1. The problem asks to rewrite the quadratic function $y = 4x^2 - 8x + 1$ in vertex form $y = a(x - h)^2 + k$ and then find the product $a \times k$. 2. Recall the vertex form of a quadratic is $y = a(x - h)^2 + k$, where $(h,k)$ is the vertex and $a$ is the same leading coefficient as in the standard form. 3. To convert to vertex form, complete the square: $$y = 4x^2 - 8x + 1 = 4(x^2 - 2x) + 1$$ 4. Complete the square inside the parentheses: Take half of the coefficient of $x$, which is $-2$, half is $-1$, square it to get $1$. Add and subtract $1$ inside the parentheses: $$y = 4(x^2 - 2x + 1 - 1) + 1 = 4((x - 1)^2 - 1) + 1$$ 5. Distribute the 4: $$y = 4(x - 1)^2 - 4 + 1 = 4(x - 1)^2 - 3$$ 6. Now the vertex form is $y = 4(x - 1)^2 - 3$, so $a = 4$ and $k = -3$. 7. The product $a \times k = 4 \times (-3) = -12$. **Final answer:** $-12$