What does f(x)=

Question

Find an exponential function of the form f(x) = b * a ^ (- x) + c that has the given horizontal asymptote and x and y-Intercepts

y = - 18 ; y-intercept 864; x - intercet 2

Answer 1

The x-intercept is where function crosses x-axis and hence y=0. The y-intercept is where function crosses y-axis and hence x=0. In other words, the x-intercept is at point (2,0) and y-intercept at point (0,864). Subsequently:

0 = b*a^(-2) + c ⇒ c= -b/(a^2).

and

864 = b*a(0) + c ⇒ 864= b + c.

The horizontal asymptote is where y =-18 when x → +∞, or equivalently: lim_x→∞ (b*a^(-x) +c) =-18, which implies c= -18. By replacing the value of c in above equation we obtain: b=882, which in turn finds a=7. The exponential function therefore is found as: f(x)= 882*7^(-x) -18.

Answer 2

For this kind of problem, we are given a function with 3 unknowns (a, b, and c) and 3 tools with which to find those 3 unknowns (Horizontal asymptote(HA), y-intercept, and x-intercept).

First, let's use the Horizontal asymptote. Consider what happens as x approaches infinity. In other words, what is the value of the limit of the function as x approaches infinity. The problem statement tells us that this value is -18. Therefore, we can find the following equation,

-18 = lim_x→∞ b*a^(-x) + c

Notice that if we calculate this limit, it is equivalent to just the variable "c",

-18 = c

So the value of "c" in our function is -18, We can use this to find "a" and "b".

Next, let's use the y-intercept

y-intercept = (0,864)

Plugging these values of x and y into the function yields,

864 = b*a^0 + c

864 = b + c

Recall that c = -18, so we can plug that in to find b,

864 = b - 18

882 = b

Now we know that b = 882

Lastly, we can use the x-intercept to find a,

x-intercept = (2,0)

Plugging these values of x and y into the function yields,

0 = b*a^(-2) + c

Now plug in the known values of "b" and "c",

0 = 882a^(-2) - 18

We can solve this using simple algebra to find that a = 7.

In conclusion, the values for the 3 unknowns are,

a = 7

b = 882

c = -18

and the function is f(x) = 882*(7)^(-x) - 18