Brianna is working on her homework. Her assignment is to come up with four representations for an exponential function of her choosing. She decides it is easiest to start by writting y=1200 (1/2) ^x. Help brianna create the other three components of the web

The second component would be :

y=1200(0.5)^x
The third component would be :
y=3^x
The fourth component would be:
y=(0.5)^x

Sure, Brianna, I'd be happy to help you create the other three components of your exponential function web! Remember, we want to keep things fun and interesting. So, here we go:

1. First, let's add a "graph" component to your web. Since you already have the equation y = 1200 (1/2)^x, let's plot some points to see its shape. How about we start with x = 0? That means y = 1200 (1/2)^0, which simplifies to y = 1200. So, the first point on your graph can be (0, 1200). Nice and easy!

2. For the second component, let's add a "real-life scenario" to your web. How about this: imagine you have a pet dinosaur named Dino. Dino has a favorite toy, a rubber bone. Every day, the rubber bone loses half of its size due to Dino's enthusiastic chewing. So, if we want to represent the size of the bone after x days, we can use your exponential function: y = 1200 (1/2)^x. This function shows how the size of the bone decreases over time.

3. The third component could be a "math joke" that relates to exponential functions. How about this: Why did the math book look sad? Because it had too many problems to solve, and the exponential ones just kept multiplying! It's always good to add a bit of humor to your work, right?

4. Lastly, let's add a "word problem" to your web. Here it is: Imagine you are a popcorn fanatic, and you find the perfect seasoning that makes your popcorn exponentially tastier. The amount of tastiness (y) you get from x scoops of seasoning can be represented by the equation y = 1200 (1/2)^x. So, if you start with 1200 units of tastiness (with no seasoning), for each scoop of seasoning you add, the tastiness will decrease by half. How many scoops of seasoning do you need to make your popcorn go from 1200 units of tastiness to just 100 units? Use your exponential function to find out!

There you have it, Brianna! Four components for your exponential function web, making it fun and engaging. Good luck with your homework!

To help Brianna create the other three components of the representation for her exponential function, let's break down the given function and explore each component step by step.

Step 1: Given function
y = 1200(1/2)^x

Step 2: Base (b)
The base (b) in an exponential function determines the rate of growth or decay. In this case, the base is 1/2.

Step 3: Exponent (x)
The exponent (x) represents the variable that the function depends on. It could be time, distance, or any other relevant factor.

Step 4: Initial value (y)
The initial value (y) is the value of the exponential function when the exponent is 0. In this case, the initial value is 1200.

Now, let's create the other three representations for Brianna's exponential function.

Representation 1: Change the base
One way to represent the same exponential function with a different base is by changing the value of the base while keeping the other components the same. Let's change the base to 2.

y = 1200 * 2^x

Representation 2: Change the exponent
Another way to create a different representation is by changing the value of the exponent while keeping the other components the same. Let's change the exponent to (x - 1).

y = 1200(1/2)^(x - 1)

Representation 3: Change the initial value
To create another representation, we can change the initial value while keeping the other components the same. Let's change the initial value to 600.

y = 600(1/2)^x

So, the four representations of Brianna's exponential function are:

1. y = 1200(1/2)^x
2. y = 1200 * 2^x
3. y = 1200(1/2)^(x - 1)
4. y = 600(1/2)^x

Brianna can use these representations to explore different aspects of the exponential function and gain a deeper understanding of its behavior.

To create the other three components of the web for the exponential function y = 1200(1/2)^x, we will use the following transformations:

1. Vertical translation: Add or subtract a constant value to the original function.
2. Horizontal translation: Replace x with (x - h), where h is a constant.
3. Vertical stretch or compression: Multiply or divide the original function by a constant value.

Let's go through each transformation one by one:

1. Vertical Translation:
To create a representation that shifts the graph vertically, you can add or subtract a constant to the original function. Let's say you want to shift it up by 500 units. Simply add 500 to the function:
y = 1200(1/2)^x + 500

2. Horizontal Translation:
To create a representation that shifts the graph horizontally, replace x with (x - h), where h is a constant representing the horizontal shift. Suppose you want to shift the graph 3 units to the right. Replace x with (x - 3):
y = 1200(1/2)^(x-3)

3. Vertical Stretch or Compression:
To create a representation that stretches or compresses the graph vertically, multiply or divide the original function by a constant value. Let's say you want to stretch the graph vertically by a factor of 2. Multiply the function by 2:
y = 2 * 1200(1/2)^x

Using these transformations, you have now created the other three components of the web for the exponential function y = 1200(1/2)^x:

1. Vertical translation: y = 1200(1/2)^x + 500
2. Horizontal translation: y = 1200(1/2)^(x-3)
3. Vertical stretch or compression: y = 2 * 1200(1/2)^x

Remember, these representations are just examples, and you can choose different values or combinations of transformations based on your assignment requirements or creative choices.

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