To determine the vertex of the quadratic equation [tex]\( y = x^2 + 8x + 17 \)[/tex], we need to follow these steps:
1. Identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] from the quadratic equation, which is in the standard form [tex]\( y = ax^2 + bx + c \)[/tex]:
[tex]\[
a = 1, \quad b = 8, \quad c = 17
\][/tex]
2. Use the formula for the x-coordinate of the vertex of a parabola, [tex]\( x = -\frac{b}{2a} \)[/tex]:
[tex]\[
x = -\frac{b}{2a} = -\frac{8}{2 \cdot 1} = -\frac{8}{2} = -4
\][/tex]
3. Substitute [tex]\( x = -4 \)[/tex] back into the original equation to find the corresponding y-coordinate:
[tex]\[
y = (-4)^2 + 8(-4) + 17
\][/tex]
Calculate each term step by step:
[tex]\[
y = 16 + (-32) + 17
\][/tex]
[tex]\[
y = 16 - 32 + 17
\][/tex]
[tex]\[
y = -16 + 17
\][/tex]
[tex]\[
y = 1
\][/tex]
4. Thus, the vertex of the quadratic equation [tex]\( y = x^2 + 8x + 17 \)[/tex] is at the point [tex]\( (-4, 1) \)[/tex].
The correct answer is [tex]\((-4, 1)\)[/tex].
