Trigonometric Graphs

NZ Level 8 (NZC) Level 3 (NCEA) [In development]

Applications of sine and cosine functions

The population (in thousands) of two different types of insects on an island can be modelled by the following functions: Butterflies: $f\left(t\right)=a+b\sin\left(mt\right)$`f`(`t`)=`a`+`b``s``i``n`(`m``t`), Crickets: $g\left(t\right)=c-d\sin\left(kt\right)$`g`(`t`)=`c`−`d``s``i``n`(`k``t`)

$t$`t` is the number of years from when the populations started being measured, and $a$`a`,$b$`b`,$c$`c`,$d$`d`,$m$`m`, and $k$`k` are positive constants. The graphs of $f$`f` and $g$`g` for the first $2$2 years are shown below.

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a

State the function $f\left(t\right)$`f`(`t`) that models the population of Butterflies over $t$`t` years.

b

State the function $g\left(t\right)$`g`(`t`) that models the population of Crickets over $t$`t` years.

c

How many times over a $18$18 year period will the population of Crickets reach its maximum value?

d

How many years after the population of Crickets first starts to increase, does it reach the same population as the Butterflies?

e

Solve for $t$`t`, the number of years it takes for the population of Butterflies to first reach $200000$200000.

Easy

Approx 19 minutes

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Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions