Question

Write an exponential function for the graph that passes through the points (0, –3) and (4, –48)

Answer 1

Given :

Two points (0 -3) and (4,-48)

To find :

Exponential function.

Solution :

The general expression for exponential function :

$\bf \: y= a \times ( {b}^{x} )$

(equation 1)

where

$a \neq 0 \\ b > 0 \\ x \in \: real \: number$

we have given two points

such that

in (0,-3)

x = 0, y = -3 ;

in (4,-48)

x =4, y = -48 ;

It is given that these two points lie on exponential curve

so these points must satisfy equation 1.

Putting (0,-3 ) in equation 1 :

$- 3 = a \times ( {b}^{0} )$

as b° = 1, so

-3 = a

so the value of a = -3 ;

Putting (4,-48) in equation 1 :

$- 48 = a \times ( {b}^{4} )$

Value of a is now known as a = -3

so

$- 48 = - 3 \times ( {b}^{4} ) \\ \\ {b}^{4} = \frac{48}{3} = 16$

so

$b ^{4} = 16 = {(2)}^{4} or {( - 2)}^{4}or {(2i)}^{4} or {( - 2i)}^{4}$

so

b can be 2 , -2 , 2i and -2i

but b should be real number so

b can be 2 and -2.

and b can not be negative (as stated above as the condition of exponential function)

So

b= 2

now

a = -3 and b = 2

On putting these values found ,in equation 1,

The required exponential function is :

$\bf \: y = \: - 3 \times ( {2}^{x} )$