# LESSON #11 - FORMS OF A LINE COMMON CORE ALGEBRA II

Save this PDF as:

Size: px
Start display at page:

Download "LESSON #11 - FORMS OF A LINE COMMON CORE ALGEBRA II"

## Transcription

1 LESSON # - FORMS OF A LINE COMMON CORE ALGEBRA II Linear functions come in a variet of forms. The two shown below have been introduced in Common Core Algebra I and Common Core Geometr. TWO COMMON FORMS OF A LINE Slope-Intercept: where m is the slope (or average rate of change) of the line and Point-Slope: represents one point on the line. Eercise #: Consider the linear function f 5. (a) Determine the -intercept of this function b evaluating. (b) Find its average rate of change over the interval. Eercise #: Consider a line whose slope is 5 and which passes through the point, 8. (a) Write the equation of this line in point-slope form,. (b) Write the equation of this line in slope-intercept form,. Eercise #: Which of the following represents an equation for the line that is parallel to passes through the point 6, 8? 7 and which () 8 6 () 8 6 () 8 6 (4) 8 6

2 Eercise #4: A line passes through the points 5, and 0, 4. (a) Determine the slope of this line in simplest rational form. (b) Write an equation of this line in point-slope form. (c) Write an equation for this line in slopeintercept form. (d) For what -value will this line pass through a -value of? Eercise #5: The graph of a linear function is shown below. (a) Write the equation of this line in m b form. (b) What must be the slope of a line perpendicular to the one shown? (c) Draw a line perpendicular to the one shown that passes through the point,. (d) Write the equation of the line ou just drew in pointslope form. (e) Does the line that ou drew contain the point 0, 5? Justif.

3 Eercise #6: For the first four problems, graph the equations. For the last problems, write an equation for each graph. Equation Graph Equation Graph 7 6 ( ) 6 ( ) Equation in Slope Intercept Form: Equation in Point Slope Form:

4 4 LESSON # - FORMS OF A LINE COMMON CORE ALGEBRA II HOMEWORK FLUENCY. Which of the following lines is perpendicular to () () 5 4 () 4 (4) Which of the following lines passes through the point 4, 8? () 8 4 () 8 4 () 8 4 (4) and has a -intercept of 4?. Which of the following equations could describe the graph of the linear function shown below? () () 4 () 4 (4) For a line whose slope is and which passes through the point 5, : (a) Write the equation of this line in point-slope m. form, (b) Write the equation of this line in slopeintercept form, m b. 5. For a line whose slope is 0.8 and which passes through the point, : (a) Write the equation of this line in point-slope m. form, (b) Write the equation of this line in slopeintercept form, m b.

5 5 REASONING 6. The two points, 6 and 6, 0 are plotted on the grid below. (a) Find an equation, in m b form, for the line passing through these two points. Use of the grid is optional. (b) Does the point 0, 6 lie on this line? Justif. 7. A linear function is graphed below along with the point,. (a) Draw a line parallel to the one shown that passes through the point,. (b) Write an equation for the line ou just drew in point-slope form. (c) Between what two consecutive integers does the -intercept of the line ou drew fall? (d) Determine the eact value of the -intercept of the line ou drew.

6 6 8. For the first two problems, graph the equations. For the last problems, write an equation for each graph. Choose the form of the equation that ou think is easiest for each of the last two problems. Equation Graph Equation Graph 5 ( ) Equation: Equation:

7 7 LESSON # - LINEAR MODELING COMMON CORE ALGEBRA II In Common Core Algebra I, ou used linear functions to model an process that had a constant rate at which one variable changes with respect to the other, or a constant slope. In this lesson we will review man of the facets of this tpe of modeling. Eercise #: Dia was driving awa from New York Cit at a constant speed of 58 miles per hour. He started 45 miles awa. (a) Write a linear function that gives Dia s distance, D, from New York Cit as a function of the number of hours, h, he has been driving. Let h = (b) If Dia s destination is 70 miles awa from New York Cit, algebraicall determine to the nearest tenth of an hour how long it will take Dia to reach his destination. Let D = Eercise #: Two students have bank accounts. Student A starts with \$600 in her bank account and takes out \$0 each month. Student B starts with \$900 in his bank account and takes out \$50 each month. (a) Create linear functions for amount of mone, in each account after months. Let A() = (b) Algebraicall determine eactl how man months it will take for Student A and Student B to have the same amount in their accounts. Let B() = Let =

8 Man times linear models have been constructed and we are asked onl to work with these models. Models in the real world can be mess and it is often convenient to use our graphing calculators to plot and investigate their behavior. Eercise #: A factor produces widgets (generic objects of no particular use). The cost, C, in dollars to produce w widgets is given b the equation C 0.8w Each widget sells for 6 cents. Thus, the revenue gained, R, from selling these widgets is given b R 0.6w. (a) Use our graphing calculator to sketch and label each of these linear functions for the interval 0 w 500. Be sure to label our - ais with its scale. Let w = Let C = Let R = 8 (b) Use our calculator s INTERSECT command to determine the number of widgets, w, that must be produced for the revenue to equal the cost. (c) If profit is defined as the revenue minus the cost, create an equation in terms of w for the profit, P. (d) Using our graphing calculator, sketch a graph of the profit over the interval 0 w 000. Use a TABLE on our calculator to determine an appropriate WINDOW for viewing. Label the and intercepts of this line on the graph. Dollars w (e) What is the minimum number of widgets that must be sold in order for the profit to reach at least \$40? Illustrate this on our graph.

9 In the previous eercises, it is clear from the contet what both the slope and the -intercept of this linear model are. Although this is often the case when constructing a linear model, sometimes the slope and a point are known, in which case, the point slope form of the a line is more appropriate. Eercise #4: Edeln is tring to model her cell-phone plan. She knows that it has a fied cost, per month, along with a \$0.5 charge per call she makes. In her last month s bill, she was charged \$.80 for making 5 calls. 9 (a) Create a linear model, in point-slope form, for the amount Edeln must pa, P, per month given the number of phone calls she makes, c. Make sure ou write our let statements. (b) How much is Edeln s fied cost? In other words, how much would she have to pa for making zero phone calls?

10 0 LESSON # - LINEAR MODELING COMMON CORE ALGEBRA II HOMEWORK APPLICATIONS. Which of the following would model the distance, D, a driver is from Chicago if the are heading towards the cit at 58 miles per hour and started 56 miles awa? () D 56t 58 () D 58t 56 () D 56 58t (4) D 58 56t. The cost, C, of producing -bikes is given b C. The revenue gained from selling -bikes is given b R 50. If the profit, P, is defined as P R C, then which of the following is an equation for P in terms of? () P 8 () P 8 () P 7 (4) P 7. The average temperature of the planet is epected to rise at an average rate of 0.04 degrees Celsius per ear due to global warming. The average temperature in the ear 000 was 4.7 degrees Celsius. The average Celsius temperature, C, is given b C , where represents the number of ears since 000. (a) What will be the average temperature in the ear 00? (b) Algebraicall determine the number of ears,, it will take for the temperature, C, to reach 0 degrees Celsius. Round to the nearest ear. (c) Sketch a graph of the average earl temperature below for the interval. Be sure to label our -ais scale as well as two points on the line (the - intercept and one additional point). (d) What does this model project to be the average global temperature in 00?

11 4. Fabio is driving west awa from Alban and towards Buffalo along Interstate 90 at a constant rate of speed of 6 miles per hour. After driving for.5 hours, Fabio is miles from Alban. (a) Write a linear model for the distance, D, that Fabio is awa from Alban as a function of the number of hours, h, that he has been driving. Write our model in point-slope form, D D m h h. (b) Rewrite this model in slope-intercept form, D mh b. (c) How far was Fabio from Alban when he started his trip? (d) If the total distance from Alban to Buffalo is 90 miles, determine how long it takes for Fabio to reach Buffalo. Round our answer to the nearest tenth of an hour. 5. A particular rocket taking off from the Earth s surface uses fuel at a constant rate of.5 gallons per minute. The rocket initiall contains 5 gallons of fuel. (Hint: When ou are using fuel our slope is negative). (a) Determine a linear model, in a b form, for the amount of fuel,, as a function of the number of minutes,, that the rocket has burned. (b) Below is a general sketch of what the graph of our model should look like. Using our calculator, determine the and intercepts of this model and label them on the graph at points A and B respectivel. A (c) The rocket must still contain 50 gallons of fuel when it hits the stratosphere. What is the maimum number of minutes the rocket can take to hit the stratosphere? Show this point on our graph b also graphing the horizontal line 50 and showing the intersection point. B

12 LESSON # - INVERSES OF LINEAR FUNCTIONS COMMON CORE ALGEBRA II The inverse of a function has all the same points as the original function, ecept that the 's and 's have been reversed. Eercise #: If our function is made up of these points: { (, 0), (, 5), (0, 4) }. Then the inverse is given b this set of points:. Eercise #: If ou are given a function on a graph, ou can also switch the s and s in the original points to plot the inverse. Use that process to graph the inverses of the two functions below. Which of the inverses is a function? How do ou know? There is a wa to determine if the inverse of a function will also be a function without graphing the inverse. What qualit of the original function determines if the inverse will be a function? Wh does this make sense?

13 Eercise #: On the grid below the linear function 4 is graphed along with the line. (a) How can ou quickl tell that 4 is a one-to-one function? (b) Graph the inverse of 4 on the same grid. Recall that this is easil done b switching the and coordinates of the original line. (c) What can be said about the graphs of 4 and its inverse with respect to the line? (d) Find the equation of the inverse in m b form. (e) Find the equation of the inverse in form. b a As we can see from part (e) in Eercise #, inverses of linear functions include the inverse operations of the original function but in reverse order. This gives rise to a simple method of finding the equation of an inverse. Simpl switch the and variables in the original equation and solve for. Eercise #4: Which of the following represents the equation of the inverse of 5 0? () () 0 () 5 0 (4)

14 Although this is a simple enough procedure, certain problems can lead to common errors when solving for. Care should be taken with each algebraic step. Eercise #5: Which of the following represents the inverse of the linear function () () 8 () (4) 8 8? 4 Eercise #6: What is the -intercept of the inverse of () 5 () 9 9? 5 () (4) 9 5 Eercise #7: Which of the following points lies on the graph of the inverse of 8 5 choice. () 8, () 0, 40 () 8, (4), 8? Eplain our Eercise #8: Which of the following linear functions would not have an inverse that is also a function? Eplain how ou made our choice. () () () (4) 5 Sometimes we are asked to work with linear functions in their point-slope form. The method of finding the inverse and plotting it, though, do not change just because the linear equation is written in a different form. Eercise #9: Which of the following would be an equation for the inverse of () 6 () () 6 (4) 4 6?

15 5 FLUENCY LESSON # - INVERSES OF LINEAR FUNCTIONS COMMON CORE ALGEBRA II HOMEWORK. The graph of a function and its inverse are alwas smmetric across which of the following lines? () 0 () () 0 (4). Which of the following represents the inverse of the linear function 4? () () 8 () 8 (4) 4 4. If the -intercept of a linear function is 8, then we know which of the following about its inverse? () Its -intercept is 8. () Its -intercept is 8. () Its -intercept is 8. (4) Its -intercept is If both were plotted, which of the following linear functions would be parallel to its inverse? Eplain our thinking. () () 5 () 4 (4) 6 5. Which of the following represents the equation of the inverse of () () 4 4 () 8 (4) Which of the following points lies on the inverse of 4 (), (), (), (4),? 4 4?

16 6 7. A linear function is graphed below. Answer the following questions based on this graph. (a) Write the equation of this linear function in m b form. (b) Sketch a graph of the inverse of this function on the same grid. (c) Write the equation of the inverse in m b form. (d) What is the intersection point of this line with its inverse? APPLICATIONS 8. A car traveling at a constant speed of 58 miles per hour has a distance of -miles from Poughkeepsie, NY, given b the equation 58 4, where represents the time in hours that the car has been traveling. (a) Find the equation of the inverse of this linear function in form. (b) Evaluate the function ou found in part (a) for an input of. (c) Give a phsical interpretation of the answer ou found in part (b). Consider what the input and output of the inverse represent in order to answer this question. REASONING 9. Given the general linear function m b, find an equation for its inverse in terms of m and b.

17 7 LESSON #4 - PIECEWISE LINEAR FUNCTIONS COMMON CORE ALGEBRA II Functions epressed algebraicall can sometimes be more complicated and involve different equations for different portions of their domains. These are known as piecewise functions (the come in pieces). If all of the pieces are linear, then the are known as piecewise linear functions. Eercise #: Consider the piecewise linear function given b the formula f (a) Evaluate each of the following b carefull appling the correct formula: (i) f (ii) f (iii) f 0 (b) Create a table of values below and graph the function. f (c) State the range of f using interval notation. Eercise #: Consider the function defined b: f (a) Graph the function f b graphing each of the two lines. (b) State the range of the function f.

18 8 Eercise #: Given the piecewise function the interval? g 6 0, what is the average rate of change over 4 0 () () () 0 (4) 4 Eercise #4: On the graph below, sketch the function (a) Graph h on the grid. h (b) State the range of h. (c) What values of solve h 0? Not onl should we be able to graph piecewise functions when we are given their equations, but we should also be able to translate the graphs of these functions into equations. Eercise #5: The function f is shown graphed below. Write a piecewise linear formula for the function. Be sure to specif both the formulas and the domain intervals over which the appl.

19 Piecewise equations can be challenging algebraicall. Sometimes information that we find from them can be misleading or incorrect. g Eercise #6: Consider the piecewise linear function (a) Determine the -intercept of this function algebraicall. Wh can a function have onl one -intercept? 5. 9 (b) Find the -intercepts of each individual linear equation. (c) Graph the piecewise linear function below. (d) Wh does our graph contradict the answers ou found in part (b)? Eercise #7: For the piecewise linear function f algebraicall. f 0 0, find all solutions to the equation 5 0

20 0 LESSON #4 - PIECEWISE LINEAR FUNCTIONS COMMON CORE ALGEBRA II HOMEWORK FLUENCY. For 5 f 8 7 answer the following questions. (a) Evaluate each of the following b carefull appling the correct formula: (i) f (ii) f 4 (iii) f (iv) f 0 (b) The three linear equations have -intercepts of, 8 and 9 respectivel. Yet, a function can have onl one -intercept. Which of these is the -intercept of this function? Eplain how ou made our choice. (c) Calculate the average rate of change of f over the interval 9. Show the calculations that lead to our answer. g. Determine the range of the function graphicall.

21 . Determine a piecewise linear equation for the function f shown below. Be sure to specif not onl the equations, but also the domain intervals over which the appl. REASONING 4. Step functions are piecewise functions that are constants (horizontal lines) over each part of their domains. Graph the following step function. f g 4 algebraicall. Justif our work b showing our algebra. Be sure to check our answers versus the domain intervals to make sure each solution is valid. 5. Find all -intercepts of the function

22 LESSON #5 - MODELLING WITH ABSOLUTE VALUE FUNCTIONS COMMON CORE ALGEBRA II In lesson 4, ou learned about piecewise functions. Toda, we will use piecewise functions to define the function for absolute value of as 0 0 The graph of this basic piecewise function consists of two ras and is V-shaped. The corner point of an absolute value graph, called the verte, occurs when the epression inside the absolute value equals zero. Eercise #: Given the absolute value equation: f ( ) (a) Find the -coordinate of the verte b setting the epression inside the absolute value equal to zero. f (b) Create a table of values with the -coordinate of the verte and three integers above and below it. Plug each -value from the table into the original absolute value equation to find each corresponding -value. (c) Graph the function.

23 Absolute value functions can be used to model real world situations where we are finding differences between values and a given number. The net two questions are common situations where this tpe of function is useful. Eercise #: A weather station is looking at dail low temperatures in Chicago. (a) Write an absolute value function, f(), to model the difference between the dail low temperature,, and 5 degrees Fahrenheit. Let f() = Let = (b) Create an absolute value graph over the interval, (Use the table on the calculator to determine the appropriate window for viewing.) (c) Wh is f(0)=0? Eplain in the contet of the problem. (d) Find f(5). Eplain the value of f(5) in the contet of the problem. (e) In Chicago, the maimum value of f() was 0 degrees Fahrenheit in Januar. Find the interval of possible low temperatures in Chicago that month.

24 Eercise #: A cereal manufacturer is measuring the weights of boes of cereal. Write a function to model the difference between the weight of each bo, w, and the goal weight of 0 ounces. 4 The cereal manufacturer has a tolerance of 0.75 ounces for each bo of cereal that is supposed to weigh 0 ounces. Use a graph of the absolute value function to find the interval of acceptable weights. w

25 5 LESSON #5 - MODELLING WITH ABSOLUTE VALUE FUNCTIONS COMMON CORE ALGEBRA II HOMEWORK. Graph the following absolute value equation and show the table of values (including the verte): 6 f. You are a qualit control inspector at a bowling pin compan. The tpical bowling pin weighs 54 ounces. (a) Write a function to model the difference between the actual weight in ounces, w, and the tpical weight of 54 ounces. Let w = Let f(w) = (b) Create an absolute value graph over the interval, 40 w 70. (Use the table on the calculator to determine the appropriate window for viewing.) Show a break in our w-ais to start at 40. (c) The acceptable weights for bowling pins should be 4 ounces from the tpical weight. Use the graph of the absolute value function to find the interval of acceptable weights.

26 . In a sample conducted b the United States Air Force, the right-hand palm widths of 4000 Air Force men were measured. The gathering of such information is useful when designing control panels, keboards, gloves, etc. (a) Write an absolute value function to model the difference between a given person s palm width, p, and the average palm width of.49 inches. 6 (b) The vast majorit of the palm widths were within 0.6 inches of the average. Use a graph of the absolute value function to find the interval of possible palm widths for the vast majorit of the population surveed. w

27 7 LESSON #6 - SYSTEMS OF LINEAR EQUATIONS COMMON CORE ALGEBRA II Sstems of equations, or more than one equation, arise frequentl in mathematics. To solve a sstem means to find all sets of values that simultaneousl make all equations true. Of special importance are sstems of linear equations. You have solved them in our last two Common Core math courses, but we will add to their compleit in this lesson. Eercise #: Solve each of the sstems of equations b elimination. (a) 9 (b) You should be ver familiar with solving two-b-two sstems of linear equations (two equations and two unknowns). These linear sstems serve as the basis for a field of math known as Linear Algebra. Eercise #: Consider the three-b-three sstem of linear equations shown below. Each equation is numbered in this first eercise to help keep track of our manipulations. () () () z 5 6 z z 4 (a) The addition propert of equalit allows us to add two equations together to produce a third valid equation. Create a sstem b adding equations () and () and () and (). Wh is this an effective strateg in this case? (b) Use this new two-b-two sstem to solve the three-b-three.

28 Just as with two b two sstems, sometimes three-b-three sstems need to be manipulated b the multiplication propert of equalit before we can eliminate an variables. Eercise #: Consider the sstem of equations shown below. Answer the following questions based on the sstem. 4 z 6 4 z 8 (b) Solve the two-b-two sstem from (a) and find the final solution to the three-b-three sstem. 5 7z 9 (a) Which variable will be easiest to eliminate? Wh? Use the multiplicative propert of equalit and elimination to reduce this sstem to a two-b-two sstem. 8 You can easil check our solution to an sstem of equations b storing our answers for each of the variables and making sure the make each of the equations true. This is especiall useful if ou get a multiple choice question on this topic. Use the storing method to solve the problem below. Eercise #4: Which ordered triple represents the solution to the sstem of equations? z 4z 5 z 8 a. (-,,-4) b. (,,-) c. (,8,) d.(,-,0)

29 9 Eercise #5: Solve the sstem of equations shown below. Show each step in our solution process. 4 z 5 z 7 4z 7 These are challenging problems onl because the are long. Be careful and ou will be able to solve each one of these more comple sstems.

30 0 LESSON 6 - SYSTEMS OF LINEAR EQUATIONS COMMON CORE ALGEBRA II HOMEWORK FLUENCY. Solve the following sstems of equations algebraicall. (a) 4 8 (b) 4 4. Show that 0, 4, and z 7 is a solution to the sstem below without solving the sstem formall. z 5 4 5z 8z. Solve the following sstem of equations. Carefull show how ou arrived at our answers. 4 z z 5z 70

31 4. Algebraicall solve the following sstem of equations. There are two variables that can be readil eliminated, but our answers will be the same no matter which ou eliminate first. 5 z 5 4z z 5. Algebraicall solve the following sstem of equations. This sstem will take more manipulation because there are no variables with coefficients equal to. z 4 5 z z 50

### LESSON #12 - FORMS OF A LINE COMMON CORE ALGEBRA II

LESSON # - FORMS OF A LINE COMMON CORE ALGEBRA II Linear functions come in a variet of forms. The two shown below have been introduced in Common Core Algebra I and Common Core Geometr. TWO COMMON FORMS

### LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II

LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,

### LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II

LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,

### Chapter 3: Linear Functions & Their Algebra

Chapter 3: Linear Functions & Their Algebra Lesson 1: Direct Variation Lesson 2: Average Rate of Change Lesson 3: Forms of a Line Lesson 4: Linear Modeling Lesson 5: Inverse of Linear Functions Lesson

### Unit 3 Functions HW #1 Mrs. Dailey

HW#1 Name Algebra II Unit Functions HW #1 Mrs. Dailey 1) In each of the following, the variable pair given are proportional to one another. Find the missing value. (a) b = 8 when a = 16 b =? when a = 18

### LESSON #1 - BASIC ALGEBRAIC PROPERTIES COMMON CORE ALGEBRA II

1 LESSON #1 - BASIC ALGEBRAIC PROPERTIES COMMON CORE ALGEBRA II Mathematics has developed a language all to itself in order to clarif concepts and remove ambiguit from the analsis of problems. To achieve

### LESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II

1 LESSON #8 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The

### LESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II

1 LESSON #4 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The

### ALGEBRA 2 NY STATE COMMON CORE

ALGEBRA NY STATE COMMON CORE Kingston High School 017-018 emathinstruction, RED HOOK, NY 1571, 015 Table of Contents U n i t 1 - Foundations of Algebra... 1 U n i t - Linear Functions, Equations, and their

### LESSON #48 - INTEGER EXPONENTS COMMON CORE ALGEBRA II

LESSON #8 - INTEGER EXPONENTS COMMON CORE ALGEBRA II We just finished our review of linear functions. Linear functions are those that grow b equal differences for equal intervals. In this unit we will

### x. 4. 2x 10 4x. 10 x

CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 1 of Reasoning with Equations and Quantities Name: Date: Understand solving equations as a process of reasoning and eplain the reasoning MCC9-1.A.REI.1 Eplain

### The Coordinate Plane and Linear Equations Algebra 1

Name: The Coordinate Plane and Linear Equations Algebra 1 Date: We use the Cartesian Coordinate plane to locate points in two-dimensional space. We can do this b measuring the directed distances the point

### Chapter 5: Systems of Equations

Chapter : Sstems of Equations Section.: Sstems in Two Variables... 0 Section. Eercises... 9 Section.: Sstems in Three Variables... Section. Eercises... Section.: Linear Inequalities... Section.: Eercises.

### Solve each system by substitution or elimination. Check your solutions. b.

Algebra: 10.3.1: Intersect or Intercept? Name Solutions Block Date Bell Work: a. = 4 2 3 = 3 2 3(4 ) = 3 2 12 + 3 = 3 5 12 = 3 5 = 15 Solve each sstem b substitution or elimination. Check our solutions.

### LESSON #17 - FACTORING COMMON CORE ALGEBRA II FACTOR TWO IMPORTANT MEANINGS

1 LESSON #17 - FACTORING COMMON CORE ALGEBRA II In the study of algebra there are certain skills that are called gateway skills because without them a student simply cannot enter into many more comple

### 8.4. If we let x denote the number of gallons pumped, then the price y in dollars can \$ \$1.70 \$ \$1.70 \$ \$1.70 \$ \$1.

8.4 An Introduction to Functions: Linear Functions, Applications, and Models We often describe one quantit in terms of another; for eample, the growth of a plant is related to the amount of light it receives,

### 3.2 Understanding Relations and Functions-NOTES

Name Class Date. Understanding Relations and Functions-NOTES Essential Question: How do ou represent relations and functions? Eplore A1.1.A decide whether relations represented verball, tabularl, graphicall,

### Ready To Go On? Skills Intervention 2-1 Solving Linear Equations and Inequalities

A Read To Go n? Skills Intervention -1 Solving Linear Equations and Inequalities Find these vocabular words in Lesson -1 and the Multilingual Glossar. Vocabular equation solution of an equation linear

### Name Date. and y = 5.

Name Date Chapter Fair Game Review Evaluate the epression when = and =.... 0 +. 8( ) Evaluate the epression when a = 9 and b =.. ab. a ( b + ) 7. b b 7 8. 7b + ( ab ) 9. You go to the movies with five

### Systems of Linear and Quadratic Equations. Check Skills You ll Need. y x. Solve by Graphing. Solve the following system by graphing.

NY- Learning Standards for Mathematics A.A. Solve a sstem of one linear and one quadratic equation in two variables, where onl factoring is required. A.G.9 Solve sstems of linear and quadratic equations

### Algebra 2 Unit 2 Practice

Algebra Unit Practice LESSON 7-1 1. Consider a rectangle that has a perimeter of 80 cm. a. Write a function A(l) that represents the area of the rectangle with length l.. A rectangle has a perimeter of

### Linear and Nonlinear Systems of Equations. The Method of Substitution. Equation 1 Equation 2. Check (2, 1) in Equation 1 and Equation 2: 2x y 5?

3330_070.qd 96 /5/05 Chapter 7 7. 9:39 AM Page 96 Sstems of Equations and Inequalities Linear and Nonlinear Sstems of Equations What ou should learn Use the method of substitution to solve sstems of linear

### Ch 3 Alg 2 Note Sheet.doc 3.1 Graphing Systems of Equations

Ch 3 Alg Note Sheet.doc 3.1 Graphing Sstems of Equations Sstems of Linear Equations A sstem of equations is a set of two or more equations that use the same variables. If the graph of each equation =.4

### 4 The Cartesian Coordinate System- Pictures of Equations

The Cartesian Coordinate Sstem- Pictures of Equations Concepts: The Cartesian Coordinate Sstem Graphs of Equations in Two Variables -intercepts and -intercepts Distance in Two Dimensions and the Pthagorean

### Lecture Guide. Math 90 - Intermediate Algebra. Stephen Toner. Intermediate Algebra, 2nd edition. Miller, O'Neill, & Hyde. Victor Valley College

Lecture Guide Math 90 - Intermediate Algebra to accompan Intermediate Algebra, 2nd edition Miller, O'Neill, & Hde Prepared b Stephen Toner Victor Valle College Last updated: 11/24/10 0 1.1 Sets of Numbers

### 1.7 Inverse Functions

71_0107.qd 1/7/0 10: AM Page 17 Section 1.7 Inverse Functions 17 1.7 Inverse Functions Inverse Functions Recall from Section 1. that a function can be represented b a set of ordered pairs. For instance,

### Ready To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions

Read To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions Find these vocabular words in Lesson 5-1 and the Multilingual Glossar. Vocabular quadratic function parabola verte

### Linear Equations and Arithmetic Sequences

CONDENSED LESSON.1 Linear Equations and Arithmetic Sequences In this lesson, ou Write eplicit formulas for arithmetic sequences Write linear equations in intercept form You learned about recursive formulas

### -5(1-5x) +5(-8x - 2) = -4x -8x. Name Date. 2. Find the product: x 3 x 2 x. 3. Solve the following equation for x.

Name Date CC Algebra 2 Period Units 1-5 Review Due Tuesday, January 3, 2017 Answer all of the following questions. The number of each question corresponds to the lesson in which it was covered. Copying

### 3.7 InveRSe FUnCTIOnS

CHAPTER functions learning ObjeCTIveS In this section, ou will: Verif inverse functions. Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.

### Study Guide and Intervention

6- NAME DATE PERID Stud Guide and Intervention Graphing Quadratic Functions Graph Quadratic Functions Quadratic Function A function defined b an equation of the form f () a b c, where a 0 b Graph of a

### Course 15 Numbers and Their Properties

Course Numbers and Their Properties KEY Module: Objective: Rules for Eponents and Radicals To practice appling rules for eponents when the eponents are rational numbers Name: Date: Fill in the blanks.

### Domain, Range, and End Behavior

Locker LESSON 1.1 Domain, Range, and End Behavior Common Core Math Standards The student is epected to: F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship

### Interpret Linear Graphs

Interpret Linear Graphs Objectives: -Interpret the meaning of the and intercepts, slope, and points on and off the line of a graph, in the contet of a real world situation. Common Core Standards: N.Q.1

Name: Date: QUADRATIC FUNCTION REVIEW Linear and eponential functions are used throughout mathematics and science due to their simplicit and applicabilit. Quadratic functions comprise another ver important

### 9-1. The Function with Equation y = ax 2. Vocabulary. Graphing y = x 2. Lesson

Chapter 9 Lesson 9-1 The Function with Equation = a BIG IDEA The graph of an quadratic function with equation = a, with a 0, is a parabola with verte at the origin. Vocabular parabola refl ection-smmetric

### MATH 152 COLLEGE ALGEBRA AND TRIGONOMETRY UNIT 1 HOMEWORK ASSIGNMENTS

0//0 MATH COLLEGE ALGEBRA AND TRIGONOMETRY UNIT HOMEWORK ASSIGNMENTS General Instructions Be sure to write out all our work, because method is as important as getting the correct answer. The answers to

### Fair Game Review. Chapter 2. and y = 5. Evaluate the expression when x = xy 2. 4x. Evaluate the expression when a = 9 and b = 4.

Name Date Chapter Fair Game Review Evaluate the epression when = and =.... 0 +. 8( ) Evaluate the epression when a = 9 and b =.. ab. a ( b + ) 7. b b 7 8. 7b + ( ab ) 9. You go to the movies with five

### Bridge-Thickness Experiment. Student 2

Applications 1. Below are some results from the bridge-thickness eperiment. Bridge-Thickness Eperiment Thickness (laers) Breaking Weight (pennies) 15 5 5 a. Plot the (thickness, breaking weight) data.

### MATH 1710 College Algebra Final Exam Review

MATH 7 College Algebra Final Eam Review MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. ) There were 80 people at a pla. The admission price was \$

### NAME DATE PERIOD. Study Guide and Intervention. Ax + By = C, where A 0, A and B are not both zero, and A, B, and C are integers with GCF of 1.

NAME DATE PERID 3-1 Stud Guide and Intervention Graphing Linear Equations Identif Linear Equations and Intercepts A linear equation is an equation that can be written in the form A + B = C. This is called

### CCSSM Algebra: Equations

CCSSM Algebra: Equations. Reasoning with Equations and Inequalities (A-REI) Eplain each step in solving a simple equation as following from the equalit of numbers asserted at the previous step, starting

### Precalculus Honors - AP Calculus A Information and Summer Assignment

Precalculus Honors - AP Calculus A Information and Summer Assignment General Information: Competenc in Algebra and Trigonometr is absolutel essential. The calculator will not alwas be available for ou

### Functions. Introduction

Functions,00 P,000 00 0 970 97 980 98 990 99 000 00 00 Figure Standard and Poor s Inde with dividends reinvested (credit "bull": modification of work b Praitno Hadinata; credit "graph": modification of

### Laurie s Notes. Overview of Section 3.5

Overview of Section.5 Introduction Sstems of linear equations were solved in Algebra using substitution, elimination, and graphing. These same techniques are applied to nonlinear sstems in this lesson.

### Essential Question How can you use a quadratic function to model a real-life situation?

3. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..A A..E A..A A..B A..C Modeling with Quadratic Functions Essential Question How can ou use a quadratic function to model a real-life situation? Work with a partner.

### 15.2 Graphing Logarithmic

_ - - - - - - Locker LESSON 5. Graphing Logarithmic Functions Teas Math Standards The student is epected to: A.5.A Determine the effects on the ke attributes on the graphs of f () = b and f () = log b

### 6. Graph each of the following functions. What do you notice? What happens when x = 2 on the graph of b?

Pre Calculus Worksheet 1. Da 1 1. The relation described b the set of points {(-,5,0,5,,8,,9 ) ( ) ( ) ( )} is NOT a function. Eplain wh. For questions - 4, use the graph at the right.. Eplain wh the graph

Chapter 11 Quadratic Functions Mathematical Overview The relationship among parabolas, quadratic functions, and quadratic equations is investigated through activities that eplore both the geometric and

### Solving Systems Using Tables and Graphs

3-1 Solving Sstems Using Tables and Graphs Vocabular Review 1. Cross out the equation that is NOT in slope-intercept form. 1 5 7 r 5 s a 5!3b 1 5 3 1 7 5 13 Vocabular Builder linear sstem (noun) LIN ee

### Fair Game Review. Chapter 2. and y = 5. Evaluate the expression when x = xy 2. 4x. Evaluate the expression when a = 9 and b = 4.

Name Date Chapter Fair Game Review Evaluate the epression when = and =.... 0 +. 8( ) Evaluate the epression when a = 9 and b =.. ab. a ( b + ) 7. b b 7 8. 7b + ( ab ) 9. You go to the movies with five

### P.4 Lines in the Plane

28 CHAPTER P Prerequisites P.4 Lines in the Plane What ou ll learn about Slope of a Line Point-Slope Form Equation of a Line Slope-Intercept Form Equation of a Line Graphing Linear Equations in Two Variables

### Maintaining Mathematical Proficiency

Name Date Chapter 3 Maintaining Mathematical Proficienc Plot the point in a coordinate plane. Describe the location of the point. 1. A( 3, 1). B (, ) 3. C ( 1, 0). D ( 5, ) 5. Plot the point that is on

### 5 Linear Graphs and Equations

Linear Graphs and Equations. Coordinates Firstl, we recap the concept of (, ) coordinates, illustrated in the following eamples. Eample On a set of coordinate aes, plot the points A (, ), B (0, ), C (,

### Mt. Douglas Secondary

Foundations of Math 11 Section.1 Review: Graphing a Linear Equation 57.1 Review: Graphing a Linear Equation A linear equation means the equation of a straight line, and can be written in one of two forms.

### Mini-Lecture 8.1 Solving Quadratic Equations by Completing the Square

Mini-Lecture 8.1 Solving Quadratic Equations b Completing the Square Learning Objectives: 1. Use the square root propert to solve quadratic equations.. Solve quadratic equations b completing the square.

### 2. Domain: The set of all abscissas (x s) of the ordered pairs (abscissa is the first element of an ordered pair)

. Relations and Functions. Relation: A set of ordered pairs E:,4,,5,,, 8,4. The set of all abscissas s of the ordered pairs abscissa is the first element of an ordered pair. Range: The set of all ordinates

### x c x c This suggests the following definition.

110 Chapter 1 / Limits and Continuit 1.5 CONTINUITY A thrown baseball cannot vanish at some point and reappear someplace else to continue its motion. Thus, we perceive the path of the ball as an unbroken

### Copyrighted by Gabriel Tang B.Ed., B.Sc. Page 1.

Chapter : Linear and Quadratic Functions Chapter : Linear and Quadratic Functions -: Points and Lines Sstem of Linear Equations: - two or more linear equations on the same coordinate grid. Solution of

### Unit 26 Solving Inequalities Inequalities on a Number Line Solution of Linear Inequalities (Inequations)

UNIT Solving Inequalities: Student Tet Contents STRAND G: Algebra Unit Solving Inequalities Student Tet Contents Section. Inequalities on a Number Line. of Linear Inequalities (Inequations). Inequalities

### 2 nd Semester Final Exam Review Block Date

Algebra 1B Name nd Semester Final Eam Review Block Date Calculator NOT Allowed Graph each function. 1 (10-1) 1. (10-1). (10-1) 3. (10-1) 4. 3 Graph each function. Identif the verte, ais of smmetr, and

88 Linear and Quadratic Functions. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions:

### Maintaining Mathematical Proficiency

Chapter Maintaining Mathematical Proficienc Find the -intercept of the graph of the linear equation. 1. = + 3. = 3 + 5 3. = 10 75. = ( 9) 5. 7( 10) = +. 5 + 15 = 0 Find the distance between the two points.

### Instructor: Imelda Valencia Course: A3 Honors Pre Calculus

Student: Date: Instructor: Imelda Valencia Course: A3 Honors Pre Calculus 01 017 Assignment: Summer Homework for those who will be taking FOCA 017 01 onl available until Sept. 15 1. Write the epression

### Math 121. Practice Questions Chapters 2 and 3 Fall Find the other endpoint of the line segment that has the given endpoint and midpoint.

Math 11. Practice Questions Chapters and 3 Fall 01 1. Find the other endpoint of the line segment that has the given endpoint and midpoint. Endpoint ( 7, ), Midpoint (, ). Solution: Let (, ) denote the

### Lesson 8.2 Exercises, pages

Lesson 8. Eercises, pages 38 A Students should verif the solutions to all equations.. Which values of are not roots of each equation? a) ƒ - 3 ƒ = 7 = 5 or =- Use mental math. 5: L.S. 7 R.S. 7 : L.S. 7

### Module 3, Section 4 Analytic Geometry II

Principles of Mathematics 11 Section, Introduction 01 Introduction, Section Analtic Geometr II As the lesson titles show, this section etends what ou have learned about Analtic Geometr to several related

### How can you write an equation of a line when you are given the slope and a point on the line? ACTIVITY: Writing Equations of Lines

.7 Writing Equations in Point-Slope Form How can ou write an equation of a line when ou are given the slope and a point on the line? ACTIVITY: Writing Equations of Lines Work with a partner. Sketch the

### For problems 1 4, evaluate each expression, if possible. Write answers as integers or simplified fractions

/ MATH 05 TEST REVIEW SHEET TO THE STUDENT: This Review Sheet gives you an outline of the topics covered on Test as well as practice problems. Answers are at the end of the Review Sheet. I. EXPRESSIONS

### Algebra 2 CPA Summer Assignment 2018

Algebra CPA Summer Assignment 018 This assignment is designed for ou to practice topics learned in Algebra 1 that will be relevant in the Algebra CPA curriculum. This review is especiall important as ou

### Solve each system by graphing. Check your solution. y =-3x x + y = 5 y =-7

Practice Solving Sstems b Graphing Solve each sstem b graphing. Check our solution. 1. =- + 3 = - (1, ). = 1 - (, 1) =-3 + 5 3. = 3 + + = 1 (, 3). =-5 = - 7. = 3-5 3 - = 0 (1, 5) 5. -3 + = 5 =-7 (, 7).

### Maintaining Mathematical Proficiency

Name Date Chapter 5 Maintaining Mathematical Proficienc Graph the equation. 1. + =. = 3 3. 5 + = 10. 3 = 5. 3 = 6. 3 + = 1 Solve the inequalit. Graph the solution. 7. a 3 > 8. c 9. d 5 < 3 10. 8 3r 5 r

### Use Properties of Exponents

4. Georgia Performance Standard(s) MMAa Your Notes Use Properties of Eponents Goal p Simplif epressions involving powers. VOCABULARY Scientific notation PROPERTIES OF EXPONENTS Let a and b be real numbers

### Essential Question How can you solve a system of linear equations? \$15 per night. Cost, C (in dollars) \$75 per Number of. Revenue, R (in dollars)

5.1 Solving Sstems of Linear Equations b Graphing Essential Question How can ou solve a sstem of linear equations? Writing a Sstem of Linear Equations Work with a partner. Your famil opens a bed-and-breakfast.

### Section 3.1 Solving Linear Systems by Graphing

Section 3.1 Solving Linear Sstems b Graphing Name: Period: Objective(s): Solve a sstem of linear equations in two variables using graphing. Essential Question: Eplain how to tell from a graph of a sstem

### SOLVING SYSTEMS OF EQUATIONS

SOLVING SYSTEMS OF EQUATIONS 4.. 4..4 Students have been solving equations even before Algebra. Now the focus on what a solution means, both algebraicall and graphicall. B understanding the nature of solutions,

### Chapter Start Thinking! For use before Activity 6.1. For use before Activity Start Thinking! For use before Lesson

. Enrichment and Etension. a =, b =. a =, b =. a =, b =. a =, b =. a =, b is an number ecept.. a =, b =. a =, b =. a =, b =. Check students work.. Puzzle PAY HIM Etension. Start Thinking! For use before

### 2.3 Solving Absolute Value Inequalities

.3 Solving Absolute Value Inequalities Essential Question: What are two was to solve an absolute value inequalit? Resource Locker Eplore Visualizing the Solution Set of an Absolute Value Inequalit You

### 9 (0, 3) and solve equations to earn full credit.

Math 0 Intermediate Algebra II Final Eam Review Page of Instructions: (6, ) Use our own paper for the review questions. For the final eam, show all work on the eam. (-6, ) This is an algebra class do not

### M122 College Algebra Review for Final Exam

M1 College Algebra Review for Final Eam Revised Fall 017 for College Algebra - Beecher All answers should include our work (this could be a written eplanation of the result, a graph with the relevant feature

### Functions. Introduction

Functions,00 P,000 00 0 70 7 80 8 0 000 00 00 Figure Standard and Poor s Inde with dividends reinvested (credit "bull": modification of work b Praitno Hadinata; credit "graph": modification of work b MeasuringWorth)

### The semester B examination for Algebra 2 will consist of two parts. Part 1 will be selected response. Part 2 will be short answer. n times per year: 1

ALGEBRA B Semester Eam Review The semester B eamination for Algebra will consist of two parts. Part 1 will be selected response. Part will be short answer. Students ma use a calculator. If a calculator

### NCC Precalculus Partnership Program Final Examination, 2004

NCC Precalculus Partnership Program Final Eamination, 2004 Part I: Answer onl 20 of the 25 questions below. Each question is worth 2 points. Place our answers on the answer sheet provided. Write the word

### Chapter 9 BUILD YOUR VOCABULARY

C H A P T E R 9 BUILD YUR VCABULARY Chapter 9 This is an alphabetical list of new vocabular terms ou will learn in Chapter 9. As ou complete the stud notes for the chapter, ou will see Build Your Vocabular

### Systems of Linear Equations

Sstems of Linear Equations Monetar Sstems Overload Lesson 3-1 Learning Targets: Use graphing, substitution, and elimination to solve sstems of linear equations in two variables. Formulate sstems of linear

### MATH 021 UNIT 1 HOMEWORK ASSIGNMENTS

MATH 01 UNIT 1 HOMEWORK ASSIGNMENTS General Instructions You will notice that most of the homework assignments for a section have more than one part. Usuall, the part (A) questions ask for eplanations,

### Systems of Linear Equations Monetary Systems Overload

Sstems of Linear Equations SUGGESTED LEARNING STRATEGIES: Shared Reading, Close Reading, Interactive Word Wall Have ou ever noticed that when an item is popular and man people want to bu it, the price

3.1 Graphing Quadratic Functions A. Quadratic Functions Completing the Square Quadratic functions are of the form. 3. It is easiest to graph quadratic functions when the are in the form using transformations.

### Algebra 2 Honors Summer Packet 2018

Algebra Honors Summer Packet 018 Solving Linear Equations with Fractional Coefficients For these problems, ou should be able to: A) determine the LCD when given two or more fractions B) solve a linear

### MATH 110: FINAL EXAM REVIEW

MATH 0: FINAL EXAM REVIEW Can you solve linear equations algebraically and check your answer on a graphing calculator? (.) () y y= y + = 7 + 8 ( ) ( ) ( ) ( ) y+ 7 7 y = 9 (d) ( ) ( ) 6 = + + Can you set

### Derivatives 2: The Derivative at a Point

Derivatives 2: The Derivative at a Point 69 Derivatives 2: The Derivative at a Point Model 1: Review of Velocit In the previous activit we eplored position functions (distance versus time) and learned

### Algebra 1 Unit 9 Quadratic Equations

Algebra 1 Unit 9 Quadratic Equations Part 1 Name: Period: Date Name of Lesson Notes Tuesda 4/4 Wednesda 4/5 Thursda 4/6 Frida 4/7 Monda 4/10 Tuesda 4/11 Wednesda 4/12 Thursda 4/13 Frida 4/14 Da 1- Quadratic

### North Carolina Community College System Diagnostic and Placement Test Sample Questions

North Carolina Communit College Sstem Diagnostic and Placement Test Sample Questions 0 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College

### Can a system of linear equations have no solution? Can a system of linear equations have many solutions?

5. Solving Special Sstems of Linear Equations Can a sstem of linear equations have no solution? Can a sstem of linear equations have man solutions? ACTIVITY: Writing a Sstem of Linear Equations Work with

### 11.1 Inverses of Simple Quadratic and Cubic Functions

Locker LESSON 11.1 Inverses of Simple Quadratic and Cubic Functions Teas Math Standards The student is epected to: A..B Graph and write the inverse of a function using notation such as f (). Also A..A,

### Name Class Date. Solving by Graphing and Algebraically

Name Class Date 16-4 Nonlinear Sstems Going Deeper Essential question: How can ou solve a sstem of equations when one equation is linear and the other is quadratic? To estimate the solution to a sstem