Step by step
The question
Question: which form of y = x² + 6x + 8 shows the lowest point?
1 Pick the form
A parabola can be written in three forms.
Each one shows one thing at a glance.
The vertex form has a square, and a square is never below 0.
So it shows the lowest point.
2 Write it in vertex form
We want a bracket squared, like (x plus something) squared.
Its number is half of the middle number, 6: 3.
The next line shows why.
Open (x + 3) squared: x squared, 3x two times, and 3 times 3, that is 9.
It starts with x² + 6x, like ours.
But it has 9 more.
So x² + 6x is (x + 3) squared, take away the extra 9.
We put that in place of x² + 6x.
The 8 stays.
The plain numbers: minus 9 plus 8 is −1.
So y = (x + 3)² − 1.
That is the vertex form.
3 Read the lowest point
A square is never below 0: a number times itself is never minus.
(x + 3) squared is smallest, 0, when x + 3 is 0: at x = −3.
There the square part is 0.
So y is just −1.
Anywhere else we add a bit more than 0, so y is bigger.
So this is the lowest point.
Careful!
Not 3.
x + 3 is 0 when x is −3: the sign in the bracket is the opposite.
So the lowest point is (−3, −1).
4 Show it on the graph
Here is the graph of y = x² + 6x + 8.
The lowest point is (−3, −1).
5 Check
Let's check in the question's own form, y = x² + 6x + 8.
x = −3 gives y = −1, the y of the lowest point.
So it is right!
Let's look back.
First: pick the form.
Then: write it in vertex form.
Then: read the lowest point.
Then: show it on the graph.
Last: check.