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LINEARITY Definition

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LINEARITY Definition SECTION 1.1 – LINEARITY Definition (Total Change) What that means: Algebraically Geometrically Example 1 At the beginning of the year, the price of gas was $3.19 per gallon. At the end of the year, the price of gas was $1.52 per gallon. What is the total change in the price of gas? Algebraically Geometrically 1 Definition (Average Rate of Change) What that means: Algebraically: Geometrically: Example 2 John collects marbles. After one year, he had 60 marbles and after 5 years, he had 140 marbles. What is the average rate of change of marbles with respect to time? Algebraically Geometrically 2 Definition (Linearly Related, Linear Model) Example 3 On an average summer day in a large city, the pollution index at 8:00 A.M. is 20 parts per million, and it increases linearly by 15 parts per million each hour until 3:00 P.M. Let P be the amount of pollutants in the air x hours after 8:00 A.M. a.) Write the linear model that expresses P in terms of x. b.) What is the air pollution index at 1:00 P.M.? c.) Graph the equation P for 0 ≤ x ≤ 7 . 3 SECTION 1.2 – GEOMETRIC & ALGEBRAIC PROPERTIES OF A LINE Definition (Slope) What that means: Algebraically Geometrically Definition (Graph of an Equation) 4 Example 1 Graph the line containing the point P with slope m. a.) P = ( ,1 1) & m = −3 c.) P = − ,1 −3 & m = 3 ( ) 5 b.) P = ,2 1 & m = 1 ( ) 2 d.) P = ,0 −2 & m = − 2 ( ) 3 5 Definition (Slope -Intercept Form) Definition (Point -Slope Form) 6 Example 2 Find the equation of the line through the points P and Q. a.) P = ( ,1 1) & Q = ( ,9 12) Using Point -Slope Form Using Slope -Intercept Form b.) P = (− ,2 −5) & Q = ( ,3 2) Using Point -Slope Form Using Slope -Intercept Form c.) P = (− ,1 2) & Q = ( ,6 −4) Using Point -Slope Form Using Slope -Intercept Form 7 d.) P = ( ,0 6) & Q = (− ,3 −2) Using Point -Slope Form Using Slope -Intercept Form Question Does it matter which points on the line we use to determine the slope of the line? Parallel vs. Perpendicular Lines Geometrically Algebraically Parallel Lines Perpendicular Lines 8 Example 3 6 a.) Find the equation of a line that is parallel to the line y = − x −19 . 7 6 b.) Find the equation of a line that is perpendicular to the line y = − x −19 . 7 6 c.) Find the equation of the line that is parallel to the line y = − x −19 and goes through the 7 point ( ,2 −1). 6 d.) Find the equation of the line that is perpendicular to the line y = − x −19 and goes through 7 the point ( ,2 −1). 9 Definition (Intercepts) Notation *In this class, all intercepts must be listed as an ordered pair (x, y). * Example 4 Find the intercepts of each of the following graphs. 10 Finding Intercepts Algebraically Example 5 Find all intercepts of the following equations. a.) x 2 + y 2 = 10 b.) y = 2x 2 + 6 c.) y = x 3 d.) y = 3 − 3x 3 11 SECTION 1.3 – FUNCTIONS AND MODELING Definition (Function, Domain, & Range) Example 1 Voting for President can be thought of as a function. Assuming that everyone who can vote does vote, then the domain is the set of American citizens who are age 18 or older and the range is the set of all Presidential nominees who receive at least one vote. Notice that each voter can only vote for one candidate. That is, for each input there is exactly one output. Example 2 Determine if each of these is a function. Also, identify the inputs, outputs, domain, and range where applicable. A A 1 1 B B 2 2 C C 3 3 D D 12 Example 3 The following table displays the number of students registered in each section of underwater basket weaving. Section # (S) 1 2 3 4 5 6 7 8 9 10 # Of Students 25 32 19 22 41 16 32 22 27 31 Enrolled (E) Is E a function of S? What are the inputs? What are the outputs? What is the domain? What is the range? Is S a function of E? Definition (Independent vs. Dependent Variables) Function Notation 13 Evaluating a Function Example 4 Let f(x) = 4x 2 + 3x + 2 . Find each of the following. f(0) = f(2) = 1 f( 2) = f(2k + 5) = f(_) = f(∆) = (tf + h)− (tf ) = h 14 Finding Domain Hint: Watch for zero in denominators and negatives under even radicals. Example 5 Find the domain of each of the following functions. f(x) = 2x + 4 g(x) = 6 2x − 3 4x + 3 h()x = x 2 −1 1+ 6x k()x = 2x 2 − 3x − 2 15 Example 6 Determine if each of the following equations describes y as a function of x, x as a function of y, or both. 4x + 3y = 2 4x 2 − 3y = 6 2x + 4 = y 2 16 Definition (Total Change, Average Rate of Change, & Difference Quotient) Example 7 A ball is thrown straight up into the air. The height of the ball in feet above the ground is described by the function h(t) = − t2 2 +14 where time is measured in seconds. Find the total change and average rate of change in height after 2 seconds and interpret each. 17 Definition (Intercepts) Example 8 Domain: Range: y-intercept: x-intercepts: Example 9 Domain: Range: y-intercept: x-intercepts: 18 Definition (Local Maximum & Local Minimum) Cost, Reven ue, Price -Demand, & Profit Functions 19 Example 10 A manufacturing company wants to make digital cameras and wholesale the cameras to retail outlets throughout the U.S. The company’s financial department has come up with the following price-demand and cost data where p is the wholesale price per camera at which x million cameras are sold and C is the cost in millions of dollars for manufacturing and selling x million cameras. x p x C (millions) ($) (millions) (thousands of $) 2 84.80 1 175.7 5 69.80 3 215.1 8 54.80 6 274.2 10 44.80 11 372.7 13 29.80 14 431.8 a.) Assuming that x and p are linearly related, write a formula that expresses p as a function of x. You may also assume that 0 ≤ x ≤ 15 . b.) Assuming that x and C are linearly related, write a formula that expresses C as a function of x. You may also assume that 0 ≤ x ≤ 15 . c.) What are the revenue and profit functions? d.) At what point(s) does the company break-even, make a profit, or experience a loss? e.) When does the company experience its maximum profits? 20 SECTION 2.1 – SIMPLE INTEREST AND COMPOUND INTEREST Definition (Principal, Interest, Simple Interest, Future Value) Simple Interest Formul a Future Value Formula (Simple Interest) 21 Example 1 Find the total amount due on a loan of $800 at 9% simple interest at the end of 4 months. Example 2 If you want to earn an annual rate of 10% on your investments, how much (to the nearest cent) should you pay for a note that will be worth $5000 in 9 months? Example 3 Treasury bills are one of the instruments the U.S. Treasury Dept uses to finance the public debt. If you buy a 180-day treasury bill with a maturity value of $10,000 for $9,893.78, what annual simple interest rate will you earn? 22 Definition (Compound Interest, Periodic Rate, Compounding Periods) Future Value Formula (Compound Interest) 23 Example 4 You want to invest $1000 at 8% interest for 5 years. a.) How much money will you have if interest is compounded semiannually? b.) How much money will you have if interest is compounded monthly? c.) How much money will you have if interest is compounded daily? Example 5 How much should you invest now at 10% interest compounded quarterly to have $8000 toward the purchase of a car in 5 years? Example 6 If money placed in a certain account triples in 2 years when interest is compounded quarterly, then what is the annual interest rate? 24 Definition (Annual Percentage Yield, Effective Annual Rate) Annual Percentage Yield Formula Example 7 A $10,000 investment in a particular growth-oriented mutual fund over a recent 10 year period would have grown to $126,000. What annual nominal rate compounded monthly would produce the same growth? What is the annual percentage yield? 25 Example 8 Find the APY’s for each of the banks in the following table and compare the CDs. CERTIFICATES OF DEPOSIT BANK RATE COMPOUNDED Advanta 4.95% Monthly DeepGreen 4.96% Daily Charter One 4.97% quarterly Example 9 A savings and loan wants to offer a CD with a monthly compounding rate that has an effective annual rate of 7.5%. What annual nominal rate compounded monthly should they use? 26 SECTION 2.2 – FUTURE VALUE OF AN ANNUITY Definition (Annuity, Ordinary Annuity) Future Value (Ordinary Annuity) Example 1 What is the value of an annuity at the end of 20 years if $2000 is deposited each year into an account earning 8.5% interest compounded annually? How much of the value is interest? 27 Example 2 A person makes monthly deposits of $100 into an ordinary annuity. After 30 years, the annuity is worth $160,000. What annual rate compounded monthly has this annuity earned during this 30- year period? Definition (Sinking Fund) Note 28 Example 3 A company estimates that it will have to replace a piece of equipment at a cost of $800,000 in 5 years.
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