Question

Determine whether the table represents an exponential decay

function, exponential growth function, negative linear function, or
positive linear function.

X 0 1 2 3
y 40 20 10 5

A) Exponential decay function
B) Exponential growth function
C) Negative linear function
D) Positive linear function

Answer 1

(A) Exponential decay fnction

Explanation:

As x increases , y decreases so it is exponential decay or negative linear

If it is linear its of the form y = mx + c where m = slope and c = y-intercepts.

m is a constant if its linear

Check if the slope m is constant:

m = (20-40) / (1 - 0) = -20

m = (10-20)/ 1 = -10

m = (5 - 10)/ 1 = -5

- not linear.

So, it is exponential decay.

The fnction is y = 40(1/2)^x.

eg when x = 3 y = 40(1/2)^3 = 40 * 1/8 = 5.

Answer 2

The non-linear nature is affirmed by negative slopes between points. The consistent fractional decrease in Y values supports an exponential decay model for the relationship in the given table.

The correct answer is option A.

Examining the given table, it's evident that as X increases from 0 to 3, there's a rapid decrease in Y from 40 to 5. To showcase that it's not a linear function, consider the slope between any two points. Taking the first two points (0, 40) and (1, 20), the slope is:

(20 - 40)/(1 - 0) = -20.

Similarly, for the next two points (1, 20) and (2, 10), the slope is

(10 - 20)/(2 - 1) = -10.

The consistent negative slope indicates a non-linear relationship.

To establish that it aligns with an exponential model, note that the ratio of Y values is constant. The ratio between consecutive Y values is

20/40

= 0.5, 10/20 = 0.5, and 5/10 = 0.5.

This demonstrates a consistent fractional decrease, characteristic of an exponential decay function.

In conclusion, the table represents a non-linear relationship, and the consistent fractional decrease in Y values suggests an exponential decay model.

Therefore, from the given options the correct one is A.