadratic functions and show

More information

Advanced Functions Unit 4

Advanced Functions Unit 4 Advanced Functions Unit 4 Absolute Value Functions Absolute Value is defined by:, 0, if if 0 0 - (), if 0 The graph of this piecewise function consists of rays, is V-shaped and opens up. To the left of

More information

Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 2 Section 15 Dividing Expressions

Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 2 Section 15 Dividing Expressions Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 2 Please watch Section 15 of this DVD before working these problems. The DVD is located at: http://www.mathtutordvd.com/products/item67.cfm

More information

Forms of Linear Equations

Forms of Linear Equations 6. 1-6.3 Forms of Linear Equations Name Sec 6.1 Writing Linear Equations in Slope-Intercept Form *Recall that slope intercept form looks like y = mx + b, where m = slope and b = y=intercept 1) Writing

More information

Welcome to Solving Systems Using the TI-Nspire

Welcome to Solving Systems Using the TI-Nspire Welcome to Solving Systems Using the TI-Nspire Presented by: Julie Speelman Madeira High School Cincinnati, Ohio jspeelman@madeiracityschools.org Please login to your calculators. Press the button. Select

More information

5.7 Solving Linear Inequalities

5.7 Solving Linear Inequalities 5.7 Solving Linear Inequalities Objectives Inequality Symbols Graphing Inequalities both simple & compound Understand a solution set for an inequality Solving & Graphing a Simple Linear Inequality Solving

More information

Lesson 08 Linear Programming

Lesson 08 Linear Programming Lesson 08 Linear Programming A mathematical approach to determine optimal (maximum or minimum) solutions to problems which involve restrictions on the variables involved. 08 - Linear Programming Applications

More information

SOLVING SYSTEMS OF EQUATIONS

SOLVING SYSTEMS OF EQUATIONS SOLVING SYSTEMS OF EQUATIONS GRAPHING System of Equations: 2 linear equations that we try to solve at the same time. An ordered pair is a solution to a system if it makes BOTH equations true. Steps to

More information

SLOPE A MEASURE OF STEEPNESS through 7.1.5

SLOPE A MEASURE OF STEEPNESS through 7.1.5 SLOPE A MEASURE OF STEEPNESS 7.1. through 7.1.5 Students have used the equation = m + b throughout this course to graph lines and describe patterns. When the equation is written in -form, the m is the

More information