Name Date. Key Math Concepts

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1 2-1 Interpret Scatterplots Key Math Concepts Bivariate data is pairs of numbers, (x,y), that represent variables. Positive correlation: the value of one variable increases as the other increases. Negative correlation: the value of one variable decreases as the other increases. Causal relationship: explanatory variable causes change in the response variable. Guided Exercises 1. Determine if the scatterplot at the right depicts a positive correlation or a negative correlation. In bivariate data, the first number, or the x-variable, is graphed on the axis. The second number, or the y-variable, is graphed on the axis. In the scatterplot at the right, the x-values begin low, close to the origin, and then. The y-values begin high, far from the origin, and then. When one variable decreases as the other increases, it is a correlation. 2. Tom is creating a scatterplot that depicts the perimeter and area of a square. (If s is the length of a side of a square, the perimeter is 4s and the area is s 2 ). Find the values of u, v, w, x, y, and z and identify the explanatory and the response variable. (10, u) (12, v) (20, w) (24, x) (36, y) (z, 49) For (x, y), x = the perimeter of the square = ; y = the area of the square =. For (10, u), 10 4 =. For (12, v), 12 4 =. So, s =, and u = or. So, s =, and v = or. For (z, 49), 49 =. So, s =, and 4u = or. Show your work for the remaining 3 ordered pairs. 49 = s 2, so 49 = s. As the x-values, the y-values. This is a correlation. Causal relationship: causes change in the. In the causal relationship, the perimeter is the variable and the area is the variable. 20 Financial Algebra Guided Practice Workbook 2-1 or duplicated, or posted to a publicly accessible website, in whole or in part.

2 Exercises 3. Let r represent the radius of a circle and let d represent the diameter. The circumference of a circle is given by the formula C = πd. The area of a circle is given by the formula A = πr 2. Find the areas and circumferences of circles with radii 4, 5, 6, and 8 using π = Round to the nearest hundredth. 4. Identify the explanatory variable and response variable in the bivariate data, amount earned in the year and income taxes paid. Use the data about Cold Spring Hills High School that are given in the table. Year Enrollment Number of Students in Recycling , , , , Let x represent the year and y represent the enrollment. Draw a scatterplot to depict the data. 6. Are the years and the enrollments negatively or positively correlated? 7. Let x represent the year and let y represent the number of students in the recycling club. Would a scatterplot of x and y show a negative or positive correlation? or duplicated, or posted to a publicly accessible website, in whole or in part. Financial Algebra Guided Practice Workbook

3 2-2 Guided Exercises Linear Regression Key Math Concepts Δx Slope = Δy The equation of a line is y = mx + b where m is the slope and b is the y-intercept. A correlation coefficient, r, between 1 and 1, shows how closely a line fits data. Positive r = positive correlation; negative r = negative correlation. The table gives the number of CDs sold at certain prices for a given online music store during December. Use the table to answer Exercises 1-2. Price (x) $8.99 $10.99 $11.99 $12.99 $13.99 $14.99 Number of CDs Sold 6,456 6,009 5,345 4,560 4,100 3, Find the correlation coefficient for the data. Round to the nearest hundredth. Interpret the correlation coefficient. Use the LIST function on a graphing calculator and input the data. L1 = x values or the. L2 = y values or the. Use the CALC function and LinReg for the values of the linear regression. The correlation coefficient is rounded to the nearest hundredth. A negative correlation coefficient indicates a correlation. A correlation coefficient with an absolute value greater than 0.75 indicates a correlation. 2. Find the equation of the regression line. Round to the nearest thousandth. If the price of CDs was raised to $17.99, predict the number of CDs that would be sold. Round to the nearest hundred. From the calculator s values: m, or the coefficient of x, is the slope. m = and b is the y-intercept. b =. y = x + When x = $17.99 then, Use the values to write the equation y = mx + b. Substitute $17.99 for x. y = + Round to the nearest hundred. At a price of $17.99, about CDs would be sold. 22 Financial Algebra Guided Practice Workbook 2-2 or duplicated, or posted to a publicly accessible website, in whole or in part.

4 Exercises 3. Find the linear regression line for a scatterplot formed by the points (10, 261), (21, 252), (42, 209), (33, 163), and (52, 98). Round to the nearest tenth. 4. The table gives the amount raised by Key Club members and the number of Key Club t-shirts sold at Meadow East High School for given years. Year Money Raised $7,456 $7,988 $8,322 $8,344 $8,901 Shirts Sold a. Draw a scatterplot on your calculator comparing money raised and shirts sold. b. Find the regression line equation correct to seven decimal places, and plot it on your scatterplot. c. Describe the correlation between the amount of money raised and the number of shirts sold. Find the correlation coefficient r for the data. 5. Describe each of the following correlation coefficients using the terms strong, moderation, weak, and negative. a. r = 0.57 b. r = Kaitlyn has sold Girl Scout cookies for the past four years. She kept data on her sales, as shown in the table. She is planning to sell again this year and plans on visiting 200 homes. Predict the number of boxes she would sell if she visited 200 homes. Round to the nearest ten boxes. Explain your work. Number of Houses Visited Number of Boxes Sold or duplicated, or posted to a publicly accessible website, in whole or in part. Financial Algebra Guided Practice Workbook

5 2-3 Supply and Demand Guided Exercises Key Math Concepts Markup = markup rate wholesale price Retail price = markup + wholesale price Demand function is q = mp + b, where q is quantity and p is price. The equilibrium price is where the functions of supply and demand intersect. 1. The graph shows supply and demand curves for the AquaPod, a digital music player for scuba divers. What is the equilibrium price? What will happen if the price is set at $35? What will happen if the price is set at $45? The equilibrium price is where the functions intersect. The equilibrium price is. q y x supply demand p At p = $35, the curve for demand is than the curve for supply. If the price is set at $35, demand will supply, and the price will be. At p = $45, the curve for supply is than the curve for demand. If the price is set at $45, supply will demand and the price will be. 2. The Knockey Corporation sells hockey sticks at a wholesale price of $103. If a store marks this up 106%, what is the retail price? Markup = 1.06 $ = Retail Price = Wholesale Price + Markup Write the markup rate as a decimal. Exercises Retail Price = + = The retail price of the hockey sticks is. 3. Wayne s Widget World sells widgets to stores for $9.20 each. A local store marks them up $8.79. What is the retail price at this store? 24 Financial Algebra Guided Practice Workbook 2-3 or duplicated, or posted to a publicly accessible website, in whole or in part.

6 4. An automobile DVD system sells to stores at a wholesale price of $497. A popular national chain store sells them for $ What is the markup? 5. A manufacturer takes a poll of several retailers to determine how many widgets they would buy at different wholesale prices. The results are shown. What is the equation of the demand function? Round values to the nearest hundredth. How many widgets, to the nearest hundred, would retailers buy at a price of $20? Wholesale Price Quantities Retailers Will Purchase (1,000s) ,000 3,450 3,100 2,550 2,000 1,900 1,750 1, The Coletti Company produces paper cups. They want to use the demand function to help them set a price. They survey dozens of retailers to get an approximation of how many cups would be demanded at each price. Wholesale Price, p (pack of 50) Quantity, q (1,000s) $5 $5.50 $6 $6.50 $7 $7.50 $8 $8.50 $9 $9.50 5,100 4,900 4,600 4,200 3,700 2,400 2,100 1, Based on the table, if the company sold 5,100,000 packages of cups at a price of $5 each, how much money would they take in? If the company sold 200,000 packages of cups at a price of $9.50 each, how much money would they take in? Compare your answers. Why is it not correct to conclude that more profit is made by selling for $5 than for $9.50? or duplicated, or posted to a publicly accessible website, in whole or in part. Financial Algebra Guided Practice Workbook

7 2-4 Fixed and Variable Expenses Key Math Concepts E = V + F, where E is total expenses, V is variable expenses, and F is fixed expenses. V = pq, where V is variable expenses, p is price to produce, and q is quantity produced. R = pq, where R is revenue, p is price of the product, and q is quantity sold. The breakeven point is when profit = loss. Guided Exercises 1. The Catania Cat Corporation manufactures litter boxes for cats. Their expense function is E = 4.18q + 82,000. Find the average cost of producing 10,000 litter boxes. E = V + F, where E is total expenses, V is variable expenses, and F is fixed expenses. So, if E = 4.18q + 82,000: the cost per unit = and the fixed expenses =. When q = 10,000; E = 4.18q + 82,000 E = 4.18( ) + 82,000 E = Total expenses to produce 10,000 litter boxes Average cost total cost = number produced Averagecost = = The average cost of producing one litter box is. 2. The Mizzi Corporation has created a demand function for one of its wrench sets. It expresses the quantity demanded in terms of the wholesale price p, and was found by surveying retailers and using linear regression. The demand function is q = 98p + 5,788. Their expense function is E = 23q + 68,000. Express the expense function as a function in terms of p. E = 23q + 68,000 E = 23( ) + 68,000 E = + + Substitute for q. Use the Distributive Property. E = + The expense function in terms of price, p, is E = Financial Algebra Guided Practice Workbook 2-4 or duplicated, or posted to a publicly accessible website, in whole or in part.

8 Exercises 3. The fixed expenses for producing widgets are $947,900. The labor and materials required for each widget produced costs $ Represent the total expenses as a function of the quantity produced. 4. The expense function for a certain item is E = 2.95q + 712,000. Express the average cost of producing q items algebraically. 5. A corporation s expense function is E = 7.50q + 34,000. The demand function was determined to be q = 5.5p + 6,000. Express the expense function in terms of the price. 6. The graph shows Expense (E) and Revenue (R) functions and several different levels of quantity produced. What are the coordinates of the break-even point? If U units are produced, will there be a profit or a loss? K p R E q U N M T W 7. The DiMonte Corporation invented a new type of sunglass lens. Their variable expenses are $12.66 per unit, and their fixed expenses are $111,200. How much does it cost them to produce one lens? 15,000 lenses? Express the expense function algebraically. What is the slope of the expense function? If the slope is interpreted as a rate, give the units to use. or duplicated, or posted to a publicly accessible website, in whole or in part. Financial Algebra Guided Practice Workbook

9 2-5 Graphs of Expense and Revenue Functions Key Math Concepts R = pq where R is revenue, p is price of the product and q is quantity sold. E = V + F, where E is total expenses, V is variable expenses, and F is fixed expenses. V = pq, where V is variable expenses, p is price to produce, and q is quantity sold. Guided Exercises 1. Geoff s company manufactures customized T-shirts. Each shirt costs $2.00 to manufacture and print. The fixed costs for this product line are $2,000. The demand function is q = 1,200p + 7,800, where p is the price for each shirt. Write the expense equation in terms of the demand q. Then write the expression in terms of price p. Write the revenue function in terms of the price. E = V + F = pq + F E = 2.00q + 2,000 Substitute for p and F. E = 2.00( + ) + 2,000 E = + R = pq = p( + ) Substitute for q to write equation in terms of p. R = + 2. Graph the revenue functions from the exercise above in an appropriate viewing window. What price yields the maximum revenue? What is the revenue at that price? In the Y= screen, input the expense function and the revenue function. Graph the functions. The y-value of the vertex (on the axis of symmetry) is the. The x-value of the vertex (on the axis of symmetry) is the at the maximum revenue. The maximum revenue is and the price at the maximum revenue is. 28 Financial Algebra Guided Practice Workbook 2-5 or duplicated, or posted to a publicly accessible website, in whole or in part.

10 Exercises 3. Identify the maximum point when R = 125p2 + 3,200p by using the axis of symmetry. Mobile Tech manufactures cellular phone accessories. A particular item in their product line costs $40 each to manufacture. The fixed costs are $120,000. The demand function is q = 120p + 8,000 where q is the quantity the public will buy given the price p. Answer Exercises Write the expense function in terms of p. 5. What is the revenue equation for this Mobile Tech product? Write the revenue equation in terms of the price. What would the revenue be if the price of the item was set at $60? 6. Graph the expense and revenue functions found above. Interpret the graph. or duplicated, or posted to a publicly accessible website, in whole or in part. Financial Algebra Guided Practice Workbook

11 2-6 Breakeven Analysis Key Math Concepts Quadratic formula = b± b 2 4 ac 2a Guided Exercises 1. Estimate the breakeven points of the expense and revenue functions in the graph. The breakeven point are where the expense and revenue functions are. This is where the lines on the graph. 250, , , ,000 y Follow the x-axis directly underneath the first intersecion: the coordinates are about (, ) 50, x Follow the x-axis directly underneath the second intersecion: the coordinates are about (, ) The coordinates of the breakeven points are (, ) and (, ). Sunset Park Equipment produces camping gear. They are considering manufacturing a new energy-efficient lantern. The expense function is E = 54,000p + 7,000,000 and the revenue function is R = 1,800p ,000p. 2. Determine the prices at the breakeven points algebraically. Round to the nearest cent. Set the expense and revenue 54,000p + 7,000,000 = 1,800p ,000p functions equal to each other. + + = 0 Rewrite as a quadratic equation with 0 on one side. Then substitute into the quadratic formula. In the quadratic formula, a = ; b = ; and c =. b + b 2 4 ac 2a = + 2 4( )( ) 2( ) b b 2 4 ac 2a = 2 4( )( ) 2( ) The prices at the breakeven points are and. 30 Financial Algebra Guided Practice Workbook 2-6 or duplicated, or posted to a publicly accessible website, in whole or in part.

12 Exercises 3. AVS Industries has determined that the combined fixed and variable expenses for the production and sale of 800,000 items are $24,000,000. What is the price at the breakeven point for this item? 4. The expense equation for the production of a certain MP3 player is E = 1,250q + 700,000 where q is the quantity demanded. At a particular price, the breakeven revenue is $3,800,000. What is the quantity demanded at the breakeven point? Baby-B-Good manufactures affordable plastic baby rattles. The expense equation is E = 3,400p + 50,000, and the revenue equation is R = 1,800p ,000p. 5. Sketch the graph of the expense and revenue functions. Circle the breakeven points. Determine the prices at the breakeven points. Round to the nearest cent. 6. Use your answers from Exercise 5 to determine the revenue and expense amounts for each of the breakeven points. Round to the nearest cent. y 7. Determine both the expense and revenue functions shown in the graph in terms of price x. $20,000 $18,000 $16,000 $14,000 $12,000 $10,000 $8,000 $6,000 $4,000 $2, x or duplicated, or posted to a publicly accessible website, in whole or in part. Financial Algebra Guided Practice Workbook

13 2-7 The Profit Equation Key Math Concepts P = R E, where P is profit, R is revenue, and E is expenses. The vertex of a profit function lies on the axis of symmetry. The vertex of a profit function contains the maximum profit. Maximum profit can be found on the graph where the difference between the revenue and expense functions is the greatest. Guided Exercises FunFleece Incorporated manufactures fleece hats. It is considering making a new type of weatherproof fleece hat. The expense and revenue functions are E = 300p + 150,000 and R = 300p ,000p. 1. Determine the profit function. P = R E P = ( 300p ,000p) ( 300p + 150,000) Substitute the expense function for E and the revenue function for R. P = + + P = + The profit function is P = Use the equation from Exercise 1. Determine the maximum profit to the nearest cent. Determine the price that yields maximum profit. Find the maximum of the profit equation. Axis of symmetry = b 2a Axis of symmetry = 2( ) Substitute the price, p, into the profit equation from Exercise 1. P = + P = The maximum profit is. 32 Financial Algebra Guided Practice Workbook 2-7 or duplicated, or posted to a publicly accessible website, in whole or in part.

14 Exercises Examine each of the following graphs and estimate the maximum profit price and the maximum profit at that price for Exercises ,000 y 150, ,000 50, x , , , , , , ,000 y 50, x 5. Listen Up sells external computer speakers. Their new product has the following expense and revenue functions: E = 828p + 400,000 and R = 38p 2 + 8,000p. Determine the profit function. Determine the price, to the nearest cent, which yields the maximum profit. Determine the maximum profit to the nearest cent. 6. Determine the expense equation given the profit and revenue equations. P = 468p ,599p 299,000 R = 468p ,000p or duplicated, or posted to a publicly accessible website, in whole or in part. Financial Algebra Guided Practice Workbook

15 2-8 Mathematically Modeling a Business Key Math Concepts Transitive property of dependence: If x depends on y and y depends on z, it follows that x depends on z. Guided Exercises 1. The graph displays the expense function (Y1), the revenue function (Y2), and the profit function (Y3). Name the coordinates of the maximum revenue point and the maximum profit point. What is the significance of the point (F, C)? A B C Y2 Y3 Y1 The maximum revenue point is the vertex of, which is the revenue function. The coordinates of the vertex of Y2 are (, ). 0 F G H The maximum profit point is the vertex of, which is the function. The coordinates of the vertex of Y3 are (, ). 2. What is the significance of the point (F, C)? Exercises Point (F, C) is a. This is where expense, revenue, and profit are. You are the CEO of Widget World Corporation. Researchers have developed a new electronic widget. The fixed costs to make the widget will be $30,000, and the variable cost will be $150 per widget. Exercises 3-7 are about the Widget World Corporation. 3. Write the expense function for the electronic widget in terms of q where q is the quantity that will be produced. 34 Financial Algebra Guided Practice Workbook 2-8 or duplicated, or posted to a publicly accessible website, in whole or in part.

16 4. The market research department conducted consumer surveys and reported the following results. In these ordered pairs, the first number represents the possible price and the second number represents the quantity demanded. The points are listed as (p, q). (75, 6,500), (100, 5,900), (125, 4,500), (150, 3,900), (175, 2,800), (200, 1,500), (225, 900) You now need to set up a demand function using the ordered pairs from the market research department. Enter these ordered pairs into graphing calculator lists. Write the linear regression equation that models this set of ordered pairs. Round each coefficient to the nearest integer. 5. The equations used are in terms of q and p. Replace y with q and replace x with p. Write the new equation for q in terms of p. 6. Recall that the revenue equation is R = pq. Substitute the quantity (demand) equation from Exercise 6 into the revenue equation and simplify. 7. Use the equation for the demand from Exercise 6. Substitute this into your expense equation from Exercise 4 to write an expense equation in terms of p rather than q. or duplicated, or posted to a publicly accessible website, in whole or in part. Financial Algebra Guided Practice Workbook