Math 002 Intermediate Algebra Spring 2012 Objectives & Assignments

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1 Math 002 Intermediate Algebra Spring 2012 Objectives & Assignments Unit 2 Equations, Inequalities, and Linear Systems I. Graphs and Functions 1. Graph equations in two variables by plotting points. 2. Graph linear equations by finding and plotting intercepts. 3. Use function notation. 4. Identify the domain, range, and function values from a graph. II. Lines and Linear Functions 5. Calculate the slope of a line through two points. 6. Graph lines using the slope and y-intercept. 7. Determine if two linear equations represent parallel or perpendicular lines. 8. Write equations for lines that a. pass through two given points. b. have a specified slope through a given point. c. are parallel or perpendicular to a given line through a specific point. d. are horizontal or vertical through a given point. 9. Solve applications involving linear relationships. III. Inequalities in Two Variables 10. Graph inequalities in two variables on the Cartesian plane. 11. Graph systems of inequalities in two variables on the Cartesian plane IV. Systems of Linear Equations 12. Solve systems of linear equations by graphing, substitution, or elimination. 13. Identify the solution to a dependent or inconsistent linear system. 14. Use systems of linear equations to solve applications. Unit 2 Topic Homework Key Problems 2.1 Graphing 2.2 Functions pg. 113: 30-42(M3), 45-48(all), 66-78(M3), 79, 81 pg. 133: 33-47(odd), 63-78(M3), 81-92(all), 96, 97 33, 39, 69, 75, 79 39, 45, 72, 78, 85, Graphing Linear Functions pg. 146: 15-60(M5), 46, 49, 61, 72, 75 20, 30, 40, 61, The Slope of a Line pg. 160: 5-55(M5), 60-75(M3) 10, 25, 50, 63, Equations of Lines 3.6 Graphing Linear Inequalities in Two Variables pg. 171:: 5-30(M5), 31,32,35-40(all), 41-51(odd) 76, 78 25, 31, 51, 52, 78 pg. 256: 10,11, 24-51(M3), 55 10, 33, 36, 45, Linear Systems pg. 278: 9, 12, 20-55(M5), 62, 81-86(all) 12, 30, 45, 62, System Applications pg. 297: 11, 12, 17, 23, 24, 25, 30, 33, 35, 36 11, 23, 24, 30, 36 1

2 Math 002 Unit 2 - Student Notes Section 2.1- Graphing Equations 1. Review Cartesian coordinate system: x axis, y axis, quadrants, plotting ordered pairs x coordinate, y coordinate. EX 1 * Note: An ordered pair is given in alphabetical order. Label the following: x- and y- axes origin Quadrants Points: A(3, -2) B(0, 3) C(-4, 1) D(-1, 0) E(-2.5, -3) F(3.5, 4.5) 2. Linear equations in two variables. Some examples: Standard Form: Example 1: Look at the following graph of. Pick a point on the graph and identify the coordinates. What does it mean for a point to be on the graph? 3. Define a solution of an equation in two variables. EX 3 solution: Example 2: Determine whether lies on the graph of. What about (5, 105)? 2

3 4. Graphing linear and nonlinear equations by hand. EX 5, EX 6, EX 8, EX 10 a. For a linear equation: Find 3 ordered pairs; plot these solutions; the line through the plotted points is the graph. b. For a nonlinear equation: Find many ordered pairs; plot the solutions; connect the plotted points with a smooth curve to sketch its graph. ; Let. x y ; Let. x y ; Let. x y 5. Graphing linear and nonlinear equations using a graphing calculator. EX 5, EX 7, EX 8, EX 9, EX 10 a. Get equation in the form y = expression. b. Enter in calculator. c. You can change the window to get an accurate picture of the graph. d. You can use the table feature to see a list of ordered pairs. 3

4 Section Introduction to Functions 1. Define relation, domain, and range. EX 1 A relation is. The following correspondences describe a relation. a) b) c) Input States Arkansas Kentucky South Carolina Texas Output # of Reps The domain of a relation is the set of all of the ordered pairs. The range of a relation is the set of all of the ordered pairs. Example 1: Determine the domain and range of each relation. 2. Define function: A relation in which each first component corresponds to exactly. Some examples: Determine whether a relation is a function. EX 2, EX 3 In Example 1 above, which relations are also functions? Use the vertical line test to determine whether a relation is a function. EX 5, EX 6 vertical line test: 3. Find the domain and range of a function. EX 6 Example 2: Find the domain and range of each relation. Determine whether the relation is also a function. 4

5 4. Use function notation. EX 7, EX 8 To denote that is a function of, we can write: What does this mean? o Dependent variable: o Independent variable Example 3: If and, find: a. b. What are the ordered pairs that correspond to these values? 5. Use a graph to identify function values, domain, and range. EX 9, EX 11 Example 4: Consider the graph of Q(x). a. Find. b. Find. c. Find. d. For what values of is e. Identify the domain: f. Identify the range: 5

6 Section 2.3 Graphing Linear Functions 1. Review graphing linear equations by making a table of values, plotting ordered pairs, and connecting with a line. Section Define and find - and - intercepts. EX 4, EX 5, EX 6 Example 1: Identify where the following graphs cross the -axis and -axis. Label the points. a) b) o - intercept: point where the graph crosses the -axis. To find an x- intercept, let or and solve for x. o - intercept: point where the graph crosses the -axis. To find a y- intercept, let and solve for y o Remember these are points. Write as an ordered pair. 3. Graphing linear equations using - and - intercepts. EX 4 Example 2: Graph the lines using the - and -intercepts. a) b) c) 4. Graphing vertical and horizontal lines. EX 7, EX 8 o Horizontal lines: o Vertical lines: 6

7 Section 2.4 The Slope of a Line 1. Find the slope of a line given two points on the line. EX 1, EX 2 Plot points (2, 1) and (4, 5). Sometimes you can look at the graph to calculate the slope of the line, but sometimes it can be difficult. Slope of a line: Given a line passing through the points, the slope m of the line is: Example 1: Find the slope of line that passes through the points and. Summarize the overall appearance of lines with positive, negative, zero, and undefined slope. H O Y V U X : EX 5, EX 6 2. Different forms of an equation of a line. Standard form: Slope-intercept form: Function notation: 3. Find the slope of a line given the equation of a line. EX 3 Write the equation in slope-intercept form:. Example 2: Find the slope and -intercept of each line. Then graph each line. a) b) c) 7

8 4. Slopes of parallel and perpendicular lines. EX 7 Two non-vertical lines are parallel if they have Two non-vertical lines are perpendicular if they have Example 3: The slope of a line is. Slope of a parallel line: Slope of a perpendicular line: Example 4: Given the line, (a) Give an equation of a line parallel to the given line: (b) Give and equation of a line perpendicular to the given line: Example 4: Determine whether the lines are parallel, perpendicular or neither. a) f ( x) 3x 6 g( x) 3x 5 b) 2x 3y 1 3x 2y 12 8

9 Section 2.5 Equations of Lines 1. Use the slope and -intercept a. to write an equation of a line. EX 1 b. to graph the equation of a line. EX 2, EX 3 2. Equations of lines. a. Forms of linear equations. You should be able to move from one form to another. i. Standard form: ii. Slope-intercept form: iii. Function notation: iv. Point-slope form: v. Horizontal line: vi. Vertical line: b. Writing equations of lines i. using slope-intercept. EX 1 Write an equation of a line with -intercept and slope 5. ii. using point-slope. EX 4 Write an equation of a line that passes through (5, 3) with slope. iii. passing though two points. EX 5 Write an equation of a line that passes through (1, -2) and (-3, -5). 9

10 iv. given the graph. EX 6 v. that are horizontal. EX 8 Write an equation of a horizontal line that passes through (-3, 1). vi. that are vertical. EX 9 Write an equation of a vertical line that passes through (2, 6). vii. that are parallel or perpendicular to a given line. EX 10, EX 11 Write an equation of the line perpendicular to that passes through (2, 1). 3. Use the point-slope form of an equation to solve linear applications. EX 7 # 76. The value of a computer bought in 2003 depreciates, or decreases, as time passes. Two years after the computer was bought, it was worth $2000; 4 years after it was bought, it was worth $800. a. If this relationship between number of years past 2003 and value of computer is linear, write an equation describing this relationship. [Use ordered pairs of the form (years past 2003, value of computer).] b. Use this equation to estimate the value of the computer in the year

11 Section 3.1 Solving Linear Equations Graphically Technology allows us to reduce the time spent on solving and aids us in verifying solutions. Consequently, by using technology we can increase the accuracy of our work. 1. Intersection-of-Graphs Method for Solving an Equation. EX 1, EX 2 STEP 1: Graph and. STEP 2: STEP 3: STEP 4: Find the point(s) of intersection of the two graphs. The x-coordinate of a point of intersection is a solution to the equation. The y-coordinate of the point of intersection is the value of both the left side and the right side of the original equation when x is replaced with the solution. Example 1 (from book p. 209): Solve the equation. Example 2: Solve using the x-intercept method. Example 3: Two appliances are being compared. Appliance A costs $395 and $0.03 per hour to run. Appliance B costs $305 and $0.07 per hour to run. Find how many hours when the appliances cost the same to run. (ans: 2250 hours; days; weeks) 11

12 Section 3.6 Graphing Linear Inequalities in Two Variables 1. The graph of a linear inequality in two variables is all the ordered pairs that satisfy the inequality. Identify: boundary line: half-plane: solutions: 1. Graphing linear inequalities. EX 1, EX 2 a) Sketch the graph of the equation that is the boundary line of the solution set. Dashed line: Solid line: b) Choose a convenient test point not on the boundary line and substitute it into the original inequality. c) If the test point satisfies the inequality, then shade the half-plane that contains the test point. Otherwise, shade the half-plane that does not include the test point. Example 1: Graph each inequality. a) b) 12

13 2. Graphing intersections or unions of linear inequalities. EX 3, EX 4 Intersection (and), : All points common to both regions (where they overlap). Union (or), : All points from both combined (everything that is shaded). Example 2: Graph the following: a) The intersection of and. b) The union of or. 13

14 Section 4.1 Solving Systems of Linear Equations in Two Variables 1. A system of equations is a collection of two or more equations in two variables. A system of two linear equations in two variables x and y consists of two equations of the following form: Ax By C Equation 1 Dx Ey F Equation 2 The following are graphs of systems of linear equations. a) b) c) A solution of a system of linear equations in two variables is an ordered pair that. EX 1 Example 1: What do you think is the solution to each of the previous systems? Example 2: Check whether the following are solutions of the following systems. x 3y 5 a) b) 2x 3y Solving Linear Systems by Graphing. EX 2, EX 3 Graph both lines in the same coordinate plane. 3. A system of two linear equations in two variables can have the following cases: CASE 1 CASE 2 CASE 3 14

15 4. Solving Linear Systems Algebraically Graphing an equation by hand is often a good method of approximating solutions of a system, but it is not a reliable method of finding exact solutions of a system. a. The substitution method: EX 4, EX 5 Step 1: Solve one of the equations for one of its variables. Step 2: Substitute the expression for the variable found in Step 1 into the other equation. Step 3: Find the value of one variable by solving the equation in Step 2. Step 4: Find the value of the other variable by substituting the value just found into the equation from Step 1. Step 5: Check the ordered pair solution in both original equations. Example 4: Use the substitution method to solve the following systems of equations. 6x 4y 10 4x 2y 5 i) ii) 3x y 3 2x y 1 b. The elimination method: EX 6, EX 7, EX 8, EX 9 Step 1: Rewrite each equation in standard form. Step 2: If necessary, multiply one or both equations by some nonzero number so that the coefficient of one variable in one equation is the opposite of its coefficient in the other equation. STEP 3: Add the equations. Step 4: Find the value of one variable by solving the equation from Step 3. Step 5: Find the value of the second variable by substituting the value found in Step 4 into either original equation. Step 6: Check the proposed solution in both original equations. Example 5: Use the elimination method to solve the following systems of equations. 3x y 1 4x 7y 10 a) c) 4x y 6 8x 14y 20 15

16 Section 4.3 Systems of Linear Equations and Problem Solving 1. Solve problems that can be modeled by a system of two linear equations. EX 1, EX 2, EX 3 Revisit and use our General Strategy for Problem Solving: o Understand the problem. Become comfortable with the problem: i. Read and reread the problem. ii. Choose and define the variables to represent the unknowns. o Translate the problem into a system of two equations. o Solve the system of equations. o Interpret the results: Check the proposed solution in the stated problem and state your conclusion. a. The sum of two numbers is -42. The first number minus the second number is 52. What are the numbers? b. One evening 1500 concert tickets were sold for the Summer Jazz Festival. Tickets cost $25 for covered pavilion seats and $15 for lawn seats. Total receipts were $28,500. How many of each type of ticket were sold? c. A mail-order whoopee cushion shipped 120 packages one day. Customers are charged $3.50 for each standard delivery package and $7.50 for each express delivery package. Total shipping charges for the day were $596. How many of each kind of package were shipped? 16

17 d. Mindi s Tee Shirt Shack sold 36 shirts one day. All short-sleeved shirts cost $12 and all longsleeved shirts cost $18. Total receipts for the day were $522. How many of each kind of shirt were sold? 2. Solving problems with cost and revenue functions. a. Define cost function: b. Define revenue function: c. Define break-even: Break-even Example: A company that manufactures boxes recently purchased $2000 worth of new equipment to make gift boxes to sell to its customers. The cost of producing a package of gift boxes is $1.50 and it is sold for $4.00. Find the number of packages that must be sold for the company to break even. 17

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