Question

How do you write an exponential function of the form \[y = a{b^x}\] the graph of which passes through \[\left( { - 1,\dfrac{4}{5}} \right)\] and \[\left( {2,100} \right)\] ?

Answer 1

Hint: The above question is based on the concept of exponential form. The main approach towards solving this question is by substituting the values of both x i.e., -1 and 2 from both the points and also y in the equation and we get the final solution in the above form.

Complete step-by-step answer:
An exponential function can describe growth or decay. An exponential function is a function of the form
 \[y = a{b^x}\]
where b is a positive real number which is not equal to 1 and the argument x occurs as an exponent.
Since we know the points given as \[\left( { - 1,\dfrac{4}{5}} \right)\] and \[\left( {2,100} \right)\] .
So, the values of x and y can be substituted in the above general equation we get
when x=-1
 \[\dfrac{4}{5} = a{\left( b \right)^{ - 1}} = \dfrac{a}{b}\]
when x=200
 \[100 = a{\left( b \right)^2}\]
In the first equation we multiply b on both the sides we get,
$\dfrac{{4b}}{5} = a$
Now substituting the value of a in the second equation,
 \[
   \Rightarrow 100 = \dfrac{{4b}}{5}{\left( b \right)^2} \\
   \Rightarrow 100 \times 5 = 4{b^3} \\
   \Rightarrow 125 = {b^3} \;
 \]
Taking cube root on both sides we get,
 \[b = 5\]
Since we got the value of b we can then find out the value of a.
 \[
  100 = a \times 25 \\
  a = 4 \;
 \]
Now by substituting the value of a in general equation we get,
 \[y = 4{\left( 5 \right)^x}\]
So, the correct answer is \[y = 4{\left( 5 \right)^x}\] ”.

Note: An important thing to note is that in an exponential form b should always be greater than zero. Since here we get the value of b as 5 that means it is a valid solution but if we would have got the value as negative then the value is not valid.