Question

question
find the coordinates of the vertex of the following parabola algebraically. write your answer as a point.

$y = x^2 + 8x + 19$

Explanation:

Step1: Recall vertex formula for parabola

For a parabola in the form \( y = ax^2+bx + c \), the x - coordinate of the vertex is given by \( x=-\frac{b}{2a} \). In the equation \( y=x^{2}+8x + 19 \), we have \( a = 1 \), \( b=8 \) and \( c = 19 \).

Step2: Calculate x - coordinate of vertex

Substitute \( a = 1 \) and \( b = 8 \) into the formula \( x=-\frac{b}{2a} \). So \( x=-\frac{8}{2\times1}=- 4 \).

Step3: Calculate y - coordinate of vertex

Substitute \( x=-4 \) into the equation \( y=x^{2}+8x + 19 \). Then \( y=(-4)^{2}+8\times(-4)+19 \). First, calculate \( (-4)^{2}=16 \), \( 8\times(-4)=-32 \). Then \( y = 16-32 + 19=3 \).

Answer:

The coordinates of the vertex are \((-4,3)\)