Give a formula for the exponential function . The two points marked on the graph are A=(−1−43) and B=(12). The red horizontal line is given by y=5, and is a horizontal asymptote of the function.
To find the formula for the exponential function given two points and a horizontal asymptote, we will use the general form of an exponential function:
f(x) = a * b^x + c,
where a, b, and c are constants.
Let's use the given points A=(-1,-43) and B=(12,y) to find the values of a, b, and c.
1. Using point A:
-43 = a * b^(-1) + c
2. Using point B:
y = a * b^12 + c
Since we are given that the red horizontal line y = 5 is a horizontal asymptote of the function, we know that as x approaches infinity, the function approaches the value 5. This implies that c = 5.
Now, we can substitute the value of c into the equations above:
1. -43 = a * b^(-1) + 5
2. y = a * b^12 + 5
To further simplify the equations, we can subtract 5 from both sides of equation 1, giving us -48 = a * b^(-1).
Now, let's solve equation 1 for b^(-1) (b raised to the power of -1):
b^(-1) = -48/a
To get rid of the negative exponent, we can take the reciprocal of both sides:
1/b = -a/48
Now, equation 2 becomes:
y = a * (1/b)^(-12) + 5
Replacing 1/b with -a/48:
y = a/((-a/48))^12 + 5
Simplifying further by removing the negative exponent and using the property of exponents:
y = a/(a/48)^12 + 5
Now, we can simplify equation 1:
b^(-1) = -48/a
Taking the reciprocal of both sides, we get:
b = -a/48
Therefore, the formula for the exponential function is:
f(x) = a * (-a/48)^x + 5,
where a is a constant and can be found by substituting point A or B into the equation.