To find the range of the function [tex]\( y = (x-3)^2 - 16 \)[/tex], we need to carefully analyze the expression and determine how the output values of [tex]\( y \)[/tex] behave as [tex]\( x \)[/tex] varies over all real numbers.
1. Understand the Form of the Function:
The given function is [tex]\( y = (x-3)^2 - 16 \)[/tex].
2. Analyze the Squared Term:
Notice that [tex]\( (x-3)^2 \)[/tex]:
- This is a squared term, which means it is always non-negative.
- That is, [tex]\( (x-3)^2 \geq 0 \)[/tex] for all [tex]\( x \)[/tex].
3. Determine When the Squared Term is at its Minimum:
The minimum value of [tex]\( (x-3)^2 \)[/tex] occurs when its argument is zero. Specifically:
- [tex]\( (x-3)^2 = 0 \)[/tex] when [tex]\( x = 3 \)[/tex].
- Substituting [tex]\( x = 3 \)[/tex] into the function, we get [tex]\( y = 0 - 16 = -16 \)[/tex].
4. Behavior of [tex]\( y \)[/tex] as [tex]\( (x-3)^2 \)[/tex] Increases:
- As [tex]\( (x-3)^2 \)[/tex] increases from its minimum value of 0, the expression [tex]\( y = (x-3)^2 - 16 \)[/tex] also increases.
- Since [tex]\( (x-3)^2 \)[/tex] can become arbitrarily large as [tex]\( x \)[/tex] moves away from 3, [tex]\( y \)[/tex] can also become correspondingly large.
5. Deducing the Range:
- The minimum value of [tex]\( y \)[/tex] is [tex]\(-16\)[/tex], which occurs at [tex]\( x = 3 \)[/tex].
- For any other value of [tex]\( x \)[/tex], [tex]\( y \)[/tex] will be greater than [tex]\(-16\)[/tex].
Putting it all together, the range of the function [tex]\( y = (x-3)^2 - 16 \)[/tex] is all real numbers that are greater than or equal to [tex]\(-16\)[/tex].
Thus, the range of the function [tex]\( y = (x-3)^2 - 16 \)[/tex] is:
[tex]\[ y \geq -16 \][/tex]
