To express the quadratic function $y = 4x^2 - 8x + 3$ in vertex form, we need to complete the square.

Steps to Convert to Vertex Form:

1. Start with the given equation: $y = 4x^2 - 8x + 3$

2. Factor out the coefficient of $x^2$ (which is 4) from the first two terms: $y = 4(x^2 - 2x) + 3$

3. To complete the square, take half of the coefficient of $x$ (which is -2), square it, and add it inside the parentheses: $\left(\frac{-2}{2}\right)^2 = 1$

4. Add and subtract 1 inside the parentheses: $y = 4(x^2 - 2x + 1 - 1) + 3$ $y = 4((x - 1)^2 - 1) + 3$

5. Distribute the 4: $y = 4(x - 1)^2 - 4 + 3$

6. Simplify the constant terms: $y = 4(x - 1)^2 - 1$

So, the function in vertex form is: $y = 4(x - 1)^2 - 1$

This represents the function in vertex form, where the vertex is $(h, k) = (1, -1)$ and $a = 4$.

Would you like further details or explanations on any of these steps?

Here are 5 related questions to expand on this topic:

1. How does completing the square help in converting a quadratic function to vertex form?
2. What is the significance of the vertex in the vertex form of a quadratic function?
3. How does the value of $a$ in the vertex form affect the graph of the function?
4. How can you determine the axis of symmetry from the vertex form of a quadratic function?
5. How can you derive the standard form of a quadratic function from its vertex form?

Tip: The vertex form of a quadratic function provides an easy way to identify the vertex, which is the highest or lowest point on the graph, depending on the sign of $a$.