Step by step
The question
Question: find the range of f(x) = −x² + 6.
f(x) = −x2 + 6
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
So x² is 0 or more.
x2 ≥ 0
−x² is minus a square.
So it is 0 or less.
It is never above 0.
−x2 ≤ 0
2 Add the 6
In f(x), + 6 comes after.
So we add 6.
We do it on both sides, so the sign stays true.
−x2 + 6 ≤ 0 + 6
0 plus 6 is 6.
The left side is f(x), the y.
So y ≤ 6.
The range is every y from 6 down.
y ≤ 6
3 See it on the graph
On the graph, the turning point is (0, 6).
The curve never goes above y = 6.
Its highest y is 6.
4 Check
Let's check.
Put in x = 0, the turning point.
We get 6, the biggest y.
x = 0: y = −02 + 6 = 0 + 6 = 6
Now x = 2.
We get 2.
2 is less than 6.
It fits.
x = 2: y = −22 + 6 = −4 + 6 = 2
Let's look back.
First: what a square can be.
Then: add the 6.
Then: see it on the graph.
Last: check.
Answer
y ≤ 6