x h x 0 1.23 2 1.54 4 1.94 The table shows the exponential relationship between

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 $h\left(x\right)$, in meters, of her best pole vault for that year. The relationship can be written as $h\left(x\right)=C{a}^x$, where $C$ and $a$ are positive constants. It's given that when $x=0$, $h\left(x\right)=1.23$. Substituting $0$ for $x$ and $1.23$ for $h\left(x\right)$ in $h\left(x\right)=C{a}^x$ yields $1.23=C{a}^0$, or $1.23=C$. Substituting $1.23$ for $C$ in $h\left(x\right)=C{a}^x$ yields $h\left(x\right)=1.23a^x$. It's also given that when $x=2$, $h\left(x\right)=1.54$. Substituting $2$ for $x$ and $1.54$ for $h\left(x\right)$ in $h\left(x\right)=1.23a^x$ yields $1.54=1.23a^2$. Dividing each side of this equation by $1.23$ yields $\frac{1.54}{1.23}=\frac{1.23a^2}{1.23}$, or $a^2$ is approximately equal to $1.252$. Since $a$ is positive, $a$ is approximately equal to $\sqrt{1.252}$, or $1.12$. Substituting $1.12$ for $a$ in $h\left(x\right)=1.23a^x$ yields $h\left(x\right)=1.23{\left(1.12\right)}^x$.

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

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

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