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

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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 COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

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. COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

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: COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

4 4 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. COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

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. COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

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: COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

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. (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. In Eercise #, 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 #: 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. (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. (b) How much is Edeln s fied cost? In other words, how much would she have to pa for making zero phone calls? COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

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 C0.8w Each widget sells for 6 cents. Thus, the revenue gained, R, from selling these widgets is given b R 0.6w. 8 (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. Dollars w (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. (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. Dollars w COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

9 9 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? () D56t 58 () D58t 56 () D56 58t (4) D58 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? () P8 () P8 () P7 (4) P7. 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). C (Celsius) (d) What does this model project to be the average global temperature in 00? COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

10 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. 0 (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. (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 COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

11 LESSON #4 - 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? COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

12 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) COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

13 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? 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? COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

14 4 FLUENCY 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? COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

15 5 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. COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

16 6 LESSON #5 - 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. COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

17 7 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. COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

18 Piecewise equations can be challenging algebraicall. Sometimes information that we find from them can be misleading or incorrect. Eercise #6: Consider the piecewise linear function g (a) Determine the -intercept of this function algebraicall. Wh can a function have onl one -intercept? 5. 8 (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 COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

19 9 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. COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

20 . 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. 0 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 COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

21 LESSON #6 - MODELLING WITH ABSOLUTE VALUE FUNCTIONS COMMON CORE ALGEBRA II In lesson 6, 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. COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

22 Eercise #: Graph the following absolute value equation and show the table of values (including the verte): g( ) 6 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 #: Write an absolute value function, f(), to model the difference between the dail low temperature,, and 5 degrees Fahrenheit. Create an absolute value graph over the interval, (Use the table on the calculator to determine the appropriate window for viewing.) In Chicago, the maimum value of f() was 0 degrees Fahrenheit in Januar. Find the range of possible low temperatures in Chicago that month. Eercise #4: 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. 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 range of acceptable weights. COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

23 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. (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 range of acceptable weights. COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

24 . 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. 4 (b) 95% of the palm widths were within 0.6 inches of the average. Use a graph of the absolute value function to find the range of possible palm widths for 95% of the population surveed. COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

25 5 LESSON #7 - 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 7 (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. COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

26 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. 6 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? 4z 5 z 8 a. (-,,-4) b. (,,-) c. (,8,) d.(,-,0) COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

27 7 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. COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

28 8 FLUENCY SYSTEMS OF LINEAR EQUATIONS COMMON CORE ALGEBRA II HOMEWORK. 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 COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

29 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 9 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 COMMON CORE ALGEBRA II, UNIT # LINEAR FUNCTIONS AND EQUATIONS

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