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4.11 Comparing exponential and quadratic functions

Interactive practice questions

Consider the functions $f\left(x\right)=2^x$f(x)=2x and $g\left(x\right)=2x^2$g(x)=2x2, for $x\ge0$x0.

a

Complete the table of values for each function.

$x$x $f\left(x\right)$f(x) Increase in $f\left(x\right)$f(x) $g\left(x\right)$g(x) Increase in $g\left(x\right)$g(x)
$0$0 $\editable{}$   $\editable{}$  
$1$1 $\editable{}$ $1$1 $\editable{}$ $2$2
$2$2 $\editable{}$ $2$2 $\editable{}$ $6$6
$3$3 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
$4$4 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
$5$5 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
$6$6 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
$7$7 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
b

Which of the following statements is true?

The exponential function increases more rapidly at first, but then the quadratic function starts to increase more rapidly.

A

For $x\ge0$x0, the exponential function increases more rapidly than the quadratic function.

B

The quadratic function increases more rapidly at first, but then the exponential function starts to increase more rapidly.

C

For $x\ge0$x0, the quadratic function increases more rapidly than the exponential function.

D
Easy
7min

The graph of the exponential function $P$P, given by $y=-4^x$y=4x is shown below.

Easy
3min

Mint Corporation’s operations are such that the total amount mined by the $n$nth week of operations is given by $M=100\left(1.15\right)^{n-1}$M=100(1.15)n1.

Crest Corporation’s operations are such that the total amount mined by the $n$nth week is given by the equation $C=100n^2$C=100n2.

Medium
5min

To accommodate for its distributing population, a country creates a new city and immediately relocates $400000$400000 of its citizens there. The city’s land is allocated such that it can immediately produce enough food for $600000$600000 people in the first year. The table shows the functions that can be used to predict the city's population ($P$P) and the number of people who can be fed ($Q$Q), after $t$t years.

Medium
6min
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Outcomes

II.F.LE.3

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. Compare linear and exponential growth to quadratic growth.

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