Let's determine which function is equivalent to the given expression [tex]\( y = x^2 - 10x + 9 \)[/tex].
Below are the options we need to verify:
1. [tex]\( y = (x + 5)^2 + 16 \)[/tex]
2. [tex]\( y = (x - 5)^2 + 16 \)[/tex]
3. [tex]\( y = (x + 5)^2 - 16 \)[/tex]
4. [tex]\( y = (x - 5)^2 - 16 \)[/tex]
To verify equivalence, we will simplify each option and compare it to the given expression.
1. Option 1: [tex]\((x + 5)^2 + 16\)[/tex]
[tex]\[
(x + 5)^2 + 16 = (x^2 + 10x + 25) + 16 = x^2 + 10x + 41
\][/tex]
Clearly, [tex]\(x^2 + 10x + 41\)[/tex] is not equivalent to [tex]\(x^2 - 10x + 9\)[/tex].
2. Option 2: [tex]\((x - 5)^2 + 16\)[/tex]
[tex]\[
(x - 5)^2 + 16 = (x^2 - 10x + 25) + 16 = x^2 - 10x + 41
\][/tex]
Clearly, [tex]\(x^2 - 10x + 41\)[/tex] is not equivalent to [tex]\(x^2 - 10x + 9\)[/tex].
3. Option 3: [tex]\((x + 5)^2 - 16\)[/tex]
[tex]\[
(x + 5)^2 - 16 = (x^2 + 10x + 25) - 16 = x^2 + 10x + 9
\][/tex]
Clearly, [tex]\(x^2 + 10x + 9\)[/tex] is not equivalent to [tex]\(x^2 - 10x + 9\)[/tex].
4. Option 4: [tex]\((x - 5)^2 - 16\)[/tex]
[tex]\[
(x - 5)^2 - 16 = (x^2 - 10x + 25) - 16 = x^2 - 10x + 9
\][/tex]
This simplifies to [tex]\(x^2 - 10x + 9\)[/tex], which is exactly the given expression.
Thus, the expression that is equivalent to [tex]\( y = x^2 - 10x + 9 \)[/tex] is:
[tex]\[
y = (x - 5)^2 - 16
\][/tex]
Therefore, the correct option is:
[tex]\[
\boxed{4}
\][/tex]
