Step by step
The question
Question: which form of y = (x + 2)² − 9 shows where it crosses the x axis?
1 Pick the form
A parabola can be written in three forms.
Each one shows one thing at a glance.
On the x axis, y is 0.
The factored form is a product, and a product is 0 when one bracket is 0.
So the factored form shows where it crosses the x axis.
2 Open the brackets
Squared means times itself.
Open the brackets: x squared, 2x two times, and 2 times 2, that is 4.
Together: x² + 4x + 4.
So y is (x² + 4x + 4), minus 9.
The plain numbers: 4 minus 9 is −5.
So y = x² + 4x − 5.
That is the standard form.
3 Write it in factored form
Factoring goes backwards.
We need two numbers.
They multiply to −5, and they add to 4.
They give the brackets.
They multiply to a minus number, so one number is plus and one is minus.
They add to a plus number, so the bigger one is plus.
We list the pairs that make 5, smallest first.
We take the smaller from the bigger.
We need 4.
1 and 5 give 4.
With the signs: −1 and 5.
The two numbers go in the brackets: y = (x + 5)(x − 1).
That is the factored form.
4 Read the x axis crossings
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 + 5 is 0 when x = −5.
x − 1 is 0 when x = 1.
Careful!
Not 5.
x + 5 is 0 when x is −5: the sign in the bracket is the opposite.
So it crosses the x axis at x = −5 and x = 1: the points (−5, 0) and (1, 0).
5 Show it on the graph
Here is the graph of y = x² + 4x − 5.
It crosses the x axis at −5 and 1.
6 Check
Let's check in the question's own form, y = (x + 2)² − 9.
x = −5 gives y = 0: on the x axis.
And x = 1 gives y = 0: on the x axis.
So it is right!
Let's look back.
First: pick the form.
Then: open the brackets.
Then: write it in factored form.
Then: read the x axis crossings.
Then: show it on the graph.
Last: check.