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Advanced Math / Nonlinear functions Difficulty: Medium

$x$ $x$	$h (x)$ $h (x)$
$0$ $0$	$1.23$ $1.23$
$2$ $2$	$1.54$ $1.54$
$4$ $4$	$1.94$ $1.94$

The table shows the exponential relationship between the number of years, $x$ , since Hana started training in pole vault, and the estimated height $h (x)$ , in meters, of her best pole vault for that year. Which of the following functions best represents this relationship, where $x less than or equals 4$ ?

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Explanation

Choice D is correct. The table shows an increasing exponential relationship between the number of years, $x$ , since Hana started training in pole vault and the estimated height $h (x)$ , in meters, of her best pole vault for that year. The relationship can be written as $h (x) = C a^{x}$ , where $C$ and $a$ are positive constants. It's given that when $x equals 0$ , $h left parenthesis x right parenthesis equals 1.23$ . Substituting $0$ for $x$ and $1.23$ for $h (x)$ in $h (x) = C a^{x}$ yields $1.23 equals upper C a Superscript 0$ , or $1.23 equals upper C$ . Substituting $1.23$ for $C$ in $h (x) = C a^{x}$ yields $h left parenthesis x right parenthesis equals 1.23 a Superscript x$ . It's also given that when $x equals 2$ , $h left parenthesis x right parenthesis equals 1.54$ . Substituting $2$ for $x$ and $1.54$ for $h (x)$ in $h left parenthesis x right parenthesis equals 1.23 a Superscript x$ yields $1.54 equals 1.23 a squared$ . Dividing each side of this equation by $1.23$ yields $\frac{1.54}{1.23} = \frac{1.23 a^{2}}{1.23}$ , or $a^{2}$ is approximately equal to $1.252$ . Since $a$ is positive, $a$ is approximately equal to $StartRoot 1.252 EndRoot$ , or $1.12$ . Substituting $1.12$ for $a$ in $h left parenthesis x right parenthesis equals 1.23 a Superscript x$ yields $h left parenthesis x right parenthesis equals 1.23 left parenthesis 1.12 right parenthesis Superscript x$ .

Choice A is incorrect. When $x equals 0$ , the value of $h (x)$ in this function is equal to $1.12$ rather than $1.23$ , and it is decreasing rather than increasing.

Choice B is incorrect. When $x equals 0$ , the value of $h (x)$ in this function is equal to $1.12$ rather than $1.23$ .

Choice C is incorrect. This function is decreasing rather than increasing.