5.05 Comparing linear and exponential relationships
Exponential functions increase at an increasing rate or decrease at a decreasing rate. Because the rate of change varies, the slope of the graph varies and the graph must be curved.
Linear functions have a constant rate of increase or decrease and therefore, their graphs are straight lines.
However, when we are looking at samples of data drawn from processes that are being investigated, the distinction between exponential and linear may be unclear. The same data set may appear to be equally well described by an exponential or a linear model, or by some other model.
One reason for this difficulty is that if we look at any small enough region of an exponential function, it looks very like a linear function. It is only when the domain of the exponential function is wide enough that we see that it diverges from the linear.
In the following pair of graphs, it is clear that between $x=-10$x=−10 and $x=10$x=10, the two functions $y=1.05^x$y=1.05x and $y=0.5x+1$y=0.5x+1 are very similar. However, as $x$x increases, the difference soon becomes apparent.
In many situations, where values of the independent variable are of interest only over a small range, it is convenient to adopt a linear model rather than an exponential model. Calculations are easier with linear models.
When the independent variable has a wider domain, another interesting comparison between an exponential model and a linear model can be made. In the following pair of graphs, we compare the functions given by $y=1.05^x$y=1.05x and $y=15x+1$y=15x+1. The linear function has a much steeper slope than the exponential function over the domain shown.
We could be forgiven for thinking that the two graphs will never cross, apart from at the point $x=0$x=0, and that the linear function will always be greater than the exponential function no matter how large $x$x grows in the positive direction.
But this is not the case!
We can see that the exponential function graph is getting steeper as $x$x increases. Eventually, it will have the same slope as the linear function. And then, when $x$x increases further, the exponential curve will be steeper than the line $y=15x+1$y=15x+1 and the two will be heading towards another crossing.
In fact, the two functions have the same slope when $x=117.4$x=117.4 and they eventually intersect when $x\approx159.46$x≈159.46. You should check that for any chosen value of $x$x greater than $159.46$159.46, the value of the linear function $y=15x+1$y=15x+1 is less than the value of the exponential function $y=1.05^x$y=1.05x.
Example
When $x=160$x=160, the linear function $y=15x+1$y=15x+1 has the value $y=2401$y=2401 but the exponential function $y=1.05^x$y=1.05x has the value $y=2456$y=2456.
We can state that any increasing exponential function will attain a value larger than any linear function with a positive slope, for large enough values of the independent variable.
This property of exponential functions explains why the phrase 'exponential growth' has entered the common vocabulary with the meaning of a very rapid or increasing rate of increase.
Worked Examples
Question 1
The scatter plot below shows the amount of ambient oxygen in the air at various altitudes.
Which of the following functions would best model the relationship between altitude and the amount of ambient oxygen?
quadratic function
Alinear function
Bexponential function
CThe function $y=160+kx$y=160+kx is to be used to model the relationship between the altitude $x$x and the oxygen level $y$y. The function has been graphed below.
Loading Graph...Determine the value of $k$k.
Altitude sickness occurs when you cannot adjust to the lower levels of oxygen at high altitudes. At an altitude of $6.5$6.5 kilometers, Luke experiences altitude sickness and decides to descend to an altitude of $6$6 kilometers. According to the model, what is the change in the amount of ambient oxygen? Express the change as a positive quantity.
According to this model, at what altitude $x$x (km) will the amount of ambient oxygen reach $0$0? Leave your answer as an exact value.
Question 2
When CTech first released a digital application (an ‘app’) onto the market, the number of sales increased slowly at first, but then the number of sales started to increase very rapidly.
Which scatter plot shows the trend in sales over time from when the app was first released?
Loading Graph...ALoading Graph...BLoading Graph...CWhat sort of function is this?
Linear
AExponential
BQuadratic
CThe function $y=1000\left(60^{\frac{t}{10}}-1\right)$y=1000(60t10−1) is used to approximate the number of sales after $t$t months, where $y$y represents the number of sales.
Complete the table of values for $y=1000\left(60^{\frac{t}{10}}-1\right)$y=1000(60t10−1).
$t$t $0$0 $10$10 $y$y $\editable{}$ $\editable{}$ $15$15 months after CTech released their app, a rival company, BTech, released a similar app with improved features.
In the month that followed, CTech’s sales dropped by $50000$50000 from their previous month's sales.
According to the model $y=1000\left(60^{\frac{t}{10}}-1\right)$y=1000(60t10−1), what were CTech’s sales one month after BTech released their new app?
Give your answer to the nearest whole number
Question 3
Consider the following table of values for the functions for $x\ge1$x≥1:
$f\left(x\right)=5^x$f(x)=5x and $g\left(x\right)=2190x$g(x)=2190x.
| $x$x | $f\left(x\right)$f(x) | $g\left(x\right)$g(x) |
|---|---|---|
| $1$1 | $5$5 | $2190$2190 |
| $2$2 | $25$25 | $4380$4380 |
| $3$3 | $125$125 | $6570$6570 |
| $4$4 | $625$625 | $8760$8760 |
| $5$5 | $3125$3125 | $10950$10950 |
Neville concludes that $g\left(x\right)$g(x) is always greater than $5^x$5x when $x\ge1$x≥1.
Choose the correct statement.
Neville is correct. $g\left(x\right)>f\left(x\right)$g(x)>f(x) for all values of $x\ge1$x≥1.
ANeville is incorrect. $f\left(x\right)>g\left(x\right)$f(x)>g(x) for some values of $x\ge1$x≥1, but $g\left(x\right)$g(x) is increasing more rapidly so at some point $g\left(x\right)$g(x) will become greater in value.
BNeville is incorrect. $g\left(x\right)>f\left(x\right)$g(x)>f(x) for some values of $x\ge1$x≥1, but $f\left(x\right)$f(x) is increasing more rapidly, so at some point $f\left(x\right)$f(x) will become greater in value.
CNeville is incorrect. $f\left(x\right)>g\left(x\right)$f(x)>g(x) for all values of $x\ge1$x≥1.
DWhich function could these points represent?
Loading Graph...$f\left(x\right)$f(x)
A$g\left(x\right)$g(x)
B