Bridgette has $1,000 to invest. She has two different investment options, shown in the table below. Each option would give her a different value over three years. Number of Years 1 2 3
Option A (amount in dollars) 1100 1200 1300
Option B (amount in dollars) 1100 1210 1331

Part A: What type of function, linear or exponential, would be best to describe the value of the investment over a fixed number of years using Option A? Explain your answer.

Part B: Write a function, a(t), to describe the value of the investment, in dollars, of Option A after t years.

Part C: What type of function, linear or exponential, would be best to describe the value of the investment over a fixed number of years using Option B? Explain your answer.

Part D: Write a function, b(t), to describe the value of the investment, in dollars, of Option B after t years.

Part E: Bridgette wants to invest in whichever option would increase her investment value by the greatest amount in 20 years. Will there be any significant difference in the value of Bridgette's investment after 20 years if she uses Option A over Option B? Explain your answer, and show the investment value after 20 years for each option.

Part A:

Option A is a linear function

Part B:

a(t) = t × 100 + 1000

Part C:

Option B is an exponential function

Part D

Part E:

There will be a significant difference in the value of Bridget's investment after 20 yeas if she uses Option A over Option B

Step-by-step explanation:

The given parameters are;

The amount Bridgette has to invest = $1,000

The value of each investment option is given as follows;

Number of years      Option A         Option B

1                                  1100                 1100

2                                 1200                1210

3                                  1300                1331

Part A:

Given that the first difference between subsequent data values are 2 - 1 = 1 and 1200 - 1100 = 100, 3 - 2 = 1 and 1300 - 1200 = 100 both of which are constant, the type of function that would be best to describe option A is a linear function

Part B:

The function a(t) for Option A is given as follows;

The rate of change of the function amount with time is (1300 - 1200)/(3 - 2) = 100

Therefore, comparing with the general form of a linear function, y = m·x + c, we have;

m = The rate of change of the function = 100

The y-intercept, the value of the amount a 0 time (the start of the investment) = c = 1000

The function in terms of t, a(t) = t × 100 + 1000

Part C:

The first difference of Option B are different but the first ratio given as follows;

1210/1100 = 1331/1210 = 1.1 = Constant which is the characteristics of an exponential variable

Therefore, the equation to describe the value of the investment in dollars, of Option B is an exponential function

Part D

We, have for an exponential function;

When t = 1, we have;

∴ 1 + r = 1100/1000 = 1.1

r = 1.1 - 1 = 0.1

We check for t = 2, to get;

We check for t = 3, to get;

The function b(t) to describe the value of the investment in dollars, of Option B after t years is

Part E:

The investment value after 20 years is given as follows;

For Option A, we have;

a(20) = 20 × 100 + 1000 = 3000

For Option B, we have;

Therefore, the difference in the investment value after 20 years for Option B and Option A is approximately $6727.5 - $3000 = $3727.5 which is significant.

here we can see that the storage increases

at certain times of the year - during the

warmer months, when the gas is not much

used for heating and reserves can be

increased - and then decreases during the

colder months because the supply is being

used. it is possible to model this with a

relatively simple trigonometric graph, but

first we examine the simplest trig curves.

065/065.html

the basics

the simplest sine and cosine curves can be generated by plotting the trig-function values

against the angles used in each function. the values can be found in the chart below:

ø (˚) 0 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360

sin ø

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0 1

2

2

2 3

2 1 3

2 2

2 1

2

0 − 1

2

− 2

2 − 3

2 − 1 − 3

2 − 2

2 − 1

2

0

cos ø

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1 3

2 2

2 1

2 0 − 1

2

− 2

2 − 3

2 − 1 − 3

2 − 2

2 − 1

2

0 1

2 2

2 3

2 1

first we plot the sine data, letting the x-axis represent the angles and the y-axis the sine

values. the result is the scatter plot below.

based on just these 17 data points, we can see

that the function is following a straightforward

pattern - the sine values are increasing to 1,

decreasing through the same values to -1, then

returning to zero. when we connect the dots, we

get the periodic curve y = sin (x) below.

this “sine curve,” with equation

y = sin(x), is the simplest of trig

graphs. the curve rises and falls by

one unit from its “axis of

oscillation” - in this case, the x-axis

- and after we cycle through 360˚ it

appears we are ready to go through

the same y-values again. we can see

this same pattern in the cosine

function on the following page.

each sine and cosine curve has two basic characteristics:

amplitude - distance the curve rises above, and falls below, its axis of oscillation (marked by

the vertical arrows on the cosine graph). the formula is

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amp = max− min

2 .

period - the length of one cycle (marked by the horizontal arrows on the cosine graph).

this can be calculated by measuring the distance between any successive maximum

points (or minimum points), or by doubling the horizontal distance between a

minimum and the next maximum.

for our first two curves, then, the amplitude is 1 and the period is 360˚ (or 2! if we are in radians).

adjusting the amplitude and frequency: y = a sin(bx), y = a cos(bx)

now that we have our “base” amplitude and frequency . . let’s change them. we can adjust both

characteristics by introducing the parameters a and b into the equations. the graph below shows the

curves y = a sin(x) for a = 1,2,3, and -0.5, with the original curve y = sin(x) being the thickest line.

clearly the amplitude of the curves has changed;

when a = 2, the amplitude is 2; when a = 3, the

amplitude is 3; and when a = -0.5 the amplitude is

0.5 (since amplitude, a distance, must be positive).

however, this last curve has also been flipped upside

down (or “reflected in the x-axis”). since the

amplitude must be positive even when a itself is

negative, we have the formula

amplitude = | a |.

the relationship between the b-value and the

period is not so straightforward. on the right

we have y = cos(bx) for b = 1,3, and 0.5. (the

thickest line is again our “base” curve.) when

b=3, the period decreases to 120˚; when

b=0.5, the period increases to 720˚ (so only

half the curve is shown here). we can describe

this reciprocal relationship with the formula

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b = 360

period or

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b = 2π

period ,

using the latter if we are working in radians.

shifting horizontally and vertically: y = sin(x-h) + k, y = cos(x-h) + k

to shift (or “translate”) a curve along either the x- or y-axis (or both), we introduce the parameters

h and k. below we have the graph of y = sin (x - 30) + 2 and y = sin(x). (note that we have returned

to an amplitude of 1 and period of 360˚). our original

curve y = sin(x) had its peak at the point (90,1); the

shifted curve has its corresponding peak at (120, 3). in

the same way, the original curve had a minimum value

at (270, -1); now the minimum is at (300,1). and

finally, the axis of oscillation has shifted vertically

from the line x = 0 to the line x =2. in other words, our

new curve has been created by shifting y = sin(x) right

30 and u p 2. and, in general, we can create the

following shifts:

horizontal shift = h (left if h < 0, right if h > 0)

vertical shift/new axis of oscillation = k.

modelling data: y = a sin[b(x-h)] + k, y = a cos[b(x-h)] + k

we have seen how the basic sine and cosine curves can be adjusted: by changing the amplitude

and period, and by shifting the curves horizontally and vertically. a summary is below.

amplitude = | a | vertical shift/axis of oscillation = k

period =

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360

b

or

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2π

b