Give a formula for the function illustrated using a vertical shift of an exponential function. The two points marked on the graph are A=(−1,17) and B=(1,2). The red horizontal line is given by y=1, and is a horizontal asymptote of the function. What is the equation?
To find the equation for the function illustrated using a vertical shift of an exponential function given the points A=(-1,17) and B=(1,2) and the horizontal asymptote y=1, we can start by considering the general form of an exponential function:
y = a * b^x
where a represents the vertical shift value and b represents the base of the exponential function.
Since we are shifting the function vertically, the equation will have the form:
y = a * b^x + k
where k is the vertical shift.
Let's first find the value of k, which is the value added to the exponential function to shift it vertically. We can use the point A=(-1,17) to solve for k:
17 = a * b^(-1) + k ...(1)
Now, let's find the value of a. Using the other given point B=(1,2), we can substitute the values into the equation:
2 = a * b^1 + k
Since k and a are the same values in both equations (1) and (2), we can subtract equation (2) from equation (1) to eliminate k:
17 - 2 = a * b^(-1) - (a * b^1)
15 = a * (1/b) - a * b
Next, we can use the horizontal asymptote y=1, which means that as x approaches infinity, the function approaches 1. This gives us:
lim (x→∞) (a * b^x) = 1
As x approaches infinity, the exponential function approaches zero. Therefore, we have:
1 = a * 0
Since any number multiplied by 0 is 0, we can conclude that a must be equal to 0.
Substituting a=0 into equation (3), we get:
15 = 0 - 0
This equation is true, and it shows that 15 equals zero. However, this doesn't make sense, so the initial assumption that a=0 must be incorrect. This means that the horizontal asymptote y=1 is not a suitable choice for the given points A=(-1,17) and B=(1,2).
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To find the formula for the function illustrated using a vertical shift of an exponential function, we need to take into account the key points A and B on the graph, as well as the horizontal asymptote y=1.
First, let's consider the standard form of an exponential function:
y = ab^x
Here, a represents the initial value or y-intercept, and b represents the base or growth/decay factor. However, the given graph is shifted vertically, so we need to modify the equation accordingly.
To shift the graph vertically, we utilize a vertical translation term: h, where h represents the vertical shift. In this case, the horizontal asymptote is y=1, which implies a vertical shift of 1 unit downward from the original exponential function.
Thus, the modified equation for the function becomes:
y = ab^x + 1
Now, we need to find the values of a and b using the given points A=(-1,17) and B=(1,2).
For point A=(-1,17), substitute x=-1 and y=17 into the equation:
17 = ab^-1 + 1
16 = ab^-1
Similarly, for point B=(1,2), substitute x=1 and y=2 into the equation:
2 = ab^1 + 1
1 = ab
Now we have a system of equations:
16 = ab^-1
1 = ab
To solve this system, let's eliminate variable a:
16 = (ab^-1) * b
16 = b^2
Solving for b, we take the square root of 16:
b = ±4
Since exponential functions cannot have negative bases, we take the positive value for b:
b = 4
Now we can substitute this value of b back into the second equation to find a:
1 = a * 4
a = 1/4
Finally, we have determined the values of a and b. Plugging them into the modified exponential equation, we get:
y = (1/4)(4^x) + 1
Simplifying further:
y = 4^(x-2) + 1
Therefore, the equation for the function illustrated is y = 4^(x-2) + 1.