Determine An Equation

Question

Determine an equation of a exponential function of base 2 with the following characteristics.

- Horizontal asymptote: y=4

- Range: y<4

- Y-Intercept (0,2)

Answer 1

Start with a function of the form y = a(2)^x, where a is a real number to be determined.  Any function of this type has horizontal asymptote y = 0 (the x-axis).

 

To get a horizontal asymptote of y = 4, translate the graph of

y = a(2)^x  four units upward.  The equation of the translated graph is                   y = a(2)^x + 4.  

 

We want this graph to have y-intercept (0, 2).   So, 2 = a(2)⁰ + 4.

 

Simplifying, we get 2 = a + 4   Thus, a = -2.

 

We have:  y = -2(2)^x + 4

 

   Note:  2^x > 0 for all x  So, -2(2)^x < 0 for all x

               Therefore, -2(2)^x + 4 < 4 for all x

               So, the range of y = -2(2)^x + 4 is y < 4

 

y = -2(2)^x + 4 is a function with the desired properties.  

 

 

We can use properties of exponents to rewrite the function as            

      y = -(2)^x+1 + 4

Answer 2

Let's start with the fact that an exponential function with a base of 2 means something of the form  2^x. Now, using the fact that the y-intercept is located at (0,2) translates to when x = 0, then y=2. But, 2^0 = 1 and 2^1 = 2, so the function could be y = f(x) = 2 ^ (x+1). Now since the range on y is that it must be less than 4, this means that X+1 must be less than 2 since 2^2 = 4. Finally if X+1 < 2, then x < 1.

 

You may use the facts of a horizontal asymptote at y = 4 and the point (0,2) is the y-intercept to sketch the graph.

 

Another point on the graph of this function is (-1,1). As x gets smaller and smaller (more negative) the value of y will approach (but never equal) 0. Thus there is another horizontal asymptote at y = 0.

 

One possible function is as follows:

 

y = f(x) = 2 ^ (x+1)  where the domain (the set of x values) is defined by x < 1

 

This will establish the range (the set of y values) as 0 < y < 4

 

This is not the only possible exponential function that has these characteristics. You should be able to think of at least one other exponential function with these same characteristics.