Question

How do you identify equations as exponential growth, exponential decay, linear growth, or linear decay $ y=2x $?

Answer 1

Hint: We start solving the problem by recalling the definition of exponential growth as a quantity undergoing growth (increase in its numbers) exponentially in the given time ‘t’. We then give a general equation to represent exponential growth. We then recall the definition of exponential decay as a quantity undergoing decay (decrease in its numbers) exponentially in the given time ‘t’. We then give a general equation to represent exponential decay. We then recall the definition of linear growth as a quantity increasing constantly for a period of time. We then give a general equation to represent linear growth. We then recall the definition of linear decay as a quantity decreasing constantly for a period of time. We then give the general equation to represent linear decay.

Complete step by step answer:
According to the problem, we are asked to tell the properties of exponential growth, exponential decay, linear growth or linear decay $ y=2x $ .
Let us recall the definition of exponential growth.
We know that if a quantity undergoes growth (increase in its numbers) exponentially to the given time ‘t’, then that growth is known as exponential growth. This is represented as follows:
 $ \Rightarrow x=a{{b}^{t}} $ .
Where x = quantity of particle at time t.
 $ \Rightarrow $ a = quantity of particle at time $ t=0 $ .
 $ \Rightarrow $ b = growth factor.
 $ \Rightarrow $ t = time.
Let us recall the definition of exponential decay.
We know that if a quantity undergoes decay (decrease in its numbers) exponentially to the given time ‘t’, then that growth is known as exponential decay. This is represented as follows:
 $ \Rightarrow x=a{{b}^{-t}} $ .
Where x = quantity of particle at time t.
 $ \Rightarrow $ a = quantity of particle at time $ t=0 $ .
 $ \Rightarrow $ b = decay factor.
 $ \Rightarrow $ t = time.
Let us recall the definition of linear growth.
We know that if a quantity increases constantly for a period of time, then that growth is known as linear growth. This is represented as follows:
 $ \Rightarrow x=a+bt $ .
Where x = quantity of particle at time t.
 $ \Rightarrow $ a = quantity of particle at time $ t=0 $ .
 $ \Rightarrow $ b = growth factor.
 $ \Rightarrow $ t = time.
From the problem, we are given $ y=2x $ is an example for linear growth.
Let us recall the definition of linear decay.
We know that if a quantity decreases constantly for a period of time, then that growth is known as linear decay. This is represented as follows:
 $ \Rightarrow x=a+bt $ .
Where x = quantity of particle at time t.
 $ \Rightarrow $ a = quantity of particle at time $ t=0 $ .
 $ \Rightarrow $ b = decay factor.
 $ \Rightarrow $ t = time.
We should know $ y=-2x $ is an example of linear decay.

Note:
 Whenever we get this type of problem, we first recall the respective definition and then give an example following that definition. We should not think that $ e $ is the only fact that can be used as a growth or decay factor in exponential growth or decay, which is a common mistake done by students. Similarly, we can expect problems to tell the properties to identify whether the given function represents cubic growth or cubic decay.