Question

Find the equation of the exponential function represented by the table below: 0 3 1.5 2 0.75 3 0.375

Answer 1

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Answer 2

Sure?

Answer 3

To find the equation of an exponential function given a table of values, we need to determine the general form of the equation, which is "y = a * b^x", where "a" is the initial value and "b" is the base.

Let's start by examining the values in the table.

When x = 0, y = 3
When x = 1.5, y = 2
When x = 0.75, y = 3
When x = 0.375, y = ?

Since the values of y decrease as x increases, we can determine that the base, "b," must be between 0 and 1.

Let's examine the first and last rows of the table:

When x = 0, y = 3
When x = 0.375, y = 0.375 * b^0 = 3 * b^1

Since both values of y have a factor of 3, we can determine that the initial value, "a," is 3.

The equation is now "y = 3 * b^x".

To find the value of "b," we can substitute the x- and y-values from any row of the table:

When x = 1.5, y = 2
2 = 3 * b^1.5
2/3 = b^1.5

Now, we need to solve for "b." To do this, we raise each side of the equation to the power of (2/3):

(2/3)^(2/3) = (b^1.5)^(2/3)
(2/3)^(2/3) = b

Therefore, b ≈ 0.743.

The equation of the exponential function represented by the table is:
y = 3 * (0.743)^x

Answer 4

I apologize for any confusion. Let's go through the process again to find the equation of the exponential function represented by the table.

When x = 0, y = 3
When x = 1.5, y = 2
When x = 0.75, y = 3
When x = 0.375, y = ?

From the given values, we can see that the values of y are decreasing as x increases. This suggests that the base, "b," must be between 0 and 1.

Let's look at the first two rows of the table:

When x = 0, y = 3
When x = 1.5, y = 2

Using the general equation of an exponential function, y = a * b^x, we can substitute these values to form two equations:

3 = a * b^0
2 = a * b^1.5

Since b^0 = 1, the first equation becomes:

3 = a * 1
3 = a

So, we have a = 3.

Plugging this value into the second equation gives us:

2 = 3 * b^1.5

Solving for b, we get:

b^1.5 = 2/3

To isolate b, we raise both sides of the equation to the power of (2/3):

(b^1.5)^(2/3) = (2/3)^(2/3)
b = (2/3)^(2/3)

Therefore, b ≈ 0.7937.

The equation of the exponential function represented by the table is:
y = 3 * (0.7937)^x

I apologize again for any mistakes in the previous response. Please let me know if you have any further questions.

Answer 5

I apologize once again for any confusion or mistakes in my previous responses. Let's carefully go through the process again to find the equation of the exponential function based on the table.

We have the following values:

x = 0, y = 3
x = 1.5, y = 2
x = 0.75, y = 3
x = 0.375, y = ?

From the given values, we can observe that as x increases, y decreases. This suggests that the base, "b," must be between 0 and 1.

Let's examine the first and last rows of the table:

When x = 0, y = 3
When x = 0.375, y = 0.375 * b^0 = 3 * b^1

Since both values of y have a factor of 3, we can determine that the initial value, "a," is 3.

The equation is now "y = 3 * b^x".

To find the value of "b," we can substitute the x- and y-values from any row of the table:

When x = 1.5, y = 2
2 = 3 * b^1.5

To isolate the base b, we can divide both sides by 3:
2/3 = b^1.5

To solve for b, we raise both sides to the power of (2/3):
(2/3)^(2/3) = (b^1.5)^(2/3)
(2/3)^(2/3) = b

Therefore, b ≈ 0.7937.

The equation of the exponential function represented by the table is:
y = 3 * (0.7937)^x

I apologize for any errors or confusion in my previous responses. Please let me know if you have any further questions.