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Determine the exponential equation for the graph below.

If a value is non-integer type is as a fraction. Some points on the graph
are (0,5),(1,1), and (2,-11)
The base is ?
The coefficient is ?
The exponent on our transformed function is ?
The constant we are adding to our function is ?

1 Answer

5 votes
To determine the exponential equation for the given graph, we can use the general form of an exponential function:

y = ab^x + c

Using the given points (0, 5), (1, 1), and (2, -11), we can substitute these values into the equation to determine the values of the base (b), coefficient (a), exponent, and constant (c).

1. Substituting (0, 5):
5 = ab^0 + c
5 = a + c

2. Substituting (1, 1):
1 = ab^1 + c
1 = ab + c

3. Substituting (2, -11):
-11 = ab^2 + c

We now have a system of three equations with three variables (a, b, c).

From equation 1, we have: a + c = 5 ----> Equation (4)

From equation 2, we have: ab + c = 1

Solving equations (4) and (2) simultaneously, we can eliminate c:

(a + c) - (ab + c) = 5 - 1
a - ab = 4 - 1
a(1 - b) = 3
a = 3 / (1 - b) ----> Equation (5)

Now, substitute the value of a from equation (5) into equation (1):

5 = (3 / (1 - b)) + c

Simplifying, we get:
5(1 - b) = 3 + c(1 - b)
5 - 5b = 3 + c - cb

Rearranging, we have:
2 = c - 5b + cb
2 = c(1 - 5b) ----> Equation (6)

Now, we have two equations (5) and (6) with two variables (b and c).

Equation (5) can be rewritten as:
a = 3 / (1 - b)

Substitute the value of a from equation (5) into equation (6):

2 = (3 / (1 - b))(1 - 5b)

Simplifying, we get:
2 = 3 - 15b - 3b^2

Rearranging, we have:
3b^2 + 15b - 1 = 0

Now, we can solve this quadratic equation for the value of b using the quadratic formula:

b = (-15 ± √(15^2 - 4 * 3 * (-1))) / (2 * 3)

Simplifying, we get:
b = (-15 ± √(225 + 12)) / 6
b = (-15 ± √(237)) / 6

The values of b obtained from this equation will be the base of the exponential equation.

Similarly, we can substitute the value of b into equation (5) to find the corresponding value of a, and substitute both b and a into equation (4) to find the value of c.

Once we have the values of a, b, and c, we can write the exponential equation in the form: y = ab^x + c.
User Amirhm
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