To find the equation of the axis of symmetry and the coordinates of the vertex for the function [tex]\( y = 4x^2 - 8x - 3 \)[/tex], follow these steps:
1. Identify the coefficients: In the quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex], the coefficients are:
[tex]\[
a = 4, \quad b = -8, \quad c = -3
\][/tex]
2. Find the axis of symmetry: The formula for the axis of symmetry of a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] is given by:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
x = -\frac{-8}{2 \cdot 4} = \frac{8}{8} = 1
\][/tex]
So, the axis of symmetry is [tex]\( x = 1 \)[/tex].
3. Find the vertex: The vertex [tex]\((h, k)\)[/tex] of the quadratic function lies on the axis of symmetry. To find the y-coordinate [tex]\( k \)[/tex] of the vertex, substitute [tex]\( x = 1 \)[/tex] back into the original equation:
[tex]\[
y = 4(1)^2 - 8(1) - 3
\][/tex]
Simplify the expression:
[tex]\[
y = 4 \cdot 1 - 8 \cdot 1 - 3 = 4 - 8 - 3 = -7
\][/tex]
Therefore, the vertex is at [tex]\( (1, -7) \)[/tex].
To summarize, the equation of the axis of symmetry is [tex]\( x = 1 \)[/tex], and the coordinates of the vertex are [tex]\( (1, -7) \)[/tex].
From the given options, the correct answer is:
[tex]\[
x = 1 \text{; vertex: } (1, -7)
\][/tex]
