explicit expression, recursive, composition of functions, arithmetic sequence, geometric sequence, domain, range

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1 Jordan-Granite-Canyons Consortium Secondary Math 1: Unit B (7 8 Weeks) Unit : Linear and Eponential Relationships In earlier grades, students define, evaluate, and compare functions, and use them to model relationships between quantities. In this unit, students will learn function notation and develop the concepts of domain and range. They move beyond viewing functions as processes that take inputs and yield outputs and start viewing functions as objects in their own right. They eplore many eamples of functions, including sequences; they interpret functions given graphically, numerically, symbolically, and verbally, translate between representations, and understand the limitations of various representations. They work with functions given by graphs and tables, keeping in mind that, depending upon the contet, these representations are likely to be approimate and incomplete. Their work includes functions that can be described or approimated by formulas as well as those that cannot. When functions describe relationships between quantities arising from a contet, students reason with the units in which those quantities are measured. Students build on and informally etend their understanding of integer eponents to consider eponential functions. They compare and contrast linear and eponential functions, distinguishing between additive and multiplicative change. They interpret arithmetic sequences as linear functions and geometric sequences as eponential functions. U Cluster 5: Build a function that models a relationship between two quantities. Limit.5.1 and.5. to linear and eponential functions. In.5., connect arithmetic sequences to linear functions and geometric sequences to eponential functions..5.1 Write a function that describes a relationship between two quantities. a. Determine an eplicit epression, a recursive process, or steps for calculation from a contet. b. Combine standard function types using arithmetic operations. For eample, build a function that models the temperature of a cooling body by adding a constant function to a decaying eponential, and relate these functions to the model..5. Write arithmetic and geometric sequences both recursively and with an eplicit formula, use them to model situations, and translate between the two forms. I can statements: I can write a function that describes a linear or eponential relationship between two quantities. I can combine different functions using addition, subtraction, multiplication, division and composition of functions to create a new function. I can write arithmetic and geometric sequences recursively. I can write an eplicit formula for arithmetic and geometric sequences. I can connect arithmetic sequences to linear functions and geometric sequences to eponential functions and eplain those connections. Vocabulary eplicit epression, recursive, composition of functions, arithmetic sequence, geometric sequence, domain, range I can write a function that describes a linear or eponential relationship between two quantities. This can be combined with other I can statements in this unit. The function could be written in algebraic form or as a verbal description of a relationship between two quantities. Students should not only be able to write a function in mathematical notation, they should also be able to eplain the linear or eponential relationship in words. WRITING A LINEAR OR EXPONENTIAL RELATIONSHIP Eample: (Linear) A battery operated car is traveling on a smooth road at a constant speed The distance the car will travel is 16 feet the first second, 48 feet the net second, 80 feet the third second, and so on in an arithmetic sequence. Write a function that describes the relationship between the number of seconds and the distance the rock has fallen. Number of seconds 1 Students could use a table or a graph to help them write the function. Distance So our function is: f() = 16 NOTE: Students will have had eperience writing linear functions in 8 th grade mathematics.

2 WRITING A LINEAR OR EXPONENTIAL RELATIONSHIP Eample: (Eponential) Maya is on vacation and is staying at Hotel Caliente. She complains to the manager that the hot tub in her hotel suite is not hot enough. The hotel tells her that they will increase the temperature by 10% each hour. If the current temperature of the hot tub is 70º F, can you write a function that describes the relationship between how many hours have elapsed and the temperature of the hot tub. The general form of an eponential function is y = a y a is the initial temperature of the hot tub. a b. In this contet, y is the temperature of the hot tub, is the number of hours and We know that the initial temperature was 70º F. The temperature is going to increase by 10%, which written as a decimal is In order to get the new temperature after an hour has elapsed, we must add the increase to the initial temperature, that is ( ). Therefore, b must be So our function is: h() = 70(1.10) I can combine different functions using addition, subtraction, multiplication, division and composition of functions to create a new function. COMBINING FUNCTIONS Students should be able to create a new function by using addition, subtraction, multiplication and division to combine two different linear and/or eponential functions. They will also need to be able to do simple compositions of functions. Discussions with students should include the impact on the domain and range when two functions are combined together. This can be done by looking at the graphs, the equations or even tables of values. Using a graphing calculator or other technology can be very useful when looking at the graphs of these combined functions. It will allow students to eamine graphs that are more difficult to create by hand and might otherwise be inaccessible. Eamples: Let f() = 4 5 and g() =. Find f ( ) g( ), f ( ) g( ), f ( ) g, and f ( ) g( ) 4 5 f ( ) g( ) 4 5 f ( ) g( ) (4 5) f ( ) 4 5 g ( ) f( ) g ( ). NOTE: The notation for combining functions can be a stumbling block for students. In particular, the notation for composition, (f g)(), is sometimes confused with the notation for the product of two functions, (f g)().

3 COMPOSING FUNCTIONS The term "composition of functions" (or "composite function") refers to the combining of functions in such a way that the output from one function becomes the input for the net function. In math terms we d say that the range (or y-values) from one function becomes the domain (or -values) of the net function. The notation used for composition is (f g)() = f(g()) and is read f composed with g of or f of g of. Notice how the letters stay in the same order in each epression for the composition. f (g()) clearly tells you to start with function g (innermost parentheses are done first). **ELL note: order here is particularly confusing for ELL students. In English we say the red car in Latin based languages (for eample) the literal translation is the car red. Hence, eplicit discussion with ELL students on this issue is important. Composition of functions can be thought of as a series of function machines to find your values. The eample below shows functions f and g working together to create the composition (f g)(). NOTE: The starting domain for function g is being limited to the four values 1,,, 4 and 5 for this eample. f() = + 1 g() = Together they create (f g)(). Domain of g() (-, ) Domain of f() (0, ) Domain of (f g)() /4 1/ 1 4 I m function g(). I ll pick up the values from the blue area, raise to that power (value), and drop them off in the yellow g() = I m function f(). I ll pick up the values from the yellow area, add 1 to each value, and drop them off in the green area. f() = + 1 1/4 1/ 1 4 5/4 / 5 Range of g() (0, ) Range of (f g)() (1, ) Just remember that (f g)() is the same as f(g()), so start with the innermost parentheses. The input or domain of g() is all real numbers. The output or range of g() is all real numbers greater than 0. The output or range of g() is now the input or domain of f(). Hence, the output or range of f() or f(g()) is all real numbers greater than 1.

4 COMPOSING FUNCTIONS (continued) Now, suppose that we wish to write our composition as an algebraic epression. f g f g f 1 1. Substitute the epression for function g (which in this case is ) in for g() in the composition. By putting this into function f as the input, it will clearly show you the order of the substitutions that will need to be made.. Now, substitute this epression ( ) into function f in place of the -value and then simplify. You will find that the concept of composite functions is widely used. Eamples: 1. Given the functions f 5 and g 1, find f g and g f. Answers: f g f g f g f g f g Notice that g f and f g do not necessarily yield the same answer. Composition of functions is not commutative.. Given the functions p and h ( ), find h p and h p Answers: h p h p where p() gives an answer of 5 and h(5) then gives an answer of. The answer is. h p h p h

5 I can write arithmetic and geometric sequences recursively. I can write an eplicit formula for arithmetic and geometric sequences. I can connect arithmetic sequences to linear functions and geometric sequences to eponential functions and eplain those connections. ARITHMETIC AND GEOMETRIC SEQUENCES Students should be able to describe an arithmetic or geometric sequence recursively as well as eplicitly. They should be able to eplain how the patterns they see in the sequence relate to the eplicit epression or function. Patterning and sequences are a great way to begin to understand the similarities and differences between linear and eponential functions. Eamples: Pattern 1: Suppose this is a pattern created by toothpicks. How many toothpicks are needed to create the 5 th step? How many toothpicks on the nth step? Use pictures, words or symbols to represent how this pattern is changing. Step 1 Step Step Step 4 Pattern : How many X in step 5? How many in step n? Use pictures, words or symbols to represent how this pattern is changing. Step 1 Step Step Step 4 Both Pattern 1 and Pattern are growing patterns. Let s eplore each of them for a moment without using a table right away. For Pattern 1 there are a few ways that you might see how the pattern grows: One way to show that is by writing the sequence of values for how many toothpicks are in each step: 4, 7, 10, (n-1) or n + 1 Other ways students may see Pattern (4 1) + (4 1) 4 + (4 1) + (4 1) + (4 1) 4 + (n 1) + (4 1) or n n + (n + 1) or n + 1 You could also write out in words: Start with 4 and then add for each successive step. a0 4 This is a verbal epression of the recursive epression: a a n n 1

6 ARITHMETIC AND GEOMETRIC SEQUENCES (continued) The above shows that students may see the pattern in a number of ways, but regardless, all valid eplications of the patterns will simplify to the same linear equation. It is important that students see the pattern from different perspectives and be comfortable algebraically representing different perspectives. In this case, all patterns simplify to: y = + 1 where y is the total number of toothpicks and is the step number. This obviously is an arithmetic sequence and the common difference is a constant rate of change which is why this pattern can be represented by a linear function. Now let s look at Pattern n Ways students may see Pattern ( + n-1 ) + n-1 or + ( n-1 ) or + ( 1 )( n-1 ) or + n (1 + n ) or + n Regardless of how the student sees the pattern, the sequence is: 4, 6, 10, 18, The pattern is not increasing by a common difference which tells us that it is not an arithmetic sequence. Again, students will see the pattern differently, but ultimately, all the patterns are the same. Now let s look at a table of values for both Pattern 1 and Pattern. Pattern 1: Pattern : Step Number Total Number of Toothpicks Difference Change Step Number Total Number of X Difference 7 Up + 6 Up 10 Up + 10 Up Up Up 8 The tables help us to see the difference between growth by equal intervals (linear) and growth by equal factors (eponential.) Notice that Pattern is increasing more rapidly than Pattern 1. There is a significant difference between linear and eponential growth. Change

7 U Cluster 6: Build new functions from eisting functions. Focus on vertical translations of graphs of linear and eponential functions. Relate the vertical translation of a linear function to its y- intercept. While applying other transformations to a linear graph is appropriate at this level, it may be difficult for students to identify or distinguish between the effects of the other transformations included in this standard..6.1 Identify the effect on the graph of replacing f() by f() + k, k f(), f(k), and f( + k) for specific values of k (both positive and negative); find the value of k given the graphs. Eperiment with cases and illustrate an eplanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic epressions for them. I can statements: I can identify and eplain (in words, pictures or with tables) the effect k on a graph of f() i.e f() + k, kf(), f(k), and f( + k). I can find the value of k given the graphs. I can recognize even and odd functions from their graphs and algebraic epressions. Vocabulary odd function, even function, transformations of a function, vertical translation I can identify and eplain (in words, pictures or with tables) the effect k on a graph of f() i.e f() + k, kf(), f(k), and f( + k). I can find the value of k given the graphs. TRANSFORMATIONS OF FUNCTIONS In this cluster, students are introduced to transformations of functions. The focus should be on helping students build a solid understanding of the vertical translations of the graphs of linear and eponential functions and making the connection to the y-intercept of the graph. Transformations should be approached from the perspective of using a constant, k, to make changes to the function. What happens when you add k to the input? What if you add it to the input? How does the graph changes if you multiply the output by k? What if you multiply the input by k? What happens when k is a negative number? Students need to eperiment with the various cases and use technology to eplore and be able to eplain the effects on the graph. Note: It is difficult to see the effects of transformations on linear and eponential functions, ecept for the vertical shift. The following link is an ecellent summary of transformations of functions. ( However, it goes above and beyond what is needed for this course. The eamples below are more along the lines of what students need to know. Eamples: In the eamples below, k = or k = -. The original functions are shown first. In subsequent graphs, the original graphs are shown in blue and the transformed graphs are shown in red. Students should also be able to find k if given a graph and the original function. ( ) 1 f ( ) g

8 TRANSFORMATIONS OF FUNCTIONS (continued) The transformation f() + k or g() + k is a vertical translation. This is the kind of transformation that should be emphasized. Notice that if k is positive the graph will shift up. If k is negative, then the graph will shift down. Attention should be paid to the effect of the vertical translation on the y-intercept of the graph of the function. Discussions should include the fact that the output of the function is being changed i.e. for f() it s decreased by and for g() increased by, hence the downward and upward translation respectively. f ( ) ( 1) g ( ) The transformation kf ( ) or kg( ) is a vertical stretch or compression. If k is negative, the graph will be reflected over the -ais. This transformation is very difficult to see with linear and eponential functions unless looking at a reflection with an eponential function. Again, the output of the function is being transformed i.e. the output is being multiplied by something, hence compressing, stretching or reflecting the output. ( ) ( 1) f ( ) g

9 I can recognize even and odd functions from their graphs and algebraic epressions. TRANSFORMATIONS OF FUNCTIONS (continued) The transformation f ( k) or g( k ) is a horizontal stretch or compression. If k is negative, the graph will be reflected over the y-ais. Again, the effects of this transformation are not easy to see with linear and eponential functions. A reflection over the y-ais with an eponential function is more apparent. Discussions of this transformation should include the fact that the input of the function is being changed. f ( ) ( ) 1 g( ) The transformation f ( k) or g( k ), is a horizontal translation. Notice that if k is positive, the graph will shift to the left. If k is negative, then the graph will shift to the right. Again, the input of the function is affected. The phrase insiders are liars does not help students make sense of the transformation. f ( ) ( ) 1 g ( )

10 EVEN AND ODD FUNCTIONS Even Functions: A function is even if f ( ) f ( ) for all in the domain of the function. Geometrically, the graph of an even function is symmetric with respect to the y-ais. That means that the graph of the function remains unchanged after reflection about the y-ais. h() = is an eample of an even function. You can see in the graph at the right that it is symmetric about the y-ais. Algebraically: h( ) h( ) ( ) Note: Linear functions are even only if they are horizontal lines. Eponential functions are never even. Eamples: Notice that the f() is symmetric about the y-ais but g() is not. f( ) 4 g( ) 4 Algebraically: f( ) 4 f( ) 4 g ( ) 4 g( ) 4 1 4

11 EVEN AND ODD FUNCTIONS (continued) Odd Functions: A function is odd if f ( ) f ( ) for all in the domain of the function. Geometrically, the graph of an odd function has rotational symmetry with respect to the origin. That means that the graph of the function will remain unchanged after a rotation of 180 about the origin. h() = is an eample of an odd function. Notice that if you rotate the graph 180 around the origin, it will match up with itself. Algebraically: h( ) h( ) h( ) ( ) Note: Linear functions are odd only if they pass through the origin. Eponential functions are never odd therefore, they are neither odd nor even. Eamples: Notice that if you rotate the graph of f() 180 about the origin, it will match up again. The graph of g() is not symmetric with respect to the origin, even though it does pass through it. f ( ) 4 g ( ) 1 Algebraically: f ( ) 4 f ( ) 4 f ( ) 4( ) 4 g ( ) 1 g ( ) ( 1) 1 1 g( ) 1 1

12 U Cluster 7: Construct and compare linear, quadratic, and eponential models and solve problems. For.7., limit to comparisons between eponential and linear models..7.1 Distinguish between situations that can be modeled with linear functions and with eponential functions. a. Prove that linear functions grow by equal differences over equal intervals; eponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another..7. Construct linear and eponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or input-output pairs (include reading these from a table)..7. Observe using graphs and tables that a quantity increasing eponentially eventually eceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. I can statements: I can distinguish between situations that can be modeled with linear and eponential functions. I can prove (using words, pictures, numbers and/or difference tables) that linear functions grow by equal differences over equal intervals. I can prove (using words, pictures, numbers and/or difference table) that eponential functions grow by equal factors over equal intervals. I can recognize situations with a constant rate of change. I can recognize situations in which a quantity either grows or decays by a constant percent rate. I can construct a linear function given an arithmetic sequence, a graph, a description of a relationship or a table of input-output pairs. I can construct an eponential function given a geometric sequence, a graph, a description of a relationship or a table of input-output pairs. I can eplain why a quantity increasing eponentially will eventually eceed a quantity increasing linearly. Vocabulary linear model, eponential model, interval, increasing eponentially, increasing linearly, equal differences, equal factors, constant rate of change, constant percent rate I can distinguish between situations that can be modeled with linear and eponential functions. I can recognize situations with a constant rate of change. I can recognize situations in which a quantity either grows or decays by a constant percent rate. MODELING WITH LINEAR AND EXPONENTIAL FUNCTIONS Students should recognize that situations that can be modeled with a linear function can be identified by having a constant rate of change, whereas situations where an eponential model is appropriate have a rate of change that is a constant percent rate. Eamples: Identify whether a linear or eponential function should be used to model each of the following situations. 1. James had surgery on his left knee. As part of his rehabilitation, the physical therapist recommends that he start jogging. James is to jog for 1 minutes each day for the first week. Each week thereafter, James is to increase the time that he jogs each day by 6 minutes. (Linear). The sum of the interior angles of a triangle is 180º, of a quadrilateral is 60º, of a pentagon is 540º and of a heagon is70º. (Linear). A culture of bacteria contains 500 individual organisms and doubles every hours. (Eponential) 4. A mine worker discovers an ore sample containing 500 mg of a radioactive material. It is discovered that the radioactive material has a half life of 1 day. (Eponential)

13 I can prove (using words, pictures, numbers and/or difference tables) that linear functions grow by equal differences over equal intervals. LINEAR FUNCTIONS GROW BY EQUAL DIFFERNCES This proof is not a formal mathematical proof, or even the two-column proof that we are used to seeing in Geometry. Rather, the students should be able to show, justify and eplain using words, pictures, graphs, numbers and difference tables. Eample: The graph at the right shows the function y 1 5. Notice that over the interval 0 10 the y value goes from 5 to 0, a difference of 5. Looking at the interval 0 10, which is the same size as the previous interval, the y value goes from 15 to 10, also a difference of 5. 5 This shows that for a linear function, the rate of change is the same over equal intervals. The same conclusion can be drawn from the difference table below y interval Difference in y

14 I can prove (using words, pictures, numbers and/or difference table) that eponential functions grow by equal factors over equal intervals. EXPONENTIAL FUNCTIONS GROW BY EQUAL FACTORS This proof is not a formal mathematical proof, or even the two-column proof that we are used to seeing in Geometry. Rather, the students should be able to show, justify and eplain using words, pictures, graphs, numbers and difference tables. Eample: The graph at the right shows the function y. Notice that over the interval 0 1the y value goes from 1 to, a difference of. For the interval 1, which is the same size as the previous interval, the y value goes from to 9, a difference of 6. This shows that for an eponential function, the rate of change is not constant over equal intervals. However, if you take the ratio of the differences in the y-values you can see that each increase is by a factor of. The same conclusion can be drawn from the difference table below. y interval Difference in y 7 9 Ratios of differences

15 I can construct a linear function given an arithmetic sequence, a graph, a description of a relationship or a table of input-output pairs. I can construct an eponential function given a geometric sequence, a graph, a description of a relationship or a table of input-output pairs. CONSTRUCTING LINEAR AND EXPONENTIAL FUNCTIONS These goals are very much connected to other I can statements in this unit. The eamples below give an idea of what you could give students and then epect them to be able to write an algebraic epression of the function. Eamples: 1. For the linear function, f ( ) 1, students could be given any of the representations below and then asked to come up with the function. Arithmetic sequence: 1, 4, 7, 10, 1, 16, 19, Graph: Verbal description: An initial quantity is 1 and it increases by a constant rate of each time. Table of input/output values: f() For the eponential function, g( ), students could be given any of the following representations in order to construct the function. Geometric sequence: 1,, 4, 8, 16,, 64, Graph: Verbal description: A certain type of bacteria reproduces by the organism dividing into two. A new bacteria culture is started by putting one cell on a new agar plate. Table of input/output values: g()

16 I can eplain why a quantity increasing eponentially will eventually eceed a quantity increasing linearly. INCREASING EXPONENTIALLY AND LINEARLY Students should be able to use all of the different representations of a function to eplain why a quantity that is increasing eponentially will eventually be much larger than one that is increasing linearly. That is they should be able to eplain it verbally, by using a graph, a table of values, etc. Eamples: 1. The graph at the right shows a linear function, y 5, in blue, and an eponential function, y, in red. In looking at the graphs you can see that the linear function has higher y-values when the -values are in between - and. However, when the -values increase above 4, the eponential function increases more quickly and has larger y- values than the linear function. y = + 5 Note: If using technology, students could find the points of intersection of the two graphs and give precise intervals when comparing the two functions. y =. Suppose a company offers you a choice in how you are paid: Option A: You can earn $10,000 a day for 0 days. Option B: You earn $1 on the first day, $ on the second day, $4 on the third day, $8 on the fourth day, and so forth. In other words, they offer to pay you $1 on the first day and then double your pay each successive day for 0 days. Which option is better? Solution: Option A is fairly straightforward; each day you earn $10,000. So at the end of 0 days, you will earn $100,000 0 = $00,000. Option B seems a little less enticing given that on day one you get $1, day two $, day three $4, and so forth, it just doesn t seem like a lot of money. But let s write out the pattern further: 1,, 4, 8, 16,, 64, 18, 56, 51, 104, 048, 4096, 819, 1684, 768 In other words, by day 15 you earn more than $10,000 in a day. As a matter of fact, if you were to continue to double the pay each day, by day 0 the pay would be $55,870,91. Hence, Option B is a far better option than Option A!

17 U Cluster 8: Interpret epressions for functions in terms of the situation they model. Limit eponential functions to those of the form f() = b + k..8.1 Interpret the parameters in a linear or eponential function in terms of a contet. I can statements: I can interpret and eplain the parameters in an eponential function in terms of a given contet (authentic situation, graph, symbolic representation.) I can interpret how parts of an epression (terms, factors, coefficients) effect the value of the entire epression (e.g behaves differently than ) Vocabulary parameter, term, factor, coefficient, epression I can interpret and eplain the parameters in an eponential function in terms of a given contet (authentic situation, graph, symbolic representation.) I can interpret how parts of an epression (terms, factors, coefficients) effect the value of the entire epression (e.g. behaves differently than ) INTERPRET PARAMETERS AND PARTS OF AN EXPRESSION This cluster is where the objectives of Unit 1 really come into play. Students should understand how each parameter in a linear or eponential function are related to one another and how they are each manifested in the symbolic representation (i.e. equation), the graph etc. In an eponential function, the general form of the equation is y = a b + k, where b > 0 and b 0. Students should understand that is the independent quantity or input of the function, while y is the dependent quantity, or output. They should also know that a is the initial value of the output, b is the common factor by which output values change and k determines the vertical shift of the graph. They should recognize these parameters in all the different representations of a function. (See the preface to Unit ). They should eplore and determine how changes in one affect the entire function. Using technology to facilitate these eplorations is highly recommended. Students should be more familiar with linear functions since they will have spent some time studying them in 8 th grade. They should be able to eplain that in a linear function of the form y = m + b that m is the slope or rate of change and b is the y-intercept of the graph, (or the initial/base value in many real-world contets). Eample: (eponential function) Peter earned $1500 last summer. He deposited the money in a bank account that earns 5% interest compounded yearly. This problem deals with interest that is compounded yearly. This means that each year the interest is calculated on the amount of money you have in the bank. That interest is added to the original amount and net year the interest is calculated on this new amount. In this way, you get paid interest on the interest. Let s write a function that describes the amount of money in the bank. The general form of an eponential function is y = a b + k. In the given problem, y is the total amount of money in the bank, is the number of years from now, and a is the initial amount that Peter deposited in the account. We know that the interest is 5% each year. The decimals equivalent of 5% is In order to get the total amount of money for the following year, we must add the interest earned on the initial amount to the initial amount ( ). Hence, we see that b must be Thus, the function that describes this problem is y = 1500(1.05). Here are some questions that could be asked to highlight the effect of changes in each of the parameters: How much money will Peter have in 5 years? (i.e. when = 5) How much money would Peter have in 5 years if his initial deposit was only $1,000? (i.e. a is now changed) What if Peter had put his money in a bank account that paid 5.5% interest? (i.e. b = 1.055) Under the current conditions when will Peter have $0,000 in his account? (i.e. when will y = $0,000?)

18 INTERPRET PARAMETERS AND PARTS OF AN EXPRESSION (continued) Eample: (linear function) Sierra pays $49.99 dollars a month for her cell phone. This gives her 700 minutes of call time and unlimited teting each month. The charge for going over her allotted call time is $0.40 per minute of usage. Just about everyone has a cell phone, and most rate plans are a linear function of some kind. Let s write a function that describes the total cost to Sierra of having a cell phone. The general form of a linear function is y m b. In the given problem, y is the total amount of money Sierra pays for her cell phone each month, is the number of minutes that Sierra uses over her allotted call time of 700 minutes, m is the rate of $0.40 per minute and b is the basic cost of $ Thus, the function that describes this problem is y = Some questions that could be asked about this scenario: How much will Sierra have to pay if she uses 750 minutes one month? (i.e. when = 50) How much would Sierra s monthly bill come to if the month that she talked 750 minutes the charge was raised to $0.45 per minute of usage? (i.e. m has now been changed) How much will Sierra s monthly bill be if the cell phone company raises their basic plan to be $5.99 a month? (i.e. b is now changed) How many minutes can Sierra talk over her allotted time and keep her phone bill under $75.00? (i.e. when will y = $75.00?)

19 Teacher Information (Unit B) Connections to previous units within this course Connections to previous courses Misconceptions (difficulties a student may have): Foundations for future learning: Teacher Resources (Unit B) Manipulatives Instructional Media (teacher), Interactive Media (student), and Suggested Lessons Website links, teacher lessons, and other resources can be found at the following link: