Step by step
The question
Question: which form of y = 2x² − 12x + 10 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
First we take 2 out of the two x parts.
Then x squared has no number in front.
The 10 stays outside.
So y is 2 times (x² − 6x), plus 10.
We want a bracket squared, like (x minus 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 2 multiplies both parts in the big bracket: 2 times minus 9 is −18.
Then −18 plus 10 is −8.
So y = 2(x − 3)² − 8.
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 −8.
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, −8).
4 Show it on the graph
Here is the graph of y = 2x² − 12x + 10.
Each square up is 2.
The lowest point is (3, −8).
5 Check
Let's check in the question's own form, y = 2x² − 12x + 10.
x = 3 gives y = −8, 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.