Here’s the question:
$x$ $g(x)$ -27 3 -9 0 21 5 The table shows three values of $x$ and their corresponding values of $g(x)$, where $g(x) = \frac{f(x)}{x+3}$ and $f$ is a linear function. What is the y-intercept of the graph $y=f(x)$ in the $xy$-plane?
This looks a lot more complicated than it actually is.
If
$$ g(x)=\frac{f(x)}{x+3} $$
then
$$ f(x) = g(x) \times (x+3) $$
Moreover, we know that $f(x)$ is linear, so if we were to theoretically input the values of the $x$s given, we could work out a gradient! From there, we could form an equation and solve for the y-intercept.
If we let $x=-27$, then $f(-27)=g(-27) \times (-27+3)$. This would simplify to $-72$. We now have one point that lies on $f(x)$. I’ll pick $x=-9$ as my second point, since $f(x)$ will come to $0$.
Our two points are:
$$(-27, -72)$$
$$ (-9, 0) $$
The gradient from these two points comes to $4$. So,
$$f(x)=4x+C$$
Solving for C, we have:
$$C=f(x)-4x$$
We can input any of our coorindates here and get a value for C. I used the first set of coordinates, and this gave me $C=36$.
The y-intercept is thus $(0, 36)$.