Question

What is the vertex of the graph of
y = −2x2 − x + 3?

Answer 1

There is an explicit formula for getting the vertex of a quadratic equation of the form
\[\Large y = \color{red}ax^2 + \color{blue}bx + \color{green}c\]

And that vertex (h,k) is given by:
\[\huge \left(-\frac{\color{blue}b}{2\color{red}a}\quad , \quad \frac{4\color{red}a\color{green}c-\color{blue}b^2}{4\color{red}a}\right)\]

Answer 2

Just plug in and be on your way ^_^

Answer 3

thank u

Answer 4

Would you mind if I ask here what exactly the vertex of a graph represents?

Answer 5

There are several ways to interpret it. I guess a nice way would be to define the vertex as the turning point of the quadratic function, where it changes from going down to going up, or going up to going down...

Answer 6

Ok, so is there any difference from finding \[\frac{df}{dx}=0?\]
Other than that this does not involve taking a derivative.

Answer 7

the top of the pynomial

Answer 8

parabola srry

Answer 9

Well, there isn't. If we differentiate the typical quadratic function:
\[\Large f(x) = \color{red}ax^2 + \color{blue}bx + \color{green}c\]
\[\Large f'(x)= \frac{d}{dx}[\color{red}ax^2 + \color{blue}bx + \color{green}c]\]
\[\Large f'(x) = 2\color{red}ax + \color{blue}b\]
And then equate it to zero, you'll find that
\[\Large 2\color{red}ax + \color{blue}b = 0 \iff x = -\frac{\color{blue}b}{2\color{red}a}\]

In other words, the function (possibly) attains an extremum at the point where
\[\Large x = -\frac{\color{blue}b}{2\color{red} a}\] or the point
\[\Large \left(-\frac{\color{blue}b}{2\color{red}a}\quad , \quad f\left(-\frac{\color{blue}b}{2\color{red}a}\right)\right)\quad =\quad\left(-\frac{\color{blue}b}{2\color{red}a}\quad , \quad \frac{4\color{red}a\color{green}c-\color{blue}b^2}{4\color{red}a}\right)\]

Answer 10

I did not look at it that way yet, thanks :) The derivative then.