Step by step
The question
Question: which form of y = (x − 1)(x − 7) 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 Open the brackets
Open the brackets: each part of the first times each part of the second.
x times x is x squared.
x times −7 is −7x.
−1 times x is −x.
−1 times −7 is 7.
The two x parts are like terms.
Together they make −8x.
So y = x² − 8x + 7.
That is the standard form.
3 Write it in vertex form
We want a bracket squared, like (x minus something) squared.
Its number is half of the middle number, −8: −4.
The next line shows why.
Open (x − 4) squared: x squared, −4x two times, and −4 times −4, that is 16.
It starts with x² − 8x, like ours.
But it has 16 more.
So x² − 8x is (x − 4) squared, take away the extra 16.
We put that in place of x² − 8x.
The 7 stays.
The plain numbers: minus 16 plus 7 is −9.
So y = (x − 4)² − 9.
That is the vertex form.
4 Read the lowest point
A square is never below 0: a number times itself is never minus.
(x − 4) squared is smallest, 0, when x − 4 is 0: at x = 4.
There the square part is 0.
So y is just −9.
Anywhere else we add a bit more than 0, so y is bigger.
So this is the lowest point.
Careful!
Not −4.
x − 4 is 0 when x is 4: the sign in the bracket is the opposite.
So the lowest point is (4, −9).
5 Show it on the graph
Here is the graph of y = x² − 8x + 7.
Each square up is 2.
The lowest point is (4, −9).
6 Check
Let's check in the question's own form, y = (x − 1)(x − 7).
x = 4 gives y = −9, the y of the lowest point.
So it is right!
Let's look back.
First: pick the form.
Then: open the brackets.
Then: write it in vertex form.
Then: read the lowest point.
Then: show it on the graph.
Last: check.