Step by step
The question
Question: which form of y = −x² + 2x + 3 shows the highest 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 highest point.
2 Write it in vertex form
First we take −1 out of the two x parts.
Then x squared has no number in front.
The 3 stays outside.
So y is minus (x² − 2x), plus 3.
We want a bracket squared, like (x minus something) squared.
Its number is half of the middle number, −2: −1.
The next line shows why.
Open (x − 1) squared: x squared, −1x two times, and −1 times −1, that is 1.
It starts with x² − 2x, like ours.
But it has 1 more.
So x² − 2x is (x − 1) squared, take away the extra 1.
We put that in place of x² − 2x.
The −1 multiplies both parts in the big bracket: −1 times minus 1 is 1.
Then 1 plus 3 is 4.
So y = −(x − 1)² + 4.
That is the vertex form.
3 Read the highest point
A square is never below 0: a number times itself is never minus.
(x − 1) squared is smallest, 0, when x − 1 is 0: at x = 1.
There the square part is 0.
So y is just 4.
The minus means we take the square away from 4.
Anywhere else we take away more than 0, so y is smaller.
So this is the highest point.
Careful!
Not −1.
x − 1 is 0 when x is 1: the sign in the bracket is the opposite.
So the highest point is (1, 4).
4 Show it on the graph
Here is the graph of y = −x² + 2x + 3.
The highest point is (1, 4).
5 Check
Let's check in the question's own form, y = −x² + 2x + 3.
x = 1 gives y = 4, the y of the highest point.
So it is right!
Let's look back.
First: pick the form.
Then: write it in vertex form.
Then: read the highest point.
Then: show it on the graph.
Last: check.