Which Exponential Function Has A Growth Factor Of 5 - T-Tees - Your Answers Await, Life's Questions Unraveled

Defining an Exponential Function

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What exactly does it mean to grow exponentially? What does the word double have in common with percent increase? People toss these words around errantly. Are these words used correctly? The words certainly appear frequently in the media.

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Percent change refers to a change based on a percent of the original amount.
Exponential growth refers to an increase based on a constant multiplicative rate of change over equal increments of time, that is, a percent increase of the original amount over time.
Exponential decay refers to a decrease based on a constant multiplicative rate of change over equal increments of time, that is, a percent decrease of the original amount over time.

For us to gain a clear understanding of exponential growth, let us contrast exponential growth with linear growth. We will construct two functions. The first function is exponential. We will start with an input of 0, and increase each input by 1. We will double the corresponding consecutive outputs. The second function is linear. We will start with an input of 0, and increase each input by 1. We will add 2 to the corresponding consecutive outputs. See (Figure).

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[latex]x[/latex] [latex]fleft(xright)={2}^{x}[/latex] [latex]gleft(xright)=2x[/latex] 0 1 0 1 2 2 2 4 4 3 8 6 4 16 8 5 32 10 6 64 12

From (Figure) we can infer that for these two functions, exponential growth dwarfs linear growth.

Exponential growth refers to the original value from the range increases by the same percentage over equal increments found in the domain.
Linear growth refers to the original value from the range increases by the same amount over equal increments found in the domain.

Apparently, the difference between “the same percentage” and “the same amount” is quite significant. For exponential growth, over equal increments, the constant multiplicative rate of change resulted in doubling the output whenever the input increased by one. For linear growth, the constant additive rate of change over equal increments resulted in adding 2 to the output whenever the input was increased by one.

The general form of the exponential function is[latex],fleft(xright)=a{b}^{x},,[/latex]where[latex],a,[/latex]is any nonzero number,[latex],b,[/latex]is a positive real number not equal to 1.

If[latex],b>1,[/latex]the function grows at a rate proportional to its size.
If[latex],0<b<1,[/latex] the function decays at a rate proportional to its size.

Let’s look at the function[latex],fleft(xright)={2}^{x},[/latex]from our example. We will create a table ((Figure)) to determine the corresponding outputs over an interval in the domain from[latex],-3,[/latex]to[latex],3.[/latex]

[latex]x[/latex] [latex]-3[/latex] [latex]-2[/latex] [latex]-1[/latex] [latex]0[/latex] [latex]1[/latex] [latex]2[/latex] [latex]3[/latex] [latex]fleft(xright)={2}^{x}[/latex] [latex]{2}^{-3}=frac{1}{8}[/latex] [latex]{2}^{-2}=frac{1}{4}[/latex] [latex]{2}^{-1}=frac{1}{2}[/latex] [latex]{2}^{0}=1[/latex] [latex]{2}^{1}=2[/latex] [latex]{2}^{2}=4[/latex] [latex]{2}^{3}=8[/latex]

Let us examine the graph of[latex],f,[/latex]by plotting the ordered pairs we observe on the table in (Figure), and then make a few observations.

Let’s define the behavior of the graph of the exponential function[latex],fleft(xright)={2}^{x},[/latex]and highlight some its key characteristics.

the domain is[latex],left(-infty ,infty right),[/latex]
the range is[latex],left(0,infty right),[/latex]
as[latex],xto infty ,fleft(xright)to infty ,[/latex]
as [latex],xto -infty ,fleft(xright)to 0,[/latex]
[latex],fleft(xright),[/latex]is always increasing,
the graph of[latex],fleft(xright),[/latex]will never touch the x-axis because base two raised to any exponent never has the result of zero.
[latex],y=0,[/latex]is the horizontal asymptote.
the y-intercept is 1.