Step by step
The question
Question: here are three forms of one parabola: y = x² + 2x − 3, (x + 3)(x − 1), and (x + 1)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: −3.
So it crosses the y axis at (0, −3).
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 + 3 is 0 when x = −3.
x − 1 is 0 when x = 1.
Careful!
Not 3.
x + 3 is 0 when x is −3: the sign in the bracket is the opposite.
So it crosses the x axis at x = −3 and x = 1: the points (−3, 0) and (1, 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 + 1) squared is smallest, 0, when x + 1 is 0: at x = −1.
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 (−1, −4).
4 Show it on the graph
Here is the graph of y = x² + 2x − 3.
It crosses the y axis at (0, −3).
It crosses the x axis at −3 and 1.
The lowest point is (−1, −4).
5 Check
Let's check that the three forms are the same.
Open the factored form: it gives x² + 2x − 3.
Open the vertex form: (x + 1) squared is x² + 2x + 1.
Minus 4: x² + 2x − 3 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.