Linear Vs. Exponential

Exponential Functions

Linear Functions

A mathematical equation in which no independent-variable is raised to a power greater than one and graphs a straight line.

Characteristics :

- There is a constant difference (adds or subtracts the same amount every time)

- Displays a straight line graphically

- Has no exponents greater tha one

- Basic function is y = x

- Domain >> (-infinite#, infinte#) ; Range >> (-infinite#, infinite#)

- Uses arithmetic sequences ∞

- Slope is found with f(x) = mx + b

Exponential Functions Graphically

Linear Vs. Exponential

An exponential function is a mathematical function of the following form: f ( x ) = a to the x power. where x is a variable, and a is a constant called the base of the function.

Let's Try It!

Situation : Mary is a barista at a local coffee shop. The bags of coffee being shipped to the coffee shop cost a dollar the first bag, $2 for 2 bags, $4 for 3 bags, and so on with the amount doubling for each bag.

Paula Dean found an online marketplace for all of her butter needs that charge 2 dollars for the first stick and continue to double the price per stick from there. Graph this.

.

70

Answer :

60

50

.

40

Price

30

.

20

.

10

.

1 2 3 4 5 6 7

Sticks of Butter

Step 1 : Identify your function

f(x) = 2 to the x-1 power represents this situation because to find the cost/f(x), you must put 2 to the x - 1 power (the number of bags), because the price of the bags will double each time.

Step 2 : Solve and plot the points

Quite simply put your x values into the equation and solve for f(x). For example, to find the cost for 4 bags, just substitute the variable and solve the funtion f(x) = 2 to the 4 - 1 power to get 8. Therefore 8 would be the cost for 4 bags of coffee. Then the last remaining step would be to plot the point on the graph. Notice how the plots will never touch the x axis. This is because in exponential forms, there are asymptotes; meaning that there will be no x intercept.

.

Characteristics :

- Follows a common ratio

- Plots will most of the time never touch the x axis (has an asymptote)

- Equations have exponents that can be greater than one

- Values change very rapidly

- Domain >> (-infinite#, infinite#) ; Range >> (0, infinite#)

- The basic function is y = ab to the x power

- Uses geometric sequences

60

50

.

40

Cost

Holly Decker & Kylee Davis

30

20

.

10

.

1 2 3 4 5 6 7

Number of Bags

Setting Up And Solving For Linear

Exponential Functions Algebraically

Graphing

Exponential Functions Numerically

Algebraically

Scenario : Kylie Jenner plans to recieve lip injections every year. Each lip injection will increase her lip size by 2 millimeters. At the moment her lips are approximately 11.5 millimeters. Hypothetically, if she were to continue recieving these treatments anually as planned, how big would her lips be in 4 years?

Guy Fieri's Fun Math Tips!!!

How Big Kylie Jenner's Lips Are

Sceneario : The population of Jacksonville was 3,732 in 2008 and continues to grow at a rate of 2.5%. If this growth continues, what will be the approximate population in 2020?

Think of converting percents to decimals like money; 30% = .30, 5% = .05, etc. Its common CENTS yo

.

22

.

20

.

18

.

16

Size (Millimeters)

.

Step 1 : Identify your variables.

In this equation, the independent (or the x/domain) value would be time in years because she is recieving the fillers annually--once a year, and the dependent (or y/range) value is however big her lips are after treatment that year in millimeters.

Step 2 : Form a formula.

Since we know that how big her lips will be is the dependent variable , we know that we will be solving for y, and because her lips are already 11.5 mm, that means that the formula will have y = and then the x term + 11.5. Now we can deduce that the next part of the equation will be x (the years) multiplied by 2 because each injection will increase her lips by 2 mm; this will serve as our slope. If we put all of this information together, we can reason that the formula we will be using is y = 2x + 11.5.

Step 3 : Detirmine where each point will go on the graph by using the formula & plot the points.

Now that we have out formula, all we have to do is plug in a number for x and solve to get y. For example, we can go ahead and fing the y intercept by plugging in 0 for x (this would mean no years had passed by because of x standing for the time). If we do so, the formula will look like this >> y = 2(0) = 11.5, which would equal 11.5. Great! Now we have our y intercept! The x intercept is merely a similar concept; to find it you'd just put 0 as the y variable and solve for x, giving you the x intercept. As the same process for each other points on the graph, simply plug in the value. Such as calculating the size of her lips after 4 years. Just substitute 4 for x and solve >> y = 2(4) + 11.5, in which y equals 19.5. And all thats really left is to plot the points an the graph!

14

.

12

Solve for x in the equation :

25 = 4x + 5

Sceneario : Imagine you have a pond in your garden. In the pond, you plant one rapidly reproducing pond lily that doubles its population daily. Use this information to predict the next three days' worth of lilys.

10

4

5

6

7

2

3

1

Answer : Approximately 5,530 poeple

Set up your equation

6,000 (1 - .04)to the 2nd power

this is a decay function so be sure to subtract one from the ratio. Solve

an = 5,530

In 2017, the population will reach about 5,530

Time (Years)

Step 1 : Convert

Take the percent and convert it into a decimal like so >> 2.5% = .025

Step 2 : Substitute into the equation

Using the equation a * (1 + r) to the t power, we can plug in 3,732 for a because that is the initial popluation; .025 for r because that is the ratio that it is growing; and 12 for t, because that is the span of years betweene 2008 and 2020. And because this is a growth function, wee will be ADDING one to r. Andd in doing so, we will get the equation an = 3,732 (1 + .025) with an exponent of 12.

Step 3 : Solve

After solving we can come to the conclusion that 5,019 will be the population in 2020 >> an = 5,019

Your Turn!

Step 1 : Geometric Sequencing Formula

Geometric sequencing is a sequence with a common ratio between two consecutive terms. The formula for this is an = a1 r to the n - 1 power. In this instance plugging in values would look like this an = 8 (2) to the 7 - 1 power. This is because 8 is our starting value, 2 is the common ratio (because the second value divided by the first value/16 divided by 8, was 2), and 7 because that is the next value we are trying to solve for.

Step 2 : Solve for each value that must be found

Now thst we have the sequencing formula set up, we must solve. After 7 days, there will be 64 lillies; 128 after 8 days and 256 after 9 days. All found by replacing x with the x value (which stands for days) in the function.

Numerically

What's the 10th term?

The key to solving functions algebraically is to isolate the variable.

Step 1 : Subtract 5

In order to get x by itself, we must first subtract both sides by 5. Because 5 is being ADDED to 4x, we need to do the inverse operation and SUBTRACT both sides by 5. At this point, the equation now looks like this >> 20 = 4x

Step 2 : Divide by 4

The next part in getting the variable alone is to DIVIDE by 4 because in the equation x is being multiplied by 4. After doing so to both sides, we now have x isolated >> 5 = x.

We have now come to the answer; x equals 5

Answer : 10th term = 430

Plug values into function an = a1 + (n-1)d

an = 214 + (10 - 1)24

solve

an = 430

430 is the tenth term

214

1

Situation : Nicholas Cage is horrible with money. His current networth is 18 million and every year his networth drops by approximatley 500,000 every year because of his tremendous debt and spending habits. Because he is ready to meet the inevitable fate that he will soon run out of money, be forgoten and fade into oblivion, he wants to figure out how much money he will have in the next 5 years.

You Try!

238

2

262

3

The population of a town is currently 6,000 in 2015 and decreases at a rate of 4%. What will the population be in 2017?

Answer : x = -1

5 x - 6 = 3 x - 8

subtract 3x from both sides

2x - 6 = - 8

add six to both sides

2x = -2

divide by 2 on both sides

x = -1

4

8

18 mil.

0

Solve for x :

5 x - 6 = 3 x - 8

5

16

Step 1 : Notice how this is an arithmetic sequence and know the variables

An arithmetic s difference between consecutive terms is constant. The formula it follows is an = a1 + (n - 1)d. A1 is the first term in the sequence, which is 18,000,000. N is the number of terms, which if we are finding years 1-5 and we already know years 1 and 2, we can start with year 3, so n in this instance will equal 3. D represents the common differnce, and because 500,000 is being subtracted everytime, -500,000 will be used as our common differece.

Step 2 : Substitute the values in the function

Because the function is an = a1 (n -1)d, all we must do is put our values in and get an = 18,000,000 + (4 - 1) -500,000 and we would get approximately 16.5 million at 3 years.

Step 3 : Continue to input the values

After continuing this process a few times, we get the full sequence for the next 5 years : 18 mil, 17.5 mil, 17 mil, 16.5 mil, 16 mil, 15.5 mil and so on

1

17.5 mil.

17 mil.

2

6

32