Step by step
The question
Question: find the range of f(x) = (x − 1)² − 4.
f(x) = (x − 1)2 − 4
The range is all the answers, the y's, that can come out.
range = the y's that can come out
1 What a square can be
A square is never below 0.
3 squared is 9.
Minus 3 squared is 9.
0 squared is 0.
3² = 9, (−3)² = 9, 0² = 0
(x − 1)² is a square too.
So it is 0 or more.
(x − 1)2 ≥ 0
2 Take away the 4
In f(x), − 4 comes after.
So we take away 4.
We do it on both sides, so the sign stays true.
(x − 1)2 − 4 ≥ 0 − 4
0 minus 4 is −4.
The left side is f(x), the y.
So y ≥ −4.
The range is every y from −4 up.
y ≥ −4
3 See it on the graph
On the graph, the turning point is (1, −4).
The curve never goes below y = −4.
Its lowest y is −4.
4 Check
Let's check.
Put in x = 1, the turning point.
We get −4, the smallest y.
x = 1: y = (1 − 1)2 − 4 = (0)2 − 4 = 0 − 4
= −4
Now x = 3.
We get 0.
0 is more than −4.
It fits.
x = 3: y = (3 − 1)2 − 4 = (2)2 − 4 = 4 − 4
= 0
Let's look back.
First: what a square can be.
Then: take away the 4.
Then: see it on the graph.
Last: check.
Answer
y ≥ −4