From the table, what is $h(0)$? Which answer choices give that same value when you plug in $x=0$?

The values of $h(x)$ increase from $x=0$ to $x=4$. For an exponential $a\cdot b^x$, should $b$ be greater than 1 or between 0 and 1?

Compare $h(2)$ to $h(0)$ using the ratio $h(2)/h(0)$. That equals $b^2$. About what value of $b$ does that suggest, and which choice matches it?

Type each option into Desmos as a separate function, for example: h1(x)=1.12*(0.23)^x, h2(x)=1.12*(1.23)^x, h3(x)=1.23*(0.12)^x, and h4(x)=1.23*(1.12)^x.

Add a table in Desmos with the three points: $(0,1.23)$, $(2,1.54)$, and $(4,1.94)$. These points represent the actual data from the problem.

Look at which of the four graphs passes through (or is extremely close to) all three plotted points from the table for $0\le x\le 4$. The function whose curve best fits all three points is the correct choice.

An exponential function that models growth usually has the form

where:

• $a$ is the initial value (the value when $x=0$), and
• $b$ is the growth factor (how much the value is multiplied by each time $x$ increases by 1).

From the table, when $x=0$, $h(0)=1.23$.

In the model $h(x)=a\cdot b^x$:

• $h(0)=a\cdot b^0=a\cdot 1=a$.

So $a$ must be $1.23$.

This immediately eliminates any choice that does not start with $1.23$ when $x=0$. That rules out the options whose first factor is $1.12$, leaving only the functions that begin with $1.23$.

Look at how the height changes as $x$ increases:

• At $x=0$, $h(x)=1.23$.
• At $x=2$, $h(x)=1.54$.
• At $x=4$, $h(x)=1.94$.

The height is increasing over time. For an exponential model $a\cdot b^x$:

• If $b>1$, the function increases (growth).
• If $0<b<1$, the function decreases (decay).

So the base $b$ must be greater than 1, which rules out any option with a base less than 1 (like $0.12$).

Use two points to find the growth over a 2-year span.

From $x=0$ to $x=2$:

• $h(0)=1.23$
• $h(2)=1.54$

In the exponential model,

So

Then $b$ is about the square root of $1.25$, which is a bit bigger than $1.1$ and very close to $1.12$. So the base should be approximately $1.12$. Among the remaining choices with $a=1.23$, the one with a base slightly greater than 1 and close to $1.12$ is the correct model:

$h(x)=1.23(1.12)^x$.