Determine the equation of the following function :

The given graph represents a hyperbola, which is a reciprocal function. The general form of such a function is $y = \frac{a}{x-p} + q$, where $x=p$ is the vertical asymptote and $y=q$ is the horizontal asymptote.

From the graph:

• The horizontal dashed line indicates the horizontal asymptote at $y=4$. So, $q=4$.
• The graph approaches the y-axis but never touches it, indicating the vertical asymptote is the y-axis, $x=0$. So, $p=0$.

Using $p=0$ and $q=4$, the equation becomes:

$$y = \frac{a}{x-0} + 4$$ $$y = \frac{a}{x} + 4$$

The graph passes through the point F, which has coordinates $(2, 10)$. Substitute $x=2$ and $y=10$ into the equation:

$$10 = \frac{a}{2} + 4$$

Subtract 4 from both sides:

$$10 - 4 = \frac{a}{2}$$ $$6 = \frac{a}{2}$$

Multiply both sides by 2:

$$a = 6 \times 2$$ $$a = 12$$

Substitute $a=12$ back into the equation from Step 2:

$$y = \frac{12}{x} + 4$$