To find the vertex of the quadratic function [tex]\( g(x) = 8x^2 - 48x + 65 \)[/tex], we will use the formula for finding the vertex of a parabola given by [tex]\( ax^2 + bx + c \)[/tex].
The general formula for the x-coordinate of the vertex is given by:
[tex]\[ x = \frac{-b}{2a} \][/tex]
In this equation:
- [tex]\( a = 8 \)[/tex]
- [tex]\( b = -48 \)[/tex]
Plugging these values into the formula:
[tex]\[ x = \frac{-(-48)}{2 \cdot 8} \][/tex]
[tex]\[ x = \frac{48}{16} \][/tex]
[tex]\[ x = 3 \][/tex]
So, the x-coordinate of the vertex is [tex]\( 3 \)[/tex].
Next, we need to find the y-coordinate of the vertex by substituting [tex]\( x = 3 \)[/tex] back into the function [tex]\( g(x) \)[/tex]:
[tex]\[ g(3) = 8(3)^2 - 48(3) + 65 \][/tex]
[tex]\[ g(3) = 8 \cdot 9 - 48 \cdot 3 + 65 \][/tex]
[tex]\[ g(3) = 72 - 144 + 65 \][/tex]
[tex]\[ g(3) = -72 + 65 \][/tex]
[tex]\[ g(3) = -7 \][/tex]
So, the y-coordinate of the vertex is [tex]\( -7 \)[/tex].
Therefore, the vertex of the function [tex]\( g(x) = 8x^2 - 48x + 65 \)[/tex] is:
[tex]\[ (3, -7) \][/tex]
From the given options, the correct answer is:
[tex]\[ (3, -7) \][/tex]
