Systems of Equations and Inequalities. Copyright Cengage Learning. All rights reserved.
|
|
- Buck Stokes
- 5 years ago
- Views:
Transcription
1 5 Systems of Equations and Inequalities Copyright Cengage Learning. All rights reserved.
2 5.5 Systems of Inequalities Copyright Cengage Learning. All rights reserved.
3 Objectives Graphing an Inequality Systems of Inequalities Systems of Linear Inequalities Application: Feasible Regions 3
4 Graphing an Inequality 4
5 Graphing an Inequality We begin by considering the graph of a single inequality. We already know that the graph of y = x 2, for example, is the parabola in Figure 1. If we replace the equal sign by the symbol, we obtain the inequality y x 2 Figure 1 5
6 Graphing an Inequality Its graph consists of not just the parabola in Figure 1, but also every point whose y-coordinate is larger than x 2. We indicate the solution in Figure 2(a) by shading the points above the parabola. y x 2 Figure 2(a) 6
7 Graphing an Inequality Similarly, the graph of y x 2 in Figure 2(b) consists of all points on and below the parabola. y x 2 Figure 2(b) 7
8 Graphing an Inequality However, the graphs of y > x 2 and y < x 2 do not include the points on the parabola itself, as indicated by the dashed curves in Figures 2(c) and 2(d). y > x 2 Figure 2(c) y < x 2 Figure 2(d) 8
9 Graphing an Inequality The graph of an inequality, in general, consists of a region in the plane whose boundary is the graph of the equation obtained by replacing the inequality sign (,, >, or < ) with an equal sign. 9
10 Graphing an Inequality To determine which side of the graph gives the solution set of the inequality, we need only check test points. 10
11 Example 1 Graphs of Inequalities Graph each inequality. (a) x 2 + y 2 < 25 (b) x + 2y 5 Solution: (a) Graph the equation. The graph of the equation x 2 + y 2 = 25 is a circle of radius 5 centered at the origin. 11
12 Example 1 Solution cont d The points on the circle itself do not satisfy the inequality because it is of the form <, so we graph the circle with a dashed curve, as shown in Figure 3. Graph of x 2 + y 2 < 25 Figure 3 12
13 Example 1 Solution cont d Graph the inequality. To determine whether the inside or the outside of the circle satisfies the inequality, we use the test points (0, 0) on the inside and (6, 0) on the outside. To do this, we substitute the coordinates of each point into the inequality and check whether the result satisfies the inequality. 13
14 Example 1 Solution cont d Note that any point inside or outside the circle can serve as a test point. We have chosen these points for simplicity. Our check shows that the points inside the circle satisfy the inequality. A graph of the inequality is shown in Figure 3. Graph of x 2 + y 2 < 25 Figure 3 14
15 Example 1 Solution cont d (b) Graph the equation. We first graph the equation of x + 2y = 5. The graph is the line shown in Figure 4. Graph of x + 2y 5 Figure 4 15
16 Example 1 Solution cont d Graph the inequality. Let s use the test points (0, 0) and (5, 5) on either sides of the line. Our check shows that the points above the line satisfy the inequality. 16
17 Example 1 Solution cont d A graph of the inequality is shown in Figure 4. Graph of x + 2y 5 Figure 4 17
18 Graphing an Inequality IMPORTANT!! We can write the inequality in Example 1 as From this form of the inequality we see that the solution consists of the points with y-values on or above the line. So the graph of the inequality is the region above the line. 18
19 Systems of Inequalities 19
20 Systems of Inequalities We now consider systems of inequalities. The solution set of a system of inequalities in two variables is the set of all points in the coordinate plane that satisfy every inequality in the system. The graph of a system of inequalities is the graph of the solution set. 20
21 Systems of Inequalities To solve a system of inequalities, we use the following guidelines. 21
22 Example 2 A System of Two Inequalities Graph the solution of the system of inequalities, and label its vertices (vertices are the intersection points). x 2 + y 2 < 25 x + 2y 5 Solution: These are the two inequalities of Example 1. Here we want to graph only those points that simultaneously satisfy both inequalities. 22
23 Example 2 Solution cont d Graph each inequality. In Figure 5(a) we graph the solutions of the two inequalities on the same axes (in different colors). x 2 + y 2 < 25 x + 2y 5 Figure 5(a) 23
24 Example 2 Solution cont d Graph the solution of the system. The solution of the system of inequalities is the intersection of the two graphs. This is the region where the two regions overlap, which is the purple region graphed in Figure 5(b). x 2 + y 2 < 25 x + 2y 5 Figure 5(b) 24
25 Example 2 Solution cont d Find the Vertices. The points ( 3, 4) and (5, 0) in Figure 5(b) are the vertices of the solution set. They are obtained by solving the system of equations x 2 + y 2 = 25 x + 2y = 5 We solve this system of equations by substitution. 25
26 Example 2 Solution cont d Solving for x in the second equation gives x = 5 2y, and substituting this into the first equation gives (5 2y) 2 + y 2 = 25 Substitute x = 5 2y (25 20y + 4y 2 ) + y 2 = 25 Expand 20y + 5y 2 = 0 Simplify Thus y = 0 or y = 4. 5y(4 y) = 0 Factor 26
27 Example 2 Solution cont d When y = 0, we have x = 5 2(0) = 5, and when y = 4, we have x = 5 2(4) = 3. So the points of intersection of these curves are (5, 0) and ( 3, 4). Note that in this case the vertices are not part of the solution set, since they don t satisfy the inequality x 2 + y 2 < 25 (so they are graphed as open circles in the figure). They simply show where the corners of the solution set lie. 27
28 Systems of Linear Inequalities 28
29 Systems of Linear Inequalities An inequality is linear if it can be put into one of the following forms: ax + by c ax + by c ax + by > c ax + by < c In the next example we graph the solution set of a system of linear inequalities. 29
30 Example 3 A System of Four Linear Inequalities Graph the solution set of the system, and label its vertices. x + 3y 12 x + y 8 x 0 y 0 30
31 Example 3 Solution Graph each inequality. In Figure 6 we first graph the lines given by the equations that correspond to each inequality. To determine the graphs of the first two inequalities, we need to check only one test point. Answer is the entire shaded area. Figure 6(a) Figure 6(b) 31
32 Example 3 Solution cont d For simplicity let s use the point (0, 0). Since (0, 0) is below the line x + 3y = 12, our check shows that the region on or below the line must satisfy the inequality. 32
33 Example 3 Solution cont d Likewise, since (0, 0) is below the line x + y = 8, our check shows that the region on or below this line must satisfy the inequality. The inequalities x 0 and y 0 say that x and y are nonnegative. 33
34 Example 3 Solution cont d These regions are sketched in Figure 6(a). Figure 6(a) 34
35 Example 3 Solution cont d Graph the solution of the system. The solution of the system of inequalities is the intersection of the graphs. This is the purple region graphed in Figure 6(b). Figure 6(b) 35
36 Example 3 Solution cont d Find the Vertices. The coordinates of each vertex are obtained by simultaneously solving the equations of the lines that intersect at that vertex. From the system x + 3y = 12 x + y = 8 we get the vertex (6, 2). The origin (0, 0) is also clearly a vertex. The other two vertices are at the x- and y-intercepts of the corresponding lines: (8, 0) and (0, 4). In this case all the vertices are part of the solution set. 36
37 Systems of Linear Inequalities A region in the plane is called bounded if it can be enclosed in a (sufficiently large) circle. A region that is not bounded is called unbounded. 37
38 Systems of Linear Inequalities For example, the region graphed in Figure 8 is bounded because it can be enclosed in a circle, as illustrated in Figure 10(a). Figure 8 A bounded region can be enclosed in a circle. Figure 10(a) 38
39 Systems of Linear Inequalities But the regions graphed in Figure 4 and 10b are unbounded, because we cannot enclose either of them in a circle as illustrated. The shaded areas go on forever, so you can t enclose them in a circle. Graph of x + 2y 5 Figure 4 An unbounded region cannot be enclosed in a circle. Figure 10(b) 39
40 Example: Graph the solution of the system of inequalities, and label its vertices, and determine whether the solution set is bounded. 40
41 Example: This is the same problem we just did, but there is one small change on it. Can you see which inequality changed? How does this affect the shading on the graph? 41
42 Example (like #63 on assignment) Graph the solution of the system of inequalities, and label its vertices, and determine whether the solution set is bounded. (This example is just like #63.) 42
43 Assignment: Section 5.5: problems 1-21 odd, odd 43
6.5. SYSTEMS OF INEQUALITIES
6.5. SYSTEMS OF INEQUALITIES What You Should Learn Sketch the graphs of inequalities in two variables. Solve systems of inequalities. Use systems of inequalities in two variables to model and solve real-life
More informationReview 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 informationGraphing Linear Inequalities in Two Variables.
Many applications of mathematics involve systems of inequalities rather than systems of equations. We will discuss solving (graphing) a single linear inequality in two variables and a system of linear
More informationLINEAR PROGRAMMING: A GEOMETRIC APPROACH. Copyright Cengage Learning. All rights reserved.
3 LINEAR PROGRAMMING: A GEOMETRIC APPROACH Copyright Cengage Learning. All rights reserved. 3.1 Graphing Systems of Linear Inequalities in Two Variables Copyright Cengage Learning. All rights reserved.
More informationChapter 3 Linear Programming: A Geometric Approach
Chapter 3 Linear Programming: A Geometric Approach Section 3.1 Graphing Systems of Linear Inequalities in Two Variables y 4x + 3y = 12 4 3 4 x 3 y 12 x y 0 x y = 0 2 1 P(, ) 12 12 7 7 1 1 2 3 x We ve seen
More informationMathematics 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 informationSection 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 informationWHAT YOU SHOULD LEARN
GRAPHS OF EQUATIONS WHAT YOU SHOULD LEARN Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs of equations. Find equations of and sketch graphs of
More information9.1 Linear Inequalities in Two Variables Date: 2. Decide whether to use a solid line or dotted line:
9.1 Linear Inequalities in Two Variables Date: Key Ideas: Example Solve the inequality by graphing 3y 2x 6. steps 1. Rearrange the inequality so it s in mx ± b form. Don t forget to flip the inequality
More informationQuestion 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 informationFinite Math Linear Programming 1 May / 7
Linear Programming Finite Math 1 May 2017 Finite Math Linear Programming 1 May 2017 1 / 7 General Description of Linear Programming Finite Math Linear Programming 1 May 2017 2 / 7 General Description of
More information2.6: Solving Systems of Linear Inequalities
Quick Review 2.6: Solving Systems of Linear Inequalities = - What is the difference between an equation and an inequality? Which one is shaded? Inequality - When is the line solid?, - When is the line
More informationWe have already studied equations of the line. There are several forms:
Chapter 13-Coordinate Geometry extended. 13.1 Graphing equations We have already studied equations of the line. There are several forms: slope-intercept y = mx + b point-slope y - y1=m(x - x1) standard
More informationnotes13.1inclass May 01, 2015
Chapter 13-Coordinate Geometry extended. 13.1 Graphing equations We have already studied equations of the line. There are several forms: slope-intercept y = mx + b point-slope y - y1=m(x - x1) standard
More information1.8 Coordinate Geometry. Copyright Cengage Learning. All rights reserved.
1.8 Coordinate Geometry Copyright Cengage Learning. All rights reserved. Objectives The Coordinate Plane The Distance and Midpoint Formulas Graphs of Equations in Two Variables Intercepts Circles Symmetry
More informationProperties of Quadratic functions
Name Today s Learning Goals: #1 How do we determine the axis of symmetry and vertex of a quadratic function? Properties of Quadratic functions Date 5-1 Properties of a Quadratic Function A quadratic equation
More informationWe have already studied equations of the line. There are several forms:
Chapter 13-Coordinate Geometry extended. 13.1 Graphing equations We have already studied equations of the line. There are several forms: slope-intercept y = mx + b point-slope y - y1=m(x - x1) standard
More informationUNIT 3 EXPRESSIONS AND EQUATIONS Lesson 3: Creating Quadratic Equations in Two or More Variables
Guided Practice Example 1 Find the y-intercept and vertex of the function f(x) = 2x 2 + x + 3. Determine whether the vertex is a minimum or maximum point on the graph. 1. Determine the y-intercept. The
More information11.3 The Tangent Line Problem
11.3 The Tangent Line Problem Copyright Cengage Learning. All rights reserved. What You Should Learn Understand the tangent line problem. Use a tangent line to approximate the slope of a graph at a point.
More informationWEEK 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 informationPolynomial and Rational Functions. Copyright Cengage Learning. All rights reserved.
2 Polynomial and Rational Functions Copyright Cengage Learning. All rights reserved. 2.1 Quadratic Functions Copyright Cengage Learning. All rights reserved. What You Should Learn Analyze graphs of quadratic
More informationTopic 6: Calculus Integration Volume of Revolution Paper 2
Topic 6: Calculus Integration Standard Level 6.1 Volume of Revolution Paper 1. Let f(x) = x ln(4 x ), for < x
More informationSection 3.1 Graphing Systems of Linear Inequalities in Two Variables
Section 3.1 Graphing Systems of Linear Inequalities in Two Variables Procedure for Graphing Linear Inequalities: 1. Draw the graph of the equation obtained for the given inequality by replacing the inequality
More informationx Boundary Intercepts Test (0,0) Conclusion 2x+3y=12 (0,4), (6,0) 0>12 False 2x-y=2 (0,-2), (1,0) 0<2 True
MATH 34 (Finite Mathematics or Business Math I) Lecture Notes MATH 34 Module 3 Notes: SYSTEMS OF INEQUALITIES & LINEAR PROGRAMMING 3. GRAPHING SYSTEMS OF INEQUALITIES Simple Systems of Linear Inequalities
More informationSection 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 informationMath 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 informationConics, Parametric Equations, and Polar Coordinates. Copyright Cengage Learning. All rights reserved.
10 Conics, Parametric Equations, and Polar Coordinates Copyright Cengage Learning. All rights reserved. 10.5 Area and Arc Length in Polar Coordinates Copyright Cengage Learning. All rights reserved. Objectives
More informationSystems of Inequalities and Linear Programming 5.7 Properties of Matrices 5.8 Matrix Inverses
5 5 Systems and Matrices Systems and Matrices 5.6 Systems of Inequalities and Linear Programming 5.7 Properties of Matrices 5.8 Matrix Inverses Sections 5.6 5.8 2008 Pearson Addison-Wesley. All rights
More informationCURVE SKETCHING EXAM QUESTIONS
CURVE SKETCHING EXAM QUESTIONS Question 1 (**) a) Express f ( x ) in the form ( ) 2 f x = x + 6x + 10, x R. f ( x) = ( x + a) 2 + b, where a and b are integers. b) Describe geometrically the transformations
More informationMath 1313 Prerequisites/Test 1 Review
Math 1313 Prerequisites/Test 1 Review Test 1 (Prerequisite Test) is the only exam that can be done from ANYWHERE online. Two attempts. See Online Assignments in your CASA account. Note the deadline too.
More informationConics, Parametric Equations, and Polar Coordinates. Copyright Cengage Learning. All rights reserved.
10 Conics, Parametric Equations, and Polar Coordinates Copyright Cengage Learning. All rights reserved. 10.5 Area and Arc Length in Polar Coordinates Copyright Cengage Learning. All rights reserved. Objectives
More informationEach point P in the xy-plane corresponds to an ordered pair (x, y) of real numbers called the coordinates of P.
Lecture 7, Part I: Section 1.1 Rectangular Coordinates Rectangular or Cartesian coordinate system Pythagorean theorem Distance formula Midpoint formula Lecture 7, Part II: Section 1.2 Graph of Equations
More informationPure Math 30: Explained!
www.puremath30.com 5 Conics Lesson Part I - Circles Circles: The standard form of a circle is given by the equation (x - h) +(y - k) = r, where (h, k) is the centre of the circle and r is the radius. Example
More informationSolve the following system of equations. " 2x + 4y = 8 # $ x 3y = 1. 1 cont d. You try:
1 Solve the following system of equations. " 2x + 4y = 8 # $ x 3y = 1 Method 1: Substitution 1. Solve for x in the second equation. 1 cont d Method 3: Eliminate y 1. Multiply first equation by 3 and second
More information10-2 Circles. Warm Up Lesson Presentation Lesson Quiz. Holt Algebra2 2
10-2 Circles Warm Up Lesson Presentation Lesson Quiz Holt Algebra2 2 Warm Up Find the slope of the line that connects each pair of points. 1. (5, 7) and ( 1, 6) 1 6 2. (3, 4) and ( 4, 3) 1 Warm Up Find
More informationThree Dimensional Geometry. Linear Programming
Three Dimensional Geometry Linear Programming A plane is determined uniquely if any one of the following is known: The normal to the plane and its distance from the origin is given, i.e. equation of a
More informationx + 2 = 0 or Our limits of integration will apparently be a = 2 and b = 4.
QUIZ ON CHAPTER 6 - SOLUTIONS APPLICATIONS OF INTEGRALS; MATH 15 SPRING 17 KUNIYUKI 15 POINTS TOTAL, BUT 1 POINTS = 1% Note: The functions here are continuous on the intervals of interest. This guarantees
More informationLinear Programming: A Geometric Approach
Chapter 3 Linear Programming: A Geometric Approach 3.1 Graphing Systems of Linear Inequalities in Two Variables The general form for a line is ax + by + c =0. The general form for a linear inequality is
More informationThis 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 informationUNIT 6 MODELLING DECISION PROBLEMS (LP)
UNIT 6 MODELLING DECISION This unit: PROBLEMS (LP) Introduces the linear programming (LP) technique to solve decision problems 1 INTRODUCTION TO LINEAR PROGRAMMING A Linear Programming model seeks to maximize
More informationChapter 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 information8.2 Graph and Write Equations of Parabolas
8.2 Graph and Write Equations of Parabolas Where is the focus and directrix compared to the vertex? How do you know what direction a parabola opens? How do you write the equation of a parabola given the
More informationFind the specific function values. Complete parts (a) through (d) below. f (x,y,z) = x y y 2 + z = (Simplify your answer.) ID: 14.1.
. Find the specific function values. Complete parts (a) through (d) below. f (x,y,z) = x y y 2 + z 2 (a) f(2, 4,5) = (b) f 2,, 3 9 = (c) f 0,,0 2 (d) f(4,4,00) = = ID: 4..3 2. Given the function f(x,y)
More informationChapter 10. Exploring Conic Sections
Chapter 10 Exploring Conic Sections Conics A conic section is a curve formed by the intersection of a plane and a hollow cone. Each of these shapes are made by slicing the cone and observing the shape
More informationLecture 14. Resource Allocation involving Continuous Variables (Linear Programming) 1.040/1.401/ESD.018 Project Management.
1.040/1.401/ESD.018 Project Management Lecture 14 Resource Allocation involving Continuous Variables (Linear Programming) April 2, 2007 Samuel Labi and Fred Moavenzadeh Massachusetts Institute of Technology
More informationSection 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 information9.3 Hyperbolas and Rotation of Conics
9.3 Hyperbolas and Rotation of Conics Copyright Cengage Learning. All rights reserved. What You Should Learn Write equations of hyperbolas in standard form. Find asymptotes of and graph hyperbolas. Use
More informationLesson 8 - Practice Problems
Lesson 8 - Practice Problems Section 8.1: A Case for the Quadratic Formula 1. For each quadratic equation below, show a graph in the space provided and circle the number and type of solution(s) to that
More information30. Constrained Optimization
30. Constrained Optimization The graph of z = f(x, y) is represented by a surface in R 3. Normally, x and y are chosen independently of one another so that one may roam over the entire surface of f (within
More informationUNIT 3B CREATING AND GRAPHING EQUATIONS Lesson 4: Solving Systems of Equations Instruction
Prerequisite Skills This lesson requires the use of the following skills: graphing multiple equations on a graphing calculator graphing quadratic equations graphing linear equations Introduction A system
More informationSection 4.5 Linear Inequalities in Two Variables
Section 4.5 Linear Inequalities in Two Variables Department of Mathematics Grossmont College February 25, 203 4.5 Linear Inequalities in Two Variables Learning Objectives: Graph linear inequalities in
More informationALGEBRA II UNIT X: Conic Sections Unit Notes Packet
Name: Period: ALGEBRA II UNIT X: Conic Sections Unit Notes Packet Algebra II Unit 10 Plan: This plan is subject to change at the teacher s discretion. Section Topic Formative Work Due Date 10.3 Circles
More informationQuadratic Functions. *These are all examples of polynomial functions.
Look at: f(x) = 4x-7 f(x) = 3 f(x) = x 2 + 4 Quadratic Functions *These are all examples of polynomial functions. Definition: Let n be a nonnegative integer and let a n, a n 1,..., a 2, a 1, a 0 be real
More informationFunctions of Several Variables
Chapter 3 Functions of Several Variables 3.1 Definitions and Examples of Functions of two or More Variables In this section, we extend the definition of a function of one variable to functions of two or
More informationLecture 5. If, as shown in figure, we form a right triangle With P1 and P2 as vertices, then length of the horizontal
Distance; Circles; Equations of the form Lecture 5 y = ax + bx + c In this lecture we shall derive a formula for the distance between two points in a coordinate plane, and we shall use that formula to
More information3.1. 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 informationPolynomial and Rational Functions. Copyright Cengage Learning. All rights reserved.
2 Polynomial and Rational Functions Copyright Cengage Learning. All rights reserved. 2.7 Graphs of Rational Functions Copyright Cengage Learning. All rights reserved. What You Should Learn Analyze and
More informationCHAPTER 4 Linear Programming with Two Variables
CHAPTER 4 Linear Programming with Two Variables In this chapter, we will study systems of linear inequalities. They are similar to linear systems of equations, but have inequalitites instead of equalities.
More informationSketching graphs of polynomials
Sketching graphs of polynomials We want to draw the graphs of polynomial functions y = f(x). The degree of a polynomial in one variable x is the highest power of x that remains after terms have been collected.
More informationAppendix F: Systems of Inequalities
A0 Appendi F Sstems of Inequalities Appendi F: Sstems of Inequalities F. Solving Sstems of Inequalities The Graph of an Inequalit The statements < and are inequalities in two variables. An ordered pair
More informationGraphing Method. Graph of x + y < > y 10. x
Graphing Method Eample: Graph the inequalities on the same plane: + < 6 and 2 - > 4. Before we graph them simultaneousl, let s look at them separatel. 10-10 10 Graph of + < 6. ---> -10 Graphing Method
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Pre-Calculus Mid Term Review. January 2014 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the graph of the function f, plotted with a solid
More informationSection 3.1 Graphing Systems of Linear Inequalities in Two Variables
Section 3.1 Graphing Systems of Linear Inequalities in Two Variables Procedure for Graphing Linear Inequalities: 1. Draw the graph of the equation obtained for the given inequality by replacing the inequality
More informationFunctions of Several Variables
Chapter 3 Functions of Several Variables 3.1 Definitions and Examples of Functions of two or More Variables In this section, we extend the definition of a function of one variable to functions of two or
More informationExample Graph the inequality 2x-3y 12. Answer - start with the = part. Graph the line 2x - 3y = 12. Linear Programming: A Geometric Approach
Linear Programming: A Geometric Approach 3.1: Graphing Systems of Linear Inequalities in Two Variables Example Graph the inequality 2x-3y 12. Answer - start with the = part. Graph the line 2x - 3y = 12.
More informationSolved Examples. Parabola with vertex as origin and symmetrical about x-axis. We will find the area above the x-axis and double the area.
Solved Examples Example 1: Find the area common to the curves x 2 + y 2 = 4x and y 2 = x. x 2 + y 2 = 4x (i) (x 2) 2 + y 2 = 4 This is a circle with centre at (2, 0) and radius 2. y = (4x-x 2 ) y 2 = x
More informationEXERCISE 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 informationConic Sections. College Algebra
Conic Sections College Algebra Conic Sections A conic section, or conic, is a shape resulting from intersecting a right circular cone with a plane. The angle at which the plane intersects the cone determines
More informationLecture 9: Linear Programming
Lecture 9: Linear Programming A common optimization problem involves finding the maximum of a linear function of N variables N Z = a i x i i= 1 (the objective function ) where the x i are all non-negative
More informationChapter 3: Section 3-2 Graphing Linear Inequalities
Chapter : Section - Graphing Linear Inequalities D. S. Malik Creighton Universit, Omaha, NE D. S. Malik Creighton Universit, Omaha, NE Chapter () : Section - Graphing Linear Inequalities / 9 Geometric
More informationTrigonometric Functions. Copyright Cengage Learning. All rights reserved.
4 Trigonometric Functions Copyright Cengage Learning. All rights reserved. 4.7 Inverse Trigonometric Functions Copyright Cengage Learning. All rights reserved. What You Should Learn Evaluate and graph
More informationAppendix F: Systems of Inequalities
Appendi F: Sstems of Inequalities F. Solving Sstems of Inequalities The Graph of an Inequalit What ou should learn The statements < and ⱖ are inequalities in two variables. An ordered pair 共a, b兲 is a
More informationQuadratic Equations Group Acitivity 3 Business Project Week #5
MLC at Boise State 013 Quadratic Equations Group Acitivity 3 Business Project Week #5 In this activity we are going to further explore quadratic equations. We are going to analyze different parts of the
More informationLINEAR PROGRAMMING. Chapter Overview
Chapter 12 LINEAR PROGRAMMING 12.1 Overview 12.1.1 An Optimisation Problem A problem which seeks to maximise or minimise a function is called an optimisation problem. An optimisation problem may involve
More informationMATH 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 information9.1 Parametric Curves
Math 172 Chapter 9A notes Page 1 of 20 9.1 Parametric Curves So far we have discussed equations in the form. Sometimes and are given as functions of a parameter. Example. Projectile Motion Sketch and axes,
More informationCLASSROOM INVESTIGATION:
MHS CLASSROOM INVESTIGATION: LINEAR PROGRAMMING LEARNING GOALS By completing this lesson you will: Practice establishing systems of inequalities Review graphing half-planes Investigate using a polygon
More informationAlgebra II Quadratic Functions
1 Algebra II Quadratic Functions 2014-10-14 www.njctl.org 2 Ta b le o f C o n te n t Key Terms click on the topic to go to that section Explain Characteristics of Quadratic Functions Combining Transformations
More informationQuadratic Functions (Section 2-1)
Quadratic Functions (Section 2-1) Section 2.1, Definition of Polynomial Function f(x) = a is the constant function f(x) = mx + b where m 0 is a linear function f(x) = ax 2 + bx + c with a 0 is a quadratic
More information6.7. Graph Linear Inequalities in Two Variables. Warm Up Lesson Presentation Lesson Quiz
6.7 Graph Linear Inequalities in Two Variables Warm Up Lesson Presentation Lesson Quiz 6.7 Warm-Up Tell whether the ordered pair is a solution of the equation. 1. x + 2y = 4; (2, 1) no 2. 4x + 3y = 22;
More informationSection 6.2: Properties of Graphs of Quadratic Functions. Vertex:
Section 6.2: Properties of Graphs of Quadratic Functions determine the vertex of a quadratic in standard form sketch the graph determine the y intercept, x intercept(s), the equation of the axis of symmetry,
More informationMath 2260 Exam #1 Practice Problem Solutions
Math 6 Exam # Practice Problem Solutions. What is the area bounded by the curves y x and y x + 7? Answer: As we can see in the figure, the line y x + 7 lies above the parabola y x in the region we care
More informationAdvanced Algebra. Equation of a Circle
Advanced Algebra Equation of a Circle Task on Entry Plotting Equations Using the table and axis below, plot the graph for - x 2 + y 2 = 25 x -5-4 -3 0 3 4 5 y 1 4 y 2-4 3 2 + y 2 = 25 9 + y 2 = 25 y 2
More information3, 10,( 2, 4) Name. CP Algebra II Midterm Review Packet Unit 1: Linear Equations and Inequalities. Solve each equation. 3.
Name CP Algebra II Midterm Review Packet 018-019 Unit 1: Linear Equations and Inequalities Solve each equation. 1. x. x 4( x 5) 6x. 8x 5(x 1) 5 4. ( k ) k 4 5. x 4 x 6 6. V lhw for h 7. x y b for x z Find
More informationMATH 1020 WORKSHEET 10.1 Parametric Equations
MATH WORKSHEET. Parametric Equations If f and g are continuous functions on an interval I, then the equations x ft) and y gt) are called parametric equations. The parametric equations along with the graph
More informationRewrite the equation in the left column into the format in the middle column. The answers are in the third column. 1. y 4y 4x 4 0 y k 4p x h y 2 4 x 0
Pre-Calculus Section 1.1 Completing the Square Rewrite the equation in the left column into the format in the middle column. The answers are in the third column. 1. y 4y 4x 4 0 y k 4p x h y 4 x 0. 3x 3y
More information(Section 6.2: Volumes of Solids of Revolution: Disk / Washer Methods)
(Section 6.: Volumes of Solids of Revolution: Disk / Washer Methods) 6.. PART E: DISK METHOD vs. WASHER METHOD When using the Disk or Washer Method, we need to use toothpicks that are perpendicular to
More informationMath 414 Lecture 2 Everyone have a laptop?
Math 44 Lecture 2 Everyone have a laptop? THEOREM. Let v,...,v k be k vectors in an n-dimensional space and A = [v ;...; v k ] v,..., v k independent v,..., v k span the space v,..., v k a basis v,...,
More informationAlgebra 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 informationGraphical Methods in Linear Programming
Appendix 2 Graphical Methods in Linear Programming We can use graphical methods to solve linear optimization problems involving two variables. When there are two variables in the problem, we can refer
More information3.1 INTRODUCTION TO THE FAMILY OF QUADRATIC FUNCTIONS
3.1 INTRODUCTION TO THE FAMILY OF QUADRATIC FUNCTIONS Finding the Zeros of a Quadratic Function Examples 1 and and more Find the zeros of f(x) = x x 6. Solution by Factoring f(x) = x x 6 = (x 3)(x + )
More informationThe diagram above shows a sketch of the curve C with parametric equations
1. The diagram above shows a sketch of the curve C with parametric equations x = 5t 4, y = t(9 t ) The curve C cuts the x-axis at the points A and B. (a) Find the x-coordinate at the point A and the x-coordinate
More informationBecause the inequality involves, graph the boundary using a solid line. Choose (0, 0) as a test point.
Graph each inequality. 12. y < x 3 y < x 3 Because the inequality involves
More informationLinear-Quadratic Inequalities
Math Objectives Students will be able to describe the solution to a linearquadratic or quadratic-quadratic system of inequalities from a geometric perspective. Students will be able to write the solution
More information3. The domain of a function of 2 or 3 variables is a set of pts in the plane or space respectively.
Math 2204 Multivariable Calculus Chapter 14: Partial Derivatives Sec. 14.1: Functions of Several Variables I. Functions and Variables A. Def n : Suppose D is a set of n-tuples of real numbers (x 1, x 2,
More informationGraphing Equations Case 1: The graph of x = a, where a is a constant, is a vertical line. Examples a) Graph: x = x
06 CHAPTER Algebra. GRAPHING EQUATIONS AND INEQUALITIES Tetbook Reference Section 6. &6. CLAST OBJECTIVE Identif regions of the coordinate plane that correspond to specific conditions and vice-versa Graphing
More information9.5 Polar Coordinates. Copyright Cengage Learning. All rights reserved.
9.5 Polar Coordinates Copyright Cengage Learning. All rights reserved. Introduction Representation of graphs of equations as collections of points (x, y), where x and y represent the directed distances
More informationChapter 6 Some Applications of the Integral
Chapter 6 Some Applications of the Integral More on Area More on Area Integrating the vertical separation gives Riemann Sums of the form More on Area Example Find the area A of the set shaded in Figure
More informationSection Graphing Systems of Linear Inequalities
Section 3.1 - Graphing Systems of Linear Inequalities Example 1: Find the graphical solution of the inequality y x 0. Example 2: Find the graphical solution of the inequality 5x 3y < 15. 1 How to find
More information2-1 Power and Radical Functions
Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 35. Evaluate the function for several x-values in
More information