| \(x\) | \(g(x)\) |
|---|---|
| -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, then 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 \(f(-27) = 3(-24)\). This comes to \(-72\). So now we know that the point \((-27, -72)\) lies on \(f(x)\). We'll repeat the same for any of the other two points. I'll pick \(x=-9\) since this will come to \(f(x)=0\).
So now we have two points on \(f(x)\). Standard fare from here. The gradient between \((-27, -72)\) and \((-9, 0)\) comes to \(4\). So \(f(x)=4x+C\), and hence \(C=f(x)-4\). We could input any of our two coordinates to get a value for \(C\). If we use the first coordinate we calculated, we'd end up with \(C=-72-4(-27)\) which comes to \(36\).
The y-intercept is hence \((0, 36)\).
This article was written on 2024-08-20. If you have any thoughts, feel free to send me an email with them. Have a nice day!