Compute how much $h(x)$ changes when $x$ goes from $2$ to $6$. Then divide the change in $h(x)$ by the change in $x$ to find the slope.

Once you know the slope $m$, write $h(x)$ in the form $h(x)=mx+b$ and plug in one of the given points to solve for $b$, the $y$-intercept.

Remember that $h(0)$ is the $y$-value when $x=0$. In the equation $h(x)=mx+b$, what does $b$ represent?

In a Desmos expression line, type 5 + (0 - 2) * (13 - 5) / (6 - 2) and press Enter. This uses the idea $h(0) = h(2) + (0-2)\cdot\dfrac{h(6)-h(2)}{6-2}$. The value that Desmos displays is $h(0)$.

A linear function $h$ whose graph is a straight line is defined by two points:

• $h(2)=5$ means the point $(2,5)$ is on the line.
• $h(6)=13$ means the point $(6,13)$ is on the line.

We want to find $h(0)$, which is the $y$-value when $x=0$ (the $y$-intercept of the line).

Use the slope formula $m = \dfrac{\Delta y}{\Delta x}$ with the two points $(2,5)$ and $(6,13)$:

So the slope of the line is $2$.

A linear function can be written in slope-intercept form $h(x)=mx+b$. We already know $m=2$, so write

Now plug in one of the known points, for example $(2,5)$, to solve for $b$:

Solve this equation for $b$. The value of $b$ will be the $y$-intercept, which is also $h(0)$. (Do not compute $h(0)$ numerically just yet.)

From the equation $5 = 2(2) + b$:

So the function is $h(x)=2x+1$. Then

Therefore, $h(0)=1$, which corresponds to answer choice D.