Question

What is the vertex of `y= 2x^2 +5x + 30`?

Answer 1

The vertex of `y` is the point `(-1.25, 26.875)`

Explanation:

For a parabola in standard form: `y=ax^2+bx+c`

the vertex is the point where `x=(-b)/(2a)`

NB: This point will be a maximum or minimum of `y` depending on the sign of `a`

In our example: `y=2x^2+5x+30 -> a=2, b=5, c=30`

`:. x_"vertex" = (-5)/(2xx2)`

`= -5/4 = -1.25`

Replacing for `x` in `y`

`y_"vertex" = 2xx(-5/4)^2+5xx(-5/4)+30`

`= 2xx25/16 - 25/4 +30`

`= 50/16 -100/16+30 = -50/16+30`

`=26.875`

The vertex of `y` is the point `(-1.25, 26.875)`

We can see this point as the minimum of `y` on the graph below.
graph{2x^2+5x+30 [-43.26, 73.74, -9.2, 49.34]}

Answer 2

To find the vertex, the easiest thing to do (besides graphing the problem) is to convert the equation into vertex form. To do that, we should "complete the square"

`y=2x^2+5x+30`

the leading coefficient must be `1`, so factor out the `2`

`y=2(x^2+5/2x+6)`

We need to find a value that changes `x^2+5/2x+6` into a perfect square.

To do that, we need to take the middle term, `5/2`, and divide it by `2`. That gives us `5/4`.

Our next step is to square the result: `(5/4)^2`, or `25/16`

`- - - - - - - - - - - - - -`

Now we have our missing value: `x^2+5/2x+6 + 25/16` WAIT We can't just add something to a problem! But, if we add something and then immediately subtract it, technically we haven't changed the equation, since they subtract out to zero

So, our problem really is `x^2+5/2x+6+25/16 -25/16`

Let's rewrite this: `x^2+5/2x+25/16+6-25/16`

`x^2+5/2x+25/16` is a perfect square. Let's rewrite it in that form: `(x+5/4)^2`

Now let's look at our equation again: `(x+5/4)^2+6-25/16`

Let's combine like-terms: `(x+5/4)^2+71/16`

Now we have the equation in vertex form, and we can find the vertex very easily from here

`(x+color(red)(5/4))^2+color(yellow)(71/16)`

`(-color(red)(x), color(yellow)(y))`

`(-color(red)(5/4), color(yellow)(71/16))`

That's the vertex.

To check our work, let's graph our equation and see the vertex
graph{y=2x^2+5x+30}

We were right! `-1.25` and `4.4375` are equivalent to `-25/16` and `71/16`