Finding a Quadratic Function from a Vertex

[cmkluza]

Hello, I'm having some trouble on this question, and I'd imagine it's just because I'm looking at it incorrectly.

The problem statement is:
The quadratic function f(x) = p + qx - x² has a maximum value of 5 when x = 3 (i.e. vertex at (3, 5), right?)

Find the value of p and the value of q.

I'll be honest, I'm completely lost. I'd imagine that since the vertex is given I'd have to try to convert the function to vertex form, but I can't see a way to do that. If I could find another point, then I could easily solve for the variables, but I don't see a way to do that. Can anyone give me a tip to get started with solving this?

Any assistance with this will be greatly appreciated. Thanks!

 

[Jameson]

Hi cmkluza,

Welcome to MHB! :)

You are correct that for a downward facing parabola, the vertex would be the maximum height. I think the following formula will be very useful. If we have $f(x)=ax^2+bx+c$, the x-coordinate of the vertex, is \(\displaystyle x=\frac{-b}{2a}\). Can you use this to find one of the variables? If you can do that then you can find the other too.

 

[cmkluza]

Hi cmkluza,

Welcome to MHB! :)

You are correct that for a downward facing parabola, the vertex would be the maximum height. I think the following formula will be very useful. If we have $f(x)=ax^2+bx+c$, the x-coordinate of the vertex, is \(\displaystyle x=\frac{-b}{2a}\). Can you use this to find one of the variables? If you can do that then you can find the other too.

Hello, Jameson!
Thanks for the welcome, and thanks for the help! It seems like I always miss the little things. I completely forgot about that equation for x under these circumstances. Thank you very much for your help!