Question

From the table below, determine whether the data shows an exponential function. Explain why or why not.

Answer 1

Answer:

Option B-  Yes; the domain values are at regular intervals and the range values have a common factor 4.

Step-by-step explanation:

Given : The data

(x) -5,   -4,  -3,   -2

(y) 0.5,  2,   8,   32

To find : The data shows an exponential function or not

Solution :

The general form of an exponential form is $y=ab^x$

To check whether the data give the exponential function we form equation with the help of two points and verify the other two points .

$y=ab^x$

Let x= -5 and y=0.5

$0.5=ab^{-5}$

$\frac{0.5}{b^{-5}}=a$ ......[1]

Let x= -4 and y=2

$2=ab^{-4}$

$\frac{2}{b^{-4}}=a$ .........[2]

Equate LHS because RHS is equal in equation [1] and [2]

$\frac{2}{b^{-4}}=\frac{0.5}{b^{-5}}$

$\frac{b^{-4}}{b^{-5}}=\frac{2}{0.5}$

$b=4$

Put back in [2]

$\frac{2}{4^{-4}}=a$ .

$a=2\times4^4$

$a=2\times256=512$

a=512 and b=4

Exponential function - $y=4(512)^x$

To verify this function put

1) x=-3

$y=512(4)^{-3}$

$y=\farc{512}{64}$

$y=8$

The point satisfied.

2) x=-2

$y=512(4)^{-2}$

$y=\farc{512}{16}$

$y=32$

The point satisfied.

Therefore, The given data is an exponential function $y=4(512)^x$

The domain values are at regular intervals and the range values have a common factor 4 because b=4 and the change happen but value of b remain same.

Hence, Option B is correct.

Yes; the domain values are at regular intervals and the range values have a common factor 4.

Answer 2

B) Notice how you can get the next value multiplying by 4 each time, while x changes by one:

0.5*5=2, 2*4=8, 8*4=32

This function is 4^(x+4)*2 = 512*4^x, 512*4^(-5)=512/1024=0.5, it works!