mini lesson lesson 2a – linear functions and … · n, years of ownership v, value of the car in...

Lesson 2a – Linear Functions and Applications MAT12x-Intermediate Algebra

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MINI LESSON

Lesson 2a – L inear Funct ions and Appl icat ions Lesson Object ives:

1. Compute AVERAGE RATE OF CHANGE 2. Explain the meaning of AVERAGE RATE OF CHANGE as it relates to a given situation 3. Interpret AVERAGE RATE OF CHANGE as SLOPE 4. Solve application problems that involve LINEAR FUNCTIONS 5. Use LINEAR REGRESSION to write LINEAR FUNCTIONS that model given data sets 6. Solve application problems that involve using LINEAR REGRESSION

This lesson will combine the concepts of FUNCTIONS and LINEAR EQUATIONS. To write a linear equation as a LINEAR FUNCTION, replace the variable y with the FUNCTION NOTATION, f(x):

f(x) = mx + b You will see additional information on what makes a linear function in the lesson to follow. Important Th ings to Remember about the L INEAR FUNCTION f(x) = mx + b

• x represents the INPUT quantity. • f(x) represents the OUTPUT quantity (where f(x), pronounced, “f of x” really just means “y”). • The graph of f(x) is a straight line with slope, m, and y-intercept (0, b). If m>0, the graph

INCREASES from left to right, if m<0, the graph DECREASES from left to right. • The DOMAIN of a Linear Function is generally ALL REAL NUMBERS unless a context or situation is

applied in which case we interpret the PRACTICAL DOMAIN in that context or situation. • One way to identify the y-intercept is to evaluate f(0). In other words, substitute 0 for input (x)

and determine output (y). Remember that y-intercept and vertical intercept are the same thing. • To find the x-intercept, solve the equation f(x) = 0 for x. In other words, set mx + b = 0 and

solve for the value of x. Then (x, 0) is your x-intercept. Remember that x-intercept and horizontal intercept are the same thing.


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Problem 1 MEDIA EXAMPLE – AVERAGE RATE OF CHANGE You have decided to purchase a new Honda Accord LX but are concerned about how quickly the value of the car will depreciate. You search the Internet and find that for the car you want, the suggested retail price is $20,025. The value of the car, at the end of two years of ownership, would be $17,255. a) Determine the average rate of change, in $/year, for the vehicle’s value between year 0 and the end of year 2 using the information above. b) You do some additional searching and find that at the end of three years, the value of your chosen car would be $15,870. Find the average rate of change, in $/year, for the vehicle’s value between year 0 and the end of year 3. c) What is the MEANING of the average rate of change in this situation? Average rate of change (often just referred to as the “rate of change”) of a function over a specified

interval is the ratio

!

change in outputchange in input

= change in ychange in x

.

Units for the Average Rate of Change are always output unitsinput units

.

Problem 2 MEDIA EXAMPLE – AVERAGE RATE OF CHANGE AS SLOPE You do some additional Internet searching and find a fairly complete depreciation table for the car from Problem 1.

n, years of Ownership V, value of the car in dollars 0 20,025 1 18,640 2 17,255 3 15,870 5 13,100 8 8,945

a) Determine the Average Rate of Change, in $/year, for the vehicle’s value between year 0 and the end of year 5 and then year 8.


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Because the Average Rate of Change is constant for these deprec iat ion data, we say that a L INEAR FUNCTION models these data best . b) Use the information from Problem 1 and from part a) of this problem to write a linear equation that describes this situation. Does AVERAGE RATE OF CHANGE look familiar? It should! Another word for “average rate of change” is SLOPE. Given any two points, (x1, y1), (x2, y2), on a line, the slope is determined by computing the following ratio:

m =y2 ! y1x2 ! x1

= change in ychange in x

Therefore, AVERAGE RATE OF CHANGE = SLOPE over a given interval. Problem 3 MEDIA EXAMPLE – APPLICATIONS OF L INEAR FUNCTIONS Let’s continue working with this example of vehicle depreciation and step back now and get a more complete picture of the situation. a) Write here the function for V(n). ______________________ b) On the grid below, draw a GOOD graph of V(n) with all appropriate labels. If using your graphing calculator, use the WINDOW x[0..15] and y[0..21,000] and see the steps in the table below:


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GRAPHING CALCULATOR STEPS: Graph V = -1385N + 20025 Step 1: Press Y= then enter -1385x + 20025 [Note: If Plot1 is Highlighted, UNhighlight it by using the arrows to move your cursor to Plot1 then press ENTER.]

Step 2: Press WINDOW then enter these values: Xmin = 0, Xmax = 15, Ymin = 0, Ymax = 21000

Step 3: Press Graph

c) Identify the INPUT and OUTPUT quantities for this function and state what they represent. d) What is the STARTING VALUE for the vehicle? Identify and plot/label on your graph. e) After how many years is the value of the car $0? Identify and plot/label on your graph. [Note: This is a “Given Output, find Input problem…set the equation equal to 0 and solve for n]. f) Write in words the meaning of the mathematical statement V(5) = $13,100 as it relates to this problem. g) After how many years is the value of the car $10,330? [Note: This is a “Given Output, find Input problem…set the equation equal to 10,330 and solve for n].


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Recap – What Have We Learned So Far in th is Lesson So far in this lesson, we have learned a couple of key items as follows:

• LINEAR FUNCTIONS are just LINEAR EQUATIONS written using FUNCTION NOTATION. So, f(x) = mx + b means the same thing as y = mx + b but the notation is slightly different.

• AVERAGE RATE of CHANGE means the same thing as SLOPE. • Data sets that have a constant average rate of change (or we can say, constant rate of change),

can best be modeled by LINEAR FUNCTIONS. G iven Input , F ind Output and G iven Output , F ind Input Quest ions When working with LINEAR FUNCTIONS, there are two main questions we will ask and solve as follows:

• Given a particular INPUT value, what is the corresponding OUTPUT value? To address this question, you will EVALUTE the function at the given input by replacing the input variable with the input value and computing the result.

• Given a particular OUTPUT value, what is the corresponding INPUT value? To address this question, you will set the equation equal to the output value and solve for the value of the input.

Problem 4 YOU TRY – APPLICATIONS OF L INEAR FUNCTIONS The data below represent your annual salary for the first four years of your current job. The data are exactly linear.

Time, t, in years 0 1 2 3 4 Salary, S, in $ 20,100 20,600 21,100 21,600 22,100

a) Identify the vertical intercept and average rate of change for the data, then use these to write the linear function model for the data. Use the indicated variables and proper function notation. b) Use your model to determine the amount of your salary in year 8. [Note: This is a “Given Input, Find Output” question.] Write your final result as a complete sentence. c) Use your model to determine how many years you would need to work to earn a yearly salary of at least $40,000. Round to the nearest whole year. [Note: This is a “Given Output, Find Input” question.]. Write your final result as a complete sentence.


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Problem 5 YOU TRY – AVERAGE RATE OF CHANGE Data below represent body weight at the start of a 5-week diet program and each week thereafter.

Time, t, in weeks 0 1 2 3 4 5 Weight, W, in lbs 196 183 180 177 174 171

a) Compute the average rate of change for the first three weeks. Be sure to include units. b) Compute the average rate of change for the 5-wk period. Be sure to include units. c) What is the meaning of the average rate of change in this situation? d) Do the data points in the table define a perfectly linear function? Why or why not? Problem 6 WORKED EXAMPLE – Scat terp lots and L inear Regress ion Models The data above are not EXACTLY linear as your results for parts a) and b) should have showed (i.e. average rate of is not constant). BUT, just because data are not EXACTLY linear does not mean we cannot write an approximate linear model for the given data set. In fact, most data in the real world are NOT exactly linear and all we can do is write models that are close to the given values. The process for writing Linear Models for data that are not perfectly linear is called LINEAR REGRESSION. If you take a statistics class, you will learn a lot more about this process. In this class, you will be introduced to the basics. This process is also called “FINDING THE LINE OF BEST FIT”. Begin by CREATING A SCATTER PLOT to v iew the data po ints on a graph Step 1: Enter the data into your ca lcu lator

• Press STAT (Second Row of Keys) • Press ENTER to access 1:Edit under EDIT menu


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Note: Be sure all data columns are cleared. To do so, use your arrows to scroll up to L1 or L2 then click CLEAR then scroll down. (DO NOT CLICK DELETE!) Once your data columns are clear, enter the input data into L1 (press ENTER after each data value to get to the next row) then right arrow to L2 and enter the output data into L2. Your result should look like this when you are finished (for L1 and L2):

Step 2: Turn on your Stat P lot

• Press Y= • Use your arrow keys to scroll up to Plot1 • Press ENTER • Scroll down and Plot1 should be highlighted as at left

Step 3: Graph the Data

• Press ZOOM • Scroll down to 9:ZoomStat and press ENTER • A graph of your data should appear in an appropriate

window so that all data points are clearly visible (see below)

Other than the first data point, our data look pretty linear. To determine a linear equation that fits the given data, we could do a variety of things. We could choose the first and last point and use those to write the equation. We could ignore the first point and just use two of the remaining points. Our calculator, however, will give us the best linear equation possible taking into account ALL the given data points. To find this equation, we use a process called LINEAR REGRESSION.


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FINDING THE LINEAR REGRESSION EQUATION Step 1: Access the L inear Regress ion sect ion of your ca lcu lator

• Press STAT • Scroll to the right one place to CALC • Scroll down to 4:LinReg(ax+b) • Your screen should look as the one at left

Step 2: Determine the l inear regress ion equat ion

• Press ENTER twice in a row to view the screen at left • The calculator computes values for slope (a) and y-

intercept (b) in what is called the equation of best-fit for your data.

• Identify these values and round to the appropriate places. Let’s say 2 decimals in this case. So, a = -4.43 and b = 191.24

• Now, replace the a and b in y = ax + b with the rounded values to write the actual equation: y = -4.43x + 191.24

• To write the equation in terms of initial variables, we would write W = -4.43t + 191.24

Once we have the equation figured out, it’s nice to graph it on top of our data to see how things match up. GRAPHING THE REGRESSION EQUATION ON TOP OF THE STAT PLOT

• Enter the Regression Equation with rounded values into Y=

• Press GRAPH • You can see from the graph that the “best fit” line does

not hit very many of the given data points. But, it will be the most accurate linear model for the overall data set.

IMPORTANT NOTE: When you are finished graphing your data, TURN OFF YOUR PLOT1. Otherwise, you will encounter an INVALID DIMENSION error when trying to graph other functions. To do this: • Press Y= • Use your arrow keys to scroll up to Plot1 • Press ENTER • Scroll down and Plot1 should be UNhighlighted


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Problem 7 YOU TRY – L INEAR REGRESSION The following table shows the growth of snowboard sales with t = 0 in 1997 and S(t) representing the total sales in each year.

t (year) 1997 1998 1999 2000 2001 S(t) (total sales in $) 180,000 250,000 300,000 430,000 450,000

a) Use your calculator to determine the equation of the regression line. Refer to the steps outlined in Problem 5 for guidance. Start by entering new t values for the table based upon the distance each year is from 1997. The first few are done for you:

t (years since 1997) 0 1 S(t) (total sales in $) 180,000 250,000 300,000 430,000 450,000

Determine the regression equation in y = ax + b form and write it here: _______________________ Rewrite the regression equation in S(t) = at + b form and write it here: _______________________ b) Write in words the meaning of the mathematical statement S(3) = $430,000 as it relates to this problem. c) Use the result from part a) to estimate the total sales in 2007 (i.e. when t =10). Show your computations here and your final result. (Hint: This is a “Given input, find output” question, so compute the value of S(10) and show your work here). Write your final result as a complete sentence.

d) What is the slope of S(t) and what is its meaning in the context of this problem?

e) Use your S(t) result from part a) to estimate the year in which snowboard sales will be $600,000. Round to the closest whole year. (Hint: This is a “Given output find input” question so solve the equation S(t) = 600,000 for t and round to the nearest year. Add your result to 1997). Write your final result as a complete sentence.


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Problem 8 YOU TRY – L INEAR REGRESSION Given the data in the table below, use your calculator to plot a scatterplot of the data and draw a rough but accurate sketch in the space below.

x 0 2 4 6 8 10 12 y 23.76 24.78 25.93 26.24 26.93 27.04 27.93

b. Use your graphing calculator to determine the equation of the regression line. Round to three decimals as needed. ___________________________ c. Using the TABLE, determine the value of y when the input is 6: _____________. Write the specific ordered pair associated with this result: ___________. d. Using your EQUATION from part b), determine the value of y when the input is 6 (Hint: Plug in 6 for x and compute the y-value): _____________. Write the specific ordered pair associated with this result: ___________. e. Your y-values for c) and d) should be different. Why is this the case (refer to the end of Problem 5 for help). f. Use the result from part b) to predict the value of x when the output is 28.14. Set up an equation and show your work to solve it here.

Write the specific ordered pair associated with this result: ___________.


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Problem 9 WORKED EXAMPLE – Mul t ip le Ways to Determine The Equat ion of a L ine

Determine if the data below represent a linear function. If so, use at least two different methods to determine the equation that best fits the given data.

x 1 5 9 13 y 75 275 475 675

Compute a few slopes to determine if the data are linear.

Between (1, 75) and (5, 275) m =275! 755!1

=2004

= 50

Between (5, 275) and (9, 475) m =475! 2759! 5

=2004

= 50

Between (9, 475 and 13, 675) m =675! 47513! 9

=2004

= 50

The data appear to be linear with a slope of 50. Method 1 to determine Linear Equation – Slope Intercept Linear Form (y=mx + b): Use the slope, m = 50, and one ordered pair, say (1, 75) to find the y-intercept 75 = 50(1) + b, so b=25. Thus the equation is given by y = 50x + 25. Method 2 to determine Linear Equation – Linear Regression: Use the steps for Linear Regression to find the equation. The steps can be used even if the data are exactly linear. Step 1: Go to STAT>EDIT>1:Edit Step 2: Clear L1 by scrolling to L1 then press CLEAR then scroll back down one row Step 3: Enter the values 1, 5, 9, 13 into the rows of L1 (pressing Enter between each one) Step 4: Right arrow then up arrow to top of L2 and Clear L2 by pressing CLEAR then scroll back down Step 5: Enter the values 75, 275, 475, 675 into the rows of L2 (pressing Enter between each one) Step 6: Go to STAT>EDIT>CALC>4:LinReg (ax + b) then press ENTER twice Step 7: Read the values a and b from the screen and use them to write the equation, y = 50x + 25


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ANSWERS – YOU TRY PROBLEMS

You Try Problem 4 a) VI (0, 20,100), AROC 500 $/yr, S(t) = 500t + 20,100 b) S(8) = $24,100 c) 40 years You Try Problem 5 a) -6.3 lbs per week b) -5 lbs per week c) average pounds lost per week over the given time period d) NO – AROC not constant You Try Problem 7 a) y = 72,000x + 178,000, S(t) = 72,000t + 178,000 b) Total sales in 2000 were $430,000 c) S(10) = $898,000 d) m = 72000; The slope is the rate of change of snowboard sales in $/year e) the year 2003 You Try Problem 8 a) sketch of data points b) y = .322x + 24.155 c) 26.24; (6, 26.24) d) 26.087; (6, 26.087) e) given data set is not exactly linear so model is an approximation or best fit f) x = 12.376; (12.376, 28.14)

mini lesson lesson 2a – linear functions and … · n, years of ownership v, value of the car in...

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