Give a possible formula for the illustrated function using a vertical shift of an exponential function. The two points marked on the graph are A=(−172) and B=(10). The red horizontal line is given by y=−3, and is a horizontal asymptote of the function

y=?

y=-3e^(x+172)

To find a possible formula for the illustrated function using a vertical shift of an exponential function, we can start by considering the horizontal asymptote.

The function has a horizontal asymptote at y = -3, which means that as x approaches infinity (or negative infinity), the y-values of the function approach -3.

Next, let's consider the two given points on the graph: A(-172) and B(10). These points provide us with some information about the vertical shift and the growth/decay of the exponential function.

Since point A is on the graph, we know that the y-value of the function at x = -172 is the same as the y-value of the horizontal line y = -3. So, we have:

f(-172) = -3

Similarly, point B tells us that the y-value of the function at x = 10 is also the same as the y-value of the horizontal line y = -3. So, we have:

f(10) = -3

Based on the given information, a possible formula for the function can be written as:

y = Ae^(kx) - 3

where A represents the initial value of the exponential function, k represents the growth/decay rate, and e is the base of the natural logarithm.

To determine the values of A and k, we can use the given points.

Using point A(-172):
-3 = Ae^(-172k)

Using point B(10):
-3 = Ae^(10k)

Simplifying these two equations, we have a system of equations:
e^(-172k) = 1/3 (equation 1)
e^(10k) = 1/3 (equation 2)

We can solve this system of equations for k by taking the natural logarithm of both sides of each equation:

ln(e^(-172k)) = ln(1/3)
-172k = ln(1/3)

ln(e^(10k)) = ln(1/3)
10k = ln(1/3)

Now, we can solve these equations for k:

-172k = ln(1/3)
k = ln(1/3) / -172

10k = ln(1/3)
k = ln(1/3) / 10

The value of k obtained from the equations is the same, so let's calculate it:

k = ln(1/3) / -172 ≈ -0.00403

Therefore, a possible formula for the illustrated function using a vertical shift of an exponential function is:

y = Ae^(-0.00403x) - 3, where A is the initial value of the exponential function.

To find a possible formula for the illustrated function using a vertical shift of an exponential function, we need to consider the given information.

First, let's understand the concept of a horizontal asymptote. A horizontal asymptote is a horizontal line that a function approaches but never touches as the input (x-values) approaches infinity or negative infinity. In this case, the red horizontal line y = -3 is a horizontal asymptote of the function.

Now, let's consider the two given points on the graph: A = (-172) and B = (10). As per the graph, these two points lie on the function.

To form an exponential function, we start with the general exponential function formula:
y = a * b^x

where "a" is the vertical scaling factor and "b" is the base of the exponential function.

To incorporate the vertical shift, we need to add or subtract a constant, "c," to the function. This vertical shift will move the graph up or down on the y-axis.

Since the given horizontal asymptote is y = -3, the graph of the function needs to approach this line as x approaches infinity or negative infinity. This means the function needs to shift down by 3 units to align with the horizontal asymptote.

Now, let's solve for the values of "a," "b," and "c" using the given points.

1. Point A = (-172):
Plug in the coordinates into the exponential function formula:
-172 = a * b^(-172) + c

2. Point B = (10):
Plug in the coordinates into the exponential function formula:
10 = a * b^10 + c

We now have two equations with two unknowns (a, b) and (c). You can solve these equations using various methods, such as substitution or elimination, to find the values of "a," "b," and "c."

Once you have determined the values of "a," "b," and "c," substitute them back into the exponential function formula:
y = a * b^x + c

This will give you a possible formula for the illustrated function using a vertical shift of an exponential function that satisfies the given points and the horizontal asymptote.