The function y=f(x) graphed above is assumed to be a vertical shift of an exponential function. The two points marked on the graph are A= (-1,-31) and B= (1,-1) . The red horizontal line is given by y= 1, and is a horizontal asymptote of the function. Give the formula for y in terms of x.
To find the formula for y in terms of x, we need to determine the equation for the exponential function that has been vertically shifted.
First, let's find the general equation for an exponential function:
y = a * e^(bx)
Since the given function has a horizontal asymptote at y = 1, this means that when x approaches positive or negative infinity, y approaches 1. In other words, the value of a in the equation above is 1.
Now, let's find the value of b by using the points A(-1, -31) and B(1, -1) on the graph.
For point A, substitute the coordinates into the equation:
-31 = 1 * e^(-1b)
-31 = e^(-b)
For point B, substitute the coordinates into the equation:
-1 = 1 * e^(1b)
-1 = e^b
Now we have a system of equations:
-31 = e^(-b)
-1 = e^b
To eliminate e, we can take the logarithm of both sides of each equation:
ln(-31) = ln(e^(-b))
ln(-1) = ln(e^b)
Using the property of logarithms (ln(e^x) = x), we have:
-31 = -b
-1 = b
Now that we have determined the value of b, we can substitute it back into one of the original equations to find the value of a:
-1 = e^b
-1 = e^-1
Therefore, the equation for y in terms of x is:
y = 1 * e^(-x)
Simplifying further:
y = e^(-x)
To find the formula for y in terms of x, let's analyze the given information. We are told that the graph above is a vertical shift of an exponential function. This means that if we can determine the equation of the underlying exponential function, we can find the equation for the shifted graph.
First, let's consider the asymptote of the function. We are given that the red horizontal line y = 1 is a horizontal asymptote. An exponential function can have a horizontal asymptote only if it approaches that value as x approaches positive or negative infinity. Since the function approaches y = 1 as x approaches infinity, this suggests that the base of the exponential function must be less than 1.
Now, let's look at the points A(-1, -31) and B(1, -1). These points represent specific values on the graph of the shifted function. We can use them to determine how the function has been shifted vertically.
Since the vertical shift is given, we need to find the difference in the y-coordinates of the points A and B. In this case, it is -31 - (-1) = -30. Therefore, the function has been shifted vertically downward by 30 units.
Now, let's assume that the original exponential function y = a*b^x. Since the graph is shifted vertically downward by 30 units, the equation for the shifted function can be written as y = a*b^x - 30.
To find the values of a and b, we can use either of the given points (A or B). Let's use point B (1, -1):
-1 = a*b^1 - 30 -> a*b - 30 = -1
Next, let's use point A (-1, -31):
-31 = a*b^(-1) - 30 -> a/b - 30 = -31
We now have a system of equations:
a*b - 30 = -1
a/b - 30 = -31
Solving this system of equations, we find a = -30 and b = 1/2.
Plugging these values back into the equation for the shifted function, we get:
y = -30*(1/2)^x - 30
So, the formula for y in terms of x is y = -30*(1/2)^x - 30.