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2. Chapter 12 Functions and Their Graphs Their Graphs

3. 12.1 Relations and Functions 12.2 Graphs of Functions and Transformations 12.3 Quadratic Functions and Their Graphs 12.4Applications of Quadratic Functions and Graphing Other Parabolas. 12.5The Algebra of Functions 12.6Variation 12 Functions and Their Graphs

4. Variation 12.6 Solve Direct Variation Problems In Section 4.6, we discussed the following situation: If you are driving on a highway at a constant speed of 60 mph, the distance you travel depends on the amount of time spent driving. Let y = the distance traveled, in miles, and let x= the number of hours spent driving. An equation relating x and y is y = 60x.  A table of values relating x and y is to the right. As the value of x increases, the value of y also increases. (The more hours you drive, the farther you will go.) Likewise, as the value of x decreases, the value of y also decreases. We can say that the distance traveled, y, is directly proportional to the time spent traveling, x. Or y varies directly as x.

5. Example 1 Suppose y varies directly as x. If y = 18 when x = 3 ; a). Find the constant of variation, k. b). Write a variation equation relating x and y using the value of k found in a). c). Find y when x = 11 Solution a). To find the constant of variation, write a general variation equation relating x and y. y varies directly as x means y = kx We are told that y = 18 when x = 3. Substitute these values into the equation and solve for k. y= kx 18 = k(3) 6 = k Substitute 3 for x and 18 for y Divide by 3

6. Example 1-continued b) The specific variation equation is the equation obtained when we substitute 6 for k: Substitute 11 for x y = kx y = 6x c). To find y when x = 11, substitute 11 for x in y = 6x and evaluate. y = 6x y = 66 y = 6(11) Multiply

7. Example 2 Solution Suppose p varies directly as the square of z. If p=12 when z = 2, find p when z = 5 Step 1: Write the general variation equation. p varies directly as the square of z means p = kz 2

8. Example 2-continued Step 2: Find k using the known values: p = 12 when z = 2. p = 3z 2 p = kz 2 12 = k(2) 2 3 = k 12 = k(4) Substitute 2 for z and 12 for p Step 3: Substitute k = 3 into p = kz 2 to get the specific variation equation : p = 3z 2 Step 4: We are asked to find p when z = 5. Substitute z= 5 into p = 3z 2 = 3(5) 2 = 3(25) = 75 Substitute 5 for z

9. Example 3 Solution General variation equation Substitute 80 for n and 3360 for R Divide by 80 A theater’s nightly revenue varies directly as the number of tickets sold. If the revenue from the sale of 80 tickets is $3360, find the revenue from the sale of 95 tickets. Let the n = number of tickets sold R = revenue We will follow the four steps for solving a variation problem. Step 1: Write the general variation equation, R = kn. Step 2: Find k using the known values: R = 3360 when n = 80. R = kn 3360 = k(80) 42 = k

10. Example 3-continued Specific variation equation Step 3: Substitute k = 42 into R = kn to get the specific variation equation : R = 42n Step 4: We must find the revenue from the sale of 95 tickets. Substitute n = 95 into R = 42n to find R. R = 42n R = 42(95) R = 3990

11. Solve Inverse Variation Problems If two quantities vary inversely (are inversely proportional), then as one value increases, the other decreases. Likewise, as one value decreases, the other increases. A good example of inverse variation is the relationship between the time, t, it takes to travel a given distance, d, as a function of the rate (or speed), r. We can define this relationship as.As the rate, r, increases, the time, t, that it takes to travel d decreases. Likewise, as r decreases, the time, t, that it takes to travel d increases. Therefore, t varies inversely as r.

12. Example 4 Suppose q varies inversely as h. If q = 4 when h = 15, find q when h = 10 Solution Step 3: Substitute k = 60 into to get the specific variation equation, Step 1: Write the general variation equation, Step 2: Find k using the known values: q = 4 when h = 15 Step 4: Substitute 10 for h in to find q.

13. Example 5 Solution The intensity of light (in lumens) varies inversely as the square of the distance from the source. If the intensity of the light is 40 lumens 5 ft from the source, what is the intensity of the light 4 ft from the source? Let d = distance from the source (in feet) I = intensity of the light (in lumens) Step 1: Write the general variation equation, Step 2: Find k using the known values: I = 40 when d = 5 General variation equation Substitute 5 for d and 40 for I Multiply by 25

14. Example 5-continued Step 3: Substitute k = 1000 into to get the specific variation equation, Step 4: Find the intensity, I, of the light 4 ft from the source. Substitute d = 4 into to find I. The intensity of the light is 62.5 lumens

15. Example 6 Solution Solve Joint Variation Problems If a variable varies directly as the product of two or more other variables, the first variable varies jointly as the other variables. For a given amount invested in a bank account (called the principal), the interest earned varies jointly as the interest rate (expressed as a decimal) and the time the principal is in the account. If Graham earns $80 in interest when he invests his money for 1 yr at 4%, how much interest would the same principal earn if he invested it at 5% for 2 yr? Let r = interest rate (as a decimal) t = the number of years the principal is invested. I = interest earned. Step 1: Write the general variation equation, I = krt

16. Example 6- continued Step 2: Find k using the known values: I = 80 when t =1 and r = 0.04 I = krt 80 = k(0.04)(1) 80 = 0.04k 2000 = k General variation equation Substitute the values into I =krt Divide by 0.04 (The amount he invested, the principal, is $2000.) Step 3: Substitute k = 2000 into I = krt to get the specific variation equation, I = 2000rt Step 4: Find the interest Graham would earn if he invested $2000 at 5% interest for 2 yr. Let r = 0.05 and t = 2. Solve for I. I = 2000(0.05)(2) Substitute 0.05 for r and 2 for t. I = 200 Multiply Graham would earn $200

17. Solve Combine Variation Problems Example 7 Suppose y varies directly as the square root of x and inversely as z. If y = 12 when x = 36 and z = 5, find y when x = 81 and z = 15 Solution Step 1: Write the general variation equation, y varies directly as the square root of x y varies inversely as z. Step 2: Find k using the known values: y = 12 when x = 36 and z = 5 Substitute the values. Multiply by 5 Divide by 6

18. Example 7-continued Step 3: Substitute k =10 into to get the specific variation equation, Step 4: Find y when x = 81 and z = 15 Substitute 81 for x and 15 for z.