Step by step
The question
Question: here are three forms of one parabola: y = −x² + 4x + 5, −(x + 1)(x − 5), and −(x − 2)² + 9.
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.
The −1 in front is never 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 highest point
Vertex form: a bracket squared, plus a number.
A square is never below 0: a number times itself is never minus.
(x − 2) squared is smallest, 0, when x − 2 is 0: at x = 2.
There the square part is 0.
So y is just 9.
The minus means we take the square away from 9.
Anywhere else we take away more than 0, so y is smaller.
So this is the highest point.
So the highest point is (2, 9).
4 Show it on the graph
Here is the graph of y = −x² + 4x + 5.
It crosses the y axis at (0, 5).
It crosses the x axis at −1 and 5.
The highest point is (2, 9).
5 Check
Let's check that the three forms are the same.
Open the factored form: it gives −x² + 4x + 5.
Open the vertex form: (x − 2) squared is x² − 4x + 4.
Plus 9, after the −1 multiplies each part: −x² + 4x + 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 highest point.
Then: show it on the graph.
Last: check.