Step by step
The question
Question: here are three forms of one parabola: y = x² − 6x + 5, (x − 1)(x − 5), and (x − 3)² − 4.
What does each form show?
1 Standard form: the y axis
Standard form: x squared, then x, then a plain number.
On the y axis, x is 0.
With x = 0, every x part is 0.
Only the plain number is left: 5.
So it crosses the y axis at (0, 5).
2 Factored form: the x axis
Factored form: a product of brackets.
On the x axis, y is 0.
So the product must be 0.
A product is 0 only when one of its parts is 0.
x − 1 is 0 when x = 1.
x − 5 is 0 when x = 5.
Careful!
Not −1.
x − 1 is 0 when x is 1: the sign in the bracket is the opposite.
So it crosses the x axis at x = 1 and x = 5: the points (1, 0) and (5, 0).
3 Vertex form: the lowest point
Vertex form: a bracket squared, plus a number.
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 −4.
Anywhere else we add a bit more than 0, so y is bigger.
So this is the lowest point.
So the lowest point is (3, −4).
4 Show it on the graph
Here is the graph of y = x² − 6x + 5.
It crosses the y axis at (0, 5).
It crosses the x axis at 1 and 5.
The lowest point is (3, −4).
5 Check
Let's check that the three forms are the same.
Open the factored form: it gives x² − 6x + 5.
Open the vertex form: (x − 3) squared is x² − 6x + 9.
Minus 4: x² − 6x + 5 too.
So it is right!
Let's look back.
First: standard form: the y axis.
Then: factored form: the x axis.
Then: vertex form: the lowest point.
Then: show it on the graph.
Last: check.