SAT Question 84dd43f8: Nonlinear Functions answer and explanation with Desmos solution

44$.It’sgiventhat$f \left( 0 \right) = 86$andthatforeachincreasein$x$by$1$,thevalueof$f \left( x \right)$decreasesby＄80\%$. Because the output of the function decreases by a constant percentage for each ＄1$-unitincreaseinthevalueof$x$,thisrelationshipcanberepresentedbyanexponentialfunctionoftheform$f \left( x \right) = a \left( b \right)^{x}$,where$a$representstheinitialvalueofthefunctionand$b$ represents the rate of decay,

expressed as a decimal. Because $f\left(0\right)=86$, the value of $a$ must be $86$. Because the value of $f\left(x\right)$ decreases by $80\%$ for each $1$ -unit increase in $x$, the value of $b$ must be $(1-0.80)$, or $0.2$. Therefore, the function $f$ can be defined by $f\left(x\right)=86{\left(0.2\right)}^x$. Substituting $2$ for $x$ in this function yields $f\left(2\right)=86{\left(0.2\right)}^2$, which is equivalent to $f\left(2\right)=86\left(0.04\right)$, or $f\left(2\right)=3.44$. Either $3.44$ or $86/25$ may be entered as the correct answer.

Alternate approach: It’s given that $f\left(0\right)=86$ and that for each increase in $x$ by $1$, the value of $f\left(x\right)$ decreases by $80\%$. Therefore, when $x=1$, the value of $f\left(x\right)$ is $(100-80)\%$, or $20\%$, of $86$, which can be expressed as $\left(0.20\right)\left(86\right)$. Since $\left(0.20\right)\left(86\right)=17.2$, the value of $f\left(1\right)$ is $17.2$. Similarly, when $x=2$, the value of $f\left(x\right)$ is $20\%$ of $17.2$, which can be expressed as $\left(0.20\right)\left(17.2\right)$. Since $\left(0.20\right)\left(17.2\right)=3.44$, the value of $f\left(2\right)$ is $3.44$. Either $3.44$ or $86/25$ may be entered as the correct answer. Note that 86/25, 3.44, and 3.440 are examples of ways to enter a correct answer.