Essential Questions. Key Terms. Algebra. Arithmetic Sequence

Transcription

1 Linear Equations and Inequalities Introduction Average Rate of Change Coefficient Constant Rate of Change Continuous Discrete Domain End Behaviors Equation Explicit Formula Expression Factor Inequality Interval Notation Linear Function Linear Model You pay $40 a month to get 5GB of data to use for your phone. Last month, you went over by 35 MB and you are charged $0.50 per MB. How much do you owe? If you can figure this problem out, then you can work with linear equations. The study of linear equations is one of the most important in all of Algebra and used to model many situations in the real world. In this unit, you will practice writing, solving, and graphing linear equations and inequalities. We will explore these functions in three different forms: as a graph, as an equation, and in a table. Connecting each of these representations will be essential for you as we work on problem solving! Essential Questions 1. How do I solve and justify my answer to an equation or inequality in one variable? 2. How do I graph a linear equation or inequality in two variables? 3. How do I use graphs to represent and solve real-world equations and inequalities? 4. What is a function and how do I use it to model real-world situations? 5. How do I interpret the parts of a function in the context of the problem? 6. How do I interpret key features of graphs in context? 7. Why are sequences functions? 8. How do I write recursive and explicit formulas for arithmetic sequences? Key Terms Algebra Arithmetic Sequence 1/18

2 Ordered Pair Parameter Range Recursive Formula Slope Substitution Term Variable X-intercept Y-intercept What to Expect In this module, you will be responsible for completing the following assignments: Assignment - Laura's Phone Plan Handout Quiz - Linear Equations & Inequalities Discussion - Life Constraints Assignment - Graphing Lines Handout Assignment - Gus's Candles Handout Quiz - Understanding Functions and Lines Quiz - Graphing Linear Inequalities & Sequences Project - Writing and Graphing Linear Inequalities Linear Equations and Inequalities Test Creating Linear Equations to Solve Problems Let's go back to our introduction question: You pay $40 a month to get 5GB of data to use for your phone. Last month, you went over by 35 MB and you are charged $0.50 per MB. How much do you owe? So we know that we owe $40 plus an additional $ so a total of Could you write an equation for any amount of data used? Let x = MB of data used over 5GB. So then you know you would owe $40 plus an additional. So our linear equation is: Notice a few things about this equation: It is linear because the degree of x is 1. We introduced a new variable, C, which represents the total cost of the bill. Now you can plug in any value for x to figure out your cost. Use the equation above to determine the following: If your bill was $60 last month, how many extra MB did you use? Let C = 60 since we know that was the total cost. The unknown, or question is how many MB did we use, which is what we defined x to be: 2/18

3 So we know that you used 40 extra MB during that month. Watch this video to try a few different examples: Write and Solve Linear Equations Practice 1. Three consecutive integers have a sum of 171. Find the first of the three numbers. SOLUTION 2. A rectangle is 12m longer than it is wide. The perimeter is 68m. Find the width. SOLUTION 3. The sum of 52 and 5 less than three times a number is 71. Find the number. SOLUTION 4. The lengths of a triangle are consecutive odd integers. The perimeter of the triangle is 33in. What is the length of the longest side? SOLUTION Solving Linear Equations When we solve an equation, we go through the steps of isolating the variable: When you are asked to solve for a variable, you follow the same steps. Sometimes you may not be able to combine like terms, but that is OK! Let's try a few more together: Solving for a Variable Practice Here are some practice problems to check your understanding: 3/18

4 1. Solve A = bh for b. Solution 2. Solve d = rt for r. Solution 3. Solve P = 2l + 2w for w. Solution 4. Solve for h. Solution Mr. Jackson would like to take his biology students on a field trip to Ziggy's Zany Zoo. Student tickets cost $5 and adult tickets cost $8. Write an equation to represent the total cost, C, of the field trip tickets. Let's discuss some important parts of the equation: If 30 students and 4 adults attend the field trip, what will the cost for tickets be? Mr. Jackson was given a budget of $300, he must take 6 adult chaperones. How many students can attend the field trip? Since Mr. Jackson cannot take 0.4 of a student, he'll have to take only 50 because he does not have enough money to take a whole other student. Assignment - Laura's Phone Plan Handout 4/18

5 It is now time to complete the Laura's Phone Plan Handout. Download the handout from the sidebar and complete. Be sure to show all work. Submit your assignment when finished. Create Linear Inequalities to Solve Problems We use inequalities in math to represent a range of numbers restricted by some sort of constraint. Let's review solving some basic inequalities: Two examples of inequalities: Let's look at these symbols and some common words associated with them: Symbol Common Phrases Examples <: less than less than >: greater than higher than, must exceed : less than or equal to at most, not higher than, does not exceed, maximum, greatest number x is less than 4 x<4 the sum of a number and 3 must exceed 7 x + 3 > 7 twice a number is at most 93 : greater than or equal to at least, minimum, is not less than you must get at least a 70 on your test We've been writing and solving linear equations, so now lets try some inequalities. But first, let's think about some common steps we can use for inequalities or equations: 1. Draw a picture, if applicable 2. Define your variable 3. Set up your equation 4. Solve your equation 5. Check to be sure you answered the question! The sum of two consecutive integers is less than 83. Find the pair of integers with the greatest sum. No picture to draw Let x = the first consecutive number, so x+1 = the second consecutive number Equation: x+x+1<83 Solve: 5/18

6 So the first of the numbers must be an integer less than 41, so x=40, which means x+1 = 41. Check to be sure these two numbers add up to less than 83: = 81 which is less than 83! Watch this video to try a few more: Write and Solve Linear Inequalities Practice Now try these problems to be sure you have it down! Rollover each solution to see the correct answer. Make sure that as you practice, you are constructing your answers in complete sentences. 1. The sum of twice a number and 7 is less than 27. Write and solve an inequality statement to figure out the constraints on this number. Solution 2. Your quiz grades are 78, 72, 87, and 90. What score on the fifth quiz will make your average quiz grade at least 82? Solution 3. Solve the inequality: Solution 4. Solve the inequality: Solution 5. You have the option between two phone plans. Option A that charges $25 a month plus $0.10 per MB of data used, or Option B that charges $10 a month plus $0.20 per MB of data used. How much data do you have to use to make Option A the better option? Solution Quiz - Linear Equations & Inequalities It is now time to complete the Linear Equations & Inequalities Quiz. You will have a limited amount of time, please plan accordingly. Discussion - Life Constraints It is now time to complete the Life Constraints Discussion. Inequalities are based on restrictions and constraints, and in the real world, we face those every day! Think about two constraints or restrictions that you face every day and write an inequality to represent those. Describe your constraints and give your equation. Example: I must sleep for 8 or more hours a night, let h describe the numbers of hours slept. The inequality is Functions A relation consists of a set of ordered pairs (x, y); a relation can also be called a mapping. The x-values are the domain and the y-values are the range. In the relation above, the domain is {1, 3, 5, 9} and the range is {2, 7, 8}. The range is only the values that have been "used" by the domain. We say the domain are the independent variables and are usually the x-values (what you input) and the range are the dependent variables and are usually the y-values (the output). A relation can be mapped onto a graph by plotting each of the ordered pairs. This graph to the right maps the relation above. A FUNCTION IS A RELATION IN WHICH EACH X - VALUE MAPS TO EXACTLY ONE Y - VALUE 6/18

7 In order for a relation to be a function, each x-value can only be associated with one y-value. It is OK if multiple x-values map to the same y-values! Function Not a Function {(3, 2) (5, 2) (1, 4) (6,3)} {(3, 2) (3, 5) (1, 7) (6, 6)} Is it a function or not a function? Let's look at the graphical representation of relations and functions: Notice that in a function, none of the points are on the same vertical line, but in the relation that is NOT a function two of the points are on the same vertical line. Watch this video to get a better idea of how the Vertical Line Test works! Function Graphs Practice Determine if each graphical representation is a function. We use functions to tell us about relationships between values. For instance, let's say your cell phone plan charges you $0.15 per MB of data used. So we can write a function for the cost (C) in terms of the amount of data used. We would say. 7/18

8 Use the rule to complete the table for the given domain values. Write the result as an ordered pair. Independent Variable: m MB's used Dependent Variable: C(m) = 0.15m (m, C (m)) 1 C(1) = 0.15(1)= 0.15 (1, 0.15) 10 C(1) = 0.15(10)= 1.50 (10, 1.50) 25 C(1) = 0.15(25)= 3.75 (25, 3.75) 100 C(1) = 0.15(100)= 15 (100, 15) 180 C(1) = 0.15(180)= 27 (180, 27) A common misconception might be to think the domain for this function is {1, 10, 25, 100, 180} however, it is not! We know that you could use any amount of MB's of data. So we must account for those continuous values, not just the values we put in the table. The domain for this function would be: We know the amount of data used must be greater than 0, because you can't use a negative amount of data. But after that there are no restrictions on what the input could be! Watch this video for a few more examples of how functions work: Input Output Practice Write a function, using appropriate notation, for each situation below: 1. You bought a plant that is 5 inches tall and you know that the plant will grow at a rate of 2 inches per week. Write a function for the height, h, of the plant after a certain number of weeks, w. Solution 2. You are selling brownies at the bake sale for $0.75 each. Write a function for the revenue, R, you've earned based on the number of brownies you've sold, b. Solution 3. Evaluate the function for each value below: f(3), f(-2), f(0), f(-5) Modeling Linear Functions Let's revisit the phone plan with a cost of $0.15 per MB used. Independent Variable: m MB's used Dependent Variable: C(m) = 0.15m (m, C (m)) 1 C(1) = 0.15(1)= 0.15 (1, 0.15) 10 C(1) = 0.15(10)= 1.50 (10, 1.50) 25 C(1) = 0.15(25)= 3.75 (25, 3.75) 100 C(1) = 0.15(100)= 15 (100, 15) 180 C(1) = 0.15(180)= 27 (180, 27) And let's look at the graphical representation of these points: 8/18

9 But, what if we used 50 MB of data? Or MB of data? We need to consider this function as a continuous line so that we know the relationship between each amount of data used and the cost of our bill. But first, we need to know how to graph lines! Understanding Slope Slope is the average rate of change of a function. For a line, the slope is considered the. We can also calculate slope algebraically using the formula: Example: Calculate the slope of the line that contains the points (1, -2) and (3, -5). 1. Let the first coordinate be and. And let the second coordinate be and. 2. Substitute into the equation: So now we know our line has negative slope which means it goes down from left to right. We also know two points on our line so we can graph it: 9/18

10 Watch this video to practice a few more: Slope Practice What is the slope of each graph? Find the slope of the line containg the given points. Graphing Lines Graphing Horizontal and Vertical Lines Practice Match the equation of each line to the appropriate graph. 10/18

11 Lines can be written in two forms: Slope-intercept: y = mx + b Standard: Ax + By = C Graphing Lines by Making a Table Like we did in the cell phone problem, we can graph lines by using a table. In order to graph a line, you always need 2 points! Let's plug in values for x, the independent variable, to find what y is the output. When we are working with graphs, we often use x and y rather than x and f(x). But the important thing to remember is that f(x) and y both represent the OUTPUT! Example: 2x + y = 7 Input (x) 1 4 Substitute for x and solve for y 2(1) + y = y = 7 y = 5 2(4) + y = y = 7 y = -1 Write your ordered pair (input, output) (1, 5) (4, -1) Ordered Pairs Practice Use the equation to complete the ordered pairs below: 3x + y = 9 (, 6) Solution (2, ) Solution (1/3, ) Solution Use the Graph to complete the order pairs below: (, -2) Solution (3, ) Solution (4, ) Solution What is the slope of the line? Solution 11/18

12 Graphing Lines by Finding X - and Y - intercepts Example: Find the x- and y- intercepts of 2x + y = /18

13 X-intercept and Y-intercept Practice Match each equation to its x- and y-intercepts Graphing by Finding Slope and Y-Intercept Example: In Slope-Intercept form, y = mx + b, m is the slope and (0, b) is the y-intercept. Use that information to graph the line: Watch this video for examples of equations in standard form: Slope and Y-Intercept Practice Match each equation to its slope and y-intercept. Assignment - Graphing Lines Handout It is now time to complete the Graphing Lines Handout. Download the handout from the sidebar and complete. Be sure to show all work. Submit your assignment when finished. 13/18

14 Writing Equations of Lines Now that you know how to graph lines, we want to learn how to write the equations of lines. First, you need to know one more formula: Point-Slope: Example: Write the equation in slope-intercept form of the line with slope 2 that contains the point (5, -3). Step 1: Substitute a point in for and and use the slope for m. Simplify as needed. Step 2: To change to slope-intercept form, isolate y. Watch this video for some more practice. Writing Equations of Lines Practice 1. Contains the points (2, 4) and (-2, 2) Solution 2. Contains the point (3, -2) and has a slope of 0 Solution 3. Contains the points (3, -2) and (3, 7) Solution 4. Contains the point (2, -5) and has a slope of -3 Solution 5. Contains the points (1, -8) and (9, 0) Solution 6. Contains the point (-4, 1) and has an undefined slope Solution Now lets imagine that you are riding in a taxi-cab. The cab charges an initial fee once you get in and then a charge per mile driven. You notice that after you have gone 3 miles, the charge is $8 and after 9 miles the charge is $17. Write a function, C, to represent the cost of the taxicab after driving a certain number of miles, m. The first step for this problem is determining which of the given information represents the independent and dependent variables: Independent Variable: miles driven in taxi; m = 3;m = 9 Dependent Variable: cost of taxi; C(3) = 8; C(9) = 17 So now lets find the rate of change, or the slope for the function: And now lets use the point-slope formula to write an equation: Use the equation we found to answer the following questions: 1. What is the initial cost for the taxi before driving any miles? Solution 2. What is the cost per mile for the taxi? Solution 3. How much will the fare be if you drive 12 miles? Solution 14/18

15 4. If you paid $14, how many miles did you drive? Solution 5. What is the domain, or input for this situation? Solution Watch this video to practice more problems like this one: Assignment - Gus' Candles Handout It is now time to complete Gus' Candles Handout. Download the handout from the sidebar and complete. Be sure to show all work. Submit your assignment when finished. Quiz - Understanding Functions and Lines It is now time to complete the "Understanding Functions and Lines Quiz". You will have a limited amount of time, please plan accordingly. Graphing Linear Inequalities The past few modules have been about graphing & writing linear equations (example: y = 2x - 3). When we graph these equations we are looking for the x- and y-values that make one side EQUAL the other side. Now, we want to look at graphing linear inequalities like: y < 2x - 3 going to be a whole lot more values! In this statement, we care about what values of x- and y- make one side less than the other side. This is The first step to graphing an inequality is to think about it like an equality. So consider how you would graph y = 2x - 3 and plot those points: Before we draw our line we need to understand a few rules about the symbols used: < or > draw a dashed line; this means that the values on the line do not make the inequality true so they cannot be included! draw a solid line; this means that the values on the line do make the inequality true so they can be included. Since y < 2x - 3 this means we should use a dashed line. Now, the points on the dashed line are not included in the solution, but there are MANY points that would work. So we must determine where those points are. To do that, pick any point on the coordinate plane that is not on the graphed line to test. Let's chose (-1, 4) and substitute: 4 < 2(-1) < /18

16 This makes an "untrue" statement so we know (-1, 4) is not a part of the solution set. So now I will shade the side opposite from the point (-1,4). Watch this video to practice a few more: Graphing Linear Inequalities Practice Arithmetic Sequences A sequence is a special type of function. Each element in a sequence is called a term, these values would be considered the range. Each term is paired with a position number, and these values would be considered the domain. The domains of sequences are the consecutive integers and usually starts at 0 or 1. Position Number Term of Sequence n Domain f(n) Range From the table, you can see that the "third term in the sequence is 6" or f(3) = 6. The explicit rule of a sequence is a rule that will allow you to determine any term in the sequence by using n, the position number. The explicit rule for the sequence above is f(n) = 2n. Arithmetic Sequences Practice Use the above table to answer the following questions. Using Explicit Rule Practice Use the explicit rule to complete the table. f(n) = 2n - 5. n 0 SOLUTION 2 3 SOLUTION 5 f(n) SOLUTION SOLUTION SOLUTION SOLUTION 3 SOLUTION A recursive rule for a sequence defines the terms of the sequence by relating it to one or more previous terms. Watch this video to help you understand further: An arithmetic sequence is a special type of sequence in which the difference between each term is constant. This difference is referred to as the common difference. Common Difference Practice Determine if the sequences below are arithmetic by looking to see if there is a constant common difference. If the sequence is arithmetic, give the common difference, d. 1. {3, 7, 11, 15} Solution 2. {1, 4, 9, 16} Solution 3. {-8, -10, -12, -14} Solution 4. {10, 7, 4, 10} Solution 16/18

17 Watch this video to learn how to write recursive and explicit rules for arithmetic sequences. Sequences Practice Write an explicit rule for each sequence. Assume the domain for the function is the set of consecutive integers starting with Solution n f(n) 2. Solution n f(n) Solution n f(n) Write a recursive rule for each sequence. Assume the domain for the function is the set of consecutive integers starting with Solution n f(n) Solution n f(n) 4 10 Quiz - Graphing Linear Inequalities & Sequences 17/18

18 It is now time to complete the "Graphing Linear Inequalities & Sequences Quiz". You will have a limited amount of time, please plan accordingly. Linear Equations and Inequalities Wrap Up In this module, you were responsible for completing the following assignments: Assignment - Laura's Phone Plan Handout Quiz - Linear Equations & Inequalities Discussion - Life Constraints Assignment - Graphing Lines Handout Assignment - Gus' Candles Handout Quiz - Understanding Functions and Lines Quiz - Graphing Linear Inequalities & Sequences Project - Writing and Graphing Linear Inequalities Linear Equations and Inequalities Final Assessments It is now time to complete Writing and Graphing Linear Inequalities Project. Download the project from the sidebar and complete. Be sure to show all work. Submit your completed project when finished. Linear Equations and Inequalities Test It is now time to complete the Linear Equations and Inequalities Test. Once you have completed all selfassessments, assignments, and the review items and feel confident in your understanding of this material, you may begin. You will have a limited amount of time to complete your test and once you begin, you will not be allowed to restart your test. Please plan accordingly. 18/18