STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA I 2 nd Nine Weeks, 2016-2017 1
OVERVIEW Algebra I Content Review Notes are designed by the High School Mathematics Steering Committee as a resource for students and parents. Each nine weeks Standards of Learning (SOLs) have been identified and a detailed explanation of the specific SOL is provided. Specific notes have also been included in this document to assist students in understanding the concepts. Sample problems allow the students to see step-by-step models for solving various types of problems. A section has also been developed to provide students with the opportunity to solve similar problems and check their answers. The answers to the found at the end of the document. problems are The document is a compilation of information found in the Virginia Department of Education (VDOE) Curriculum Framework, Enhanced Scope and Sequence, and Released Test items. In addition to VDOE information, Prentice Hall textbook series and resources have been used. Finally, information from various websites is included. The websites are listed with the information as it appears in the document. Supplemental online information can be accessed by scanning QR codes throughout the document. These will take students to video tutorials and online resources. In addition, a self-assessment is available at the end of the document to allow students to check their readiness for the nine-weeks test. The Algebra I Blueprint Summary Table is listed below as a snapshot of the reporting categories, the number of questions per reporting category, and the corresponding SOLs. Algebra I Blueprint Summary Table Reporting Categories No. of Items SOL Expressions & Operations 12 A.1 A.2a c A.3 Equations & Inequalities 18 A.4a f A.5a d A.6a b Functions & Statistics 20 A.7a f A.8 A.9 A.10 A.11 Total Number of Operational Items 50 Field-Test Items* 10 Total Number of Items 60 * These field-test items will not be used to compute the students scores on the test. It is the Mathematics Instructors desire that students and parents will use this document as a tool toward the students success on the end-of-year assessment. 2
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Functions A.7 The student will investigate and analyze function (linear and quadratic) families and their characteristics both algebraically and graphically, including a) determining whether a relation is a function; The set of all solutions of an equation forms the graph of an equation. You can determine if a relation is a function if every element in the domain is paired with exactly one element in the range. Relation Function Not a Function T-Chart or Table x y 1 2 3 4 5 10 Graph Passes the Vertical Line Test x y 2 1 2 3 2 5 Does Not Pass the Vertical Line Test Mapping Real World A grocery store is selling peaches for $0.98 per pound. Brandon buys 4 old CDs for $8.00, and Sharon buys 3 old CDs for $6.00. Pete talks to the seller and makes a deal to buy 4 old CDs for $7.00. 4
Scan this QR code to go to a video tutorial on determining whether a relation is a function. Relations and Functions 1. Draw a T-Chart for the relation (Set of Ordered Pairs) and determine whether it is a function.,. 2. Which is the graph of a function and why? A. B. 3. Draw a Mapping for the following relation. Is it a function? x y 3-4 -2 0 3 5 4. Jalen is buying tickets for the basketball game at $8 each. Is this a function? 5
A.7 The student will investigate and analyze function (linear and quadratic) families and their characteristics both algebraically and graphically, including b) domain and range; T-Chart or Table Domain Range 1 2 3 4 5 10 Ordered Pairs (1,-2), (3, 4), (5, 10) Domain {1, 3, 5} Range {-2, 4, 10} Mapping Domain Range Domain Read the line Left to Right Graph Example #1 Range Read the line Bottom to Top Domain Read the line Left to Right Graph Example #2 All real numbers Range Read the line Bottom to Top All Real Numbers 6
Scan this QR code to go to a video tutorial on determining domain and range of a linear function. Domain & Range 1. What is the domain and range for the set of ordered pairs? {, } 2. What is the domain and range given the graph of this function? A. B. A.7 The student will investigate and analyze function (linear and quadratic) families and their characteristics both algebraically and graphically, including c) zeros of a function; The zeros of a linear function are all the independent (x-values) that make the dependent value (y-values) equal to zero. To find out this value algebraically, substitute zero for y in the function y = -x +5 and solve for x. In the graph it is the point where the graph crosses the x-axis. In the chart (t-table) you will see an ordered pair in the format of (x,0). 7
Scan this QR code to go to a video tutorial on finding the zeros of a linear function. Zeros 1. What are the zeros for each function? A. B. 2. What is the zero of the function represented in the table (t-chart)? Functions and Graphs A.6 The student will graph linear equations and linear inequalities in two variables including a) determining the slope of a line when given an equation of the line, the graph of the line, or two points on the line. Slope will be described as rate of change and will be positive, negative, zero, or undefined; A.6 The student will graph linear equations and linear inequalities in two variables including b) writing the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line. The set of all solutions of an equation forms the graph of an equation. x y -2 16-1 -12 0-8 1-4 2 0 As an example, if the equation is, you can probably think of a lot of values that could satisfy x and y. Some examples are listed below in the t-chart. x y 5 0 0 5 2 3 4 1 7-2 We will now plot each of these points on the coordinate plane to reveal the graph of. We will find that this graph forms a straight line, because this is a linear equation. This line passes through all of the solutions to this linear equation, including the ones that we did not plot, such as (3, 2), (1, 4), (-1, 6) and (6, -1). 8
To graph an equation, you can simply create a table of values by choosing values to plug in for x, and solve for the resulting y-values. Once you have x and y values, you can plot the ordered pairs and connect them with a line or curve that best fits. Example 1: Graph Example 2: Graph You could start by transforming the equation for y. Choose x-values that will work out nicely with the fraction (i.e. even numbers in this case) Scan this QR code to go to a video tutorial on graphing equations using a T-Chart. 9
Functions and Graphs 1. Graph 2. Complete the function table for the function 3. Which of these is closest to the graph of x -1 0 1 y A B C Slope A.6 The student will graph linear equations and linear inequalities in two variables including a) determining the slope of a line when given an equation of the line, the graph of the line, or two points on the line. Slope will be described as rate of change and will be positive, negative, zero, or undefined The slope of a line is determined by the vertical change divided by the horizontal change (or rise over run). Slope can be positive, negative, zero, or undefined. Positive Negative Zero Undefined You can determine slope by counting rise over run, or using the formula: 10
Example 1: Find the slope of the line that passes through (-3, 9) and (2, 4). Example 2: Determine the slope of the line graphed below. By counting rise over run, we can see that the graph goes up 5, and to the right 2. Example 3: Find the slope of the line that passes through (-4, 3) and (-4, 0). You may also be asked to find the missing coordinate of a point given the slope and a different point on the line. To do this, plug in all of the given values into the slope formula, and solve for the missing value. Example 4: The slope of the line passing through and is. Solve for. and Write the slope in a ratio format. Now you can cross multiply! 11
Example 5: The slope of the line passing through and is. Solve for. and Scan this QR code to go to a video tutorial on slope. Now you can cross multiply! Slope 1. Find the slope of the line that passes through and. 2. Find the slope of the line that passes through and 3. The slope of the line passing through and is. Solve for. 4. The slope of the line passing through and is 3. Solve for. A.6 The student will graph linear equations and linear inequalities in two variables including b) writing the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line. A.7 The student will investigate and analyze function (linear and quadratic) families and their characteristics both algebraically and graphically, including d) x- and y-intercepts; Slope-Intercept and Transformations If you are asked to translate (or shift) a line, just move two points on the line the specified units and draw a new line. Remember, the new line should be parallel to the original line. This means the slope should remain the same. Only the y-intercept will change. Example 1: Shift the line down 5 units and right 2 units. 12
Choose two points on the line. From each point, count down 5 units and right 2 units. Plot a new point. Then, draw a line through the two new points. The equation of the translated line is. Notice that the slopes of the lines are the same. Example 2: The function. Choose two points on the function f(x), which is. From each point, count up 3 units. Plot a new point. Then, draw a line through the two new points. The equation of the translated line is. Notice that the slopes of the lines are the same. is displayed on the graph below. Graph the function Slope-Intercept The intercept of a graph is where it crosses the axis. The x-intercept is where a graph crosses the x-axis, and the y-intercept is where a graph crosses the y-axis. y-intercept x-intercept A special form of a linear equation is called slope-intercept form:. Where is the slope of the line and is the y-intercept (0, ). Example 1: What are the slope and y-intercept of? 13
The slope is the coefficient of x, and the y-intercept is the constant. The slope of the line is, and the y-intercept is 4. Example 2: What is the equation of the line with a slope of and a y-intercept of? It is often easier to graph an equation when it is written in slope-intercept form. You can start by plotting a point on the y-axis for the y-intercept, and then count the slope as rise over run, from the y-intercept. Scan this QR code to go to a video tutorial on graphing linear equations in slope-intercept form. Example 3: Graph the equation To graph the equation above, put a point on the y-axis at 2, and then count the slope by going down 1 and to the right 3. Put a second point there and draw a line through the two points. Sometimes, you will be asked to graph an equation that is not in slope-intercept form. You can transform an equation into slope-intercept form by solving for y. 14
Example 4: Put in slope-intercept form, state the slope and the y-intercept, and graph. The slope is 4, and the y-intercept is (0, -3). To graph the equation above, put a point on the y-axis at -3, and then count the slope by going up 4 and to the right 1. Put a second point there and draw a line through the two points. You can find the equation of a line given two points on the line by first finding the slope. Once you have the slope and one point, you can plug this information into slopeintercept form and solve for the y-intercept ( ). Example 5: What is the equation of the line that passes through and? First find the slope: Now you can use the slope and either of the points to solve for b. Now that you have and you can write an equation in slope intercept form. 15
Slope-Intercept Form 1. What is the slope and y-intercept of? 2. What is the equation of a line whose y-intercept is and slope is? 3. What is the equation of the line that passes through and whose slope is? 4. What is the y-intercept of the equation? 5. Graph 6. Which of these equations has a slope of -2? A. B. C. D. 7. What is the equation of the line that passes through (3, 4) and has a slope of -2? 8. What is the equation of the line that passes through (3, -2) and (9, -4)? 9. Graph 10. Graph 11. The function is displayed on the graph below. What is the equation of the function? 16
Direct and Inverse Variation A.8 The student, given a situation in a real-world context, will analyze a relation to determine whether a direct or inverse variation exists, and represent a direct variation algebraically and graphically and an inverse variation algebraically. If the ratio between two variables is a constant, then a direct variation exists. A direct variation can be written in the form, where is the constant of variation. If the product of two variables is a constant, then an inverse variation exists. An inverse variation can be written in the form or. Example 1: Determine if each relation is a direct variation, inverse variation, or neither. x y 1 3 2 4 3 5 x y -1 2 0 0 1-2 First check the ratios: Does the ratio? NO! Therefore this is NOT a direct variation! Next check the products: Does? NO! Therefore this is NOT an inverse variation! First check the ratios: Does the ratio? YES! Therefore this IS a direct variation! Notice that we did not use the ordered pair to check the ratios. It is impossible to divide by zero, therefore we used the other points. x y 3 4 1 12-2 -6 First check the ratios: Does the ratio? NO! Therefore this is NOT a direct variation! Next check the products: Does? Does this also equal? YES! Therefore this IS an inverse variation! To write an equation of a direct variation, use a given point (other than plug into, to solve for. To write an equation of an inverse variation, use a given point to plug into, to solve for. ) to 17
Example 2: Suppose varies directly with, and when. What direct variation equation relates and? Start with. We are given a value for and, so plug those in and solve for. This is the constant of variation! This is the direct variation equation! Once you have a direct variation equation, you can use that equation to determine other values. Example 3: The distance that you jog,, varies directly with the amount of time you jog,. If you can jog 9 miles in 1.5 hours, how long will it take you to jog 4 miles? Jogging varies directly with time Now we need to solve for k in order to write a direct variation equation. Use the values that are related to one another. This is the constant of variation! This is the direct variation equation! Now, we can use this equation to solve for the time it takes to jog 4 miles. We are given that you jog 4 miles. This will be plugged in for j. Then, solve for t. Therefore, it would take you of an hour to jog 4 miles. Scan this QR code to go to a video tutorial on direct and inverse variations. 18
Direct and Inverse Variation 1. Determine if each of the following relations are a direct variation, an inverse variation, or neither. A x y 2 4 0 0 x y -4-20 5 16 2. Suppose varies 1 8 8 10 directly with, and when. What direct variation equation relates and? 3. Suppose varies directly with, and when. What will be when? 4. The amount of money spent at the mall varies directly with the amount of time you spend shopping. If you spend $90 when you are in the mall for 2.5 hours, how much time did you spend in the mall when you spent $340? 5. Jason s earnings ( ) during his summer job is directly proportional to the amount of hours he worked ( ). When,, what is the constant of variation? 6. The time it takes to complete a job ( ) is inversely proportional to the amount of workers assigned to the job ( ). What value would accurately represent this relationship? Time (t) B Amount of Workers (w) 3 10 5? 15 2 19
Systems of Equations A.4 The student will solve multistep linear and quadratic equations in two variables, including e) solving systems of two linear equations in two variables algebraically and graphically; and f) solving real-world problems involving equations and systems of equations. A system of equations is two or more equations, whose solution is any point that each of the equations has in common. This can be seen on a graph as the intersection point of the lines. Systems of two linear equations can have no solutions, one solution, or infinitely many solutions. No Solutions One Solution Infinitely Many Solutions Two lines that are parallel. These lines have the same slope, but different y-intercepts. They will never intersect. Therefore, there is no solution. Two lines that intersect. These lines have different slopes, which causes them to intersect in one place. Therefore, there is one solution. In this example, the solution is (2, 3). Two lines that are the same. These lines have the same slope and the same y-intercept. This means they are the same line and will share all points. Therefore, there are infinitely many solutions. 20
Example 1: What is the solution to the system of equations pictured here? The graphs intersect at the point. Therefore, the solution is. Systems of equations can also be solved algebraically by substitution or elimination. It is often easier to use substitution when one of the equations has a variable on the side by itself. If this is the case, you can substitute the value of that variable into the other equation. This will allow you to solve for one variable. Example 2: Solve the system of equations by substitution. Since, you can replace the in the second equation with! Remember that the solution to a system of equations is an ordered pair! You have a y-value, so use that to help you solve for x. Remember that an ordered pair is always (x, y)! 21
Example 3: Solve the system of equations by substitution. You can substitute the value of y from the first equation into the second equation. Remember that the solution to a system of equations is an ordered pair! You have an x-value, so use that to help you solve for y. Another method of solving a system of equations is called elimination. This is often easier when both equations are written in standard form ( ). To use the elimination method, you will add or subtract the two equations, or some multiple of them, to get one of the variables to cancel out. Example 4: Solve this system of equations by elimination. Notice that the y terms have equal and opposite coefficients! If you add these two equations together the y-term will cancel out! + Don t forget to solve for y! Plug the x-value back into one of the original equations. 22
You can always check your work when solving a system of equations by transforming both equations and graphing them in your calculator. The two lines should intersect at the ordered pair that you found. Below, you will see the calculator check for Example 4. The calculated intersection is Which is the same thing as You can also check your work by plugging your values in for x and y to verify that both equations are true for those values. Scan this QR code to go to a video tutorial on solving systems of equations. Sometimes the system of equations will not have variables that immediately cancel out (like the y-term did in Example 4). When this happens, you may have to multiply one or both of the equations by a constant to get two variables to have equal and opposite coefficients. Example 5: Solve this system of equations by elimination. Notice that the x terms have opposite coefficient signs. What can you do to make the coefficients equal? Multiply the first equation by 2, then the x-terms will have equal and opposite coefficients. + Don t forget to solve for x! Plug the y-value back into one of the original equations. 23
Example 6: Solve this system of equations by elimination. What can you do to make two of the coefficients equal? Multiply the first equation by 3, and the second equation by 2, then the y-terms will have equal coefficients. Now you can subtract to get the y s to cancel! _ Don t forget to solve for y! Plug the x-value back into one of the original equations. Systems of equations are often presented as word problems. In these cases, you will often not be given the equations, and you will be responsible for setting those up. Once you have two equations set up, you can solve the system of equations using any method that you prefer. 24
Example 7: A class of 148 students went on a field trip. They took 10 vehicles, some cars and some buses. Find the number of cars and buses they took if each car holds 4 students and each bus holds 40 students. We know total number of vehicles, and total number of students. We can set up two equations where those are our totals. The number of cars plus the number of busses equals 10 total vehicles. 4 students per car plus 40 students per bus equals 148 total students. Now we have two equations. We can solve this system using any method that we ve learned. Multiply the first equation by, then the c-terms will have equal and opposite coefficients. + So now we know that they took 3 busses. We can plug this value into one of the other equations to solve for the number of cars! They took 3 busses and 7 cars! Don t forget that you can check your work by graphing! Just solve both equations for one of the variables. Scan this QR code to go to a video tutorial on solving systems of equations word problems. 25
Example 8: Lauren is raising pot-bellied pigs and ostriches for fun. Among her animals, she has 17 heads and 56 legs in all. How many of each animal does she have? We know total number of heads, and total number of legs. We can set up two equations where those are our totals. Each pig has one head plus each ostrich has one head equals 17 total heads. 4 legs per pig, plus 2 legs per ostrich equals 56 legs total. Be careful using as a variable! Don t confuse it with 0. Now we have two equations. We can solve this equation using any method that we ve learned. Multiply the first equation by, then the -terms will have equal and opposite coefficients. + So now we know that she has 11 pigs. We can plug this value into one of the other equations to solve for the number of ostriches! She has 11 pigs and 6 ostriches! Systems of Equations Solve each system using whatever method you prefer. 1. 2. 3. 4. The admission fee at a small fair is $1.75 for children and $3.00 for adults. On a certain day, 1700 people enter the fair and $3375 is collected. How many children and how many adults attended? 5. Kris spent $144 on shirts. Dress shirts cost $19 and t-shirts cost $7. If he bought a total of 12, then how many of each kind did he buy? 26
Systems of Inequalities A.5 The student will solve multistep linear inequalities in two variables, including c) solving real-world problems involving inequalities; and d) solving systems of inequalities. A linear inequality can be formed by replacing the equal sign in any linear equation with an inequality symbol. The solutions for a linear inequality are any ordered pairs that make it a true statement. Example 1: Identify which ordered pairs are a solution of. This is true, therefore is a solution. This is false, therefore is NOT a solution. This is true, therefore is a solution. As you can see from the last example, linear inequalities will have more than one solution. In fact, they will have infinitely many solutions. The graph of a linear inequality will indicate all of the solutions, and it is called a half-plane, and is bounded by a boundary line. All of the points on one side of this boundary are solutions, while all of the points on the other side of the boundary are not solutions. You graph a linear inequality the same way that you graph a linear equation. The line that you graph will either be dashed or solid depending on the inequality symbol. Dashed lines are used for < and >. This indicates that the points on the line are not part of the solution set. Solid lines are used for the solution set. and. This indicates that the points on the line are part of To determine which half-plane to shade in, you can select one point that is not on the graph and determine if it is a solution or not. If it is a solution, shade that side of the boundary. If it is not a solution, shade the other side. The point (0, 0) is often an easy point to check with. 27
Example 2: Graph First, determine if you will be using a dashed or solid line (dashed in this case because you have < ) Then, graph the equation of the line by plotting the y-intercept and counting the slope as rise over run. Once you have a couple of points graphed, connect them with a dashed line. Finally, figure out which half-plane to color in. Select a point to see if it works. This is false. Therefore, is NOT a solution. So, we will shade the other side. All of the points on the shaded side will satisfy the inequality. Scan this QR code to go to a video tutorial on graphing linear inequalities. 28
Example 3: Graph First, determine if you will be using a dashed or solid line (solid in this case because you have greater than or = equal to. Before we graph this equation, we should put it in slope intercept form! Remember that if you multiply or divide by a negative number you will have to switch the inequality symbol s direction. You divided by here! Don t forget to switch the sign! Now we can put a point at the y-intercept, and count the slope as rise over run. Finally, figure out which half-plane to color in. Select a point to see if it works. You can plug this point into the original equation or the transformed equation. This is true. Therefore, is a solution. So, we will shade that side. All of the points on the shaded side will satisfy the inequality. 29
To solve a system of linear inequalities, you will graph both inequalities on the same coordinate plane. The solution is any area that is shaded for both inequalities. Example 4: Solve the system of inequalities by graphing. First, graph the first equation with a solid line and determine which half-plane you should shade by choosing a point and verifying. Does the point (0, 0) work? No, so shade the other side! Now, graph the second equation on the same graph and determine which side to shade. Does the point (0, 0) work? No, therefore shade the other side! The solutions to the system of inequalities are all of the points that are shaded from both inequalities. Scan this QR code to go to a video tutorial on systems of inequalities. 30
Systems of Inequalities 1. Is a solution to the inequality? 2. Is a solution to the inequality? 3. Is (2, 2) a solution to this system of inequalities? 4. Graph. 5. Graph. 6. Write the inequality that is graphed here 7. Solve the system of inequalities by graphing. 31
Answers to the Relations and Functions 1. problems: x y 4 4-3 4 0 4 It is a function because none of the x-values are repeated Functions and Graphs 1. 2. A. Yes. Passes the vertical line test B. No. Does not pass the vertical line test. 3. It is not a function 4. Yes 2. 3. B Slope x y -1-3 0-5 1-3 Domain & Range 1.Domain {-2,0,5} Range {4,3,-6} 2. A. Domain {All Real Numbers} Range B. Domain Rangel Zeros 1. A. x=4 B. x=2 2. 2 1. 2. 3. 4. Slope-Intercept and Transformations 1. 2. 3. 4. 32
5. Direct and Inverse Variation 1. A ; B 2. 3. 4. 5. 6. 6. B 7. 8. 9. Systems of Equations 1. 2. 3. 4. 1380 Children 320 Adults 5. 5 dress shirts 7 t-shirts Systems of Inequalities 10. 1. No 2. Yes 3. Yes 4. 11. 33
5. 6. 7. 34