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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 \( a = 4 \).

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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 \( a \).

Express the function y = 4x^2 - 8x + 3 in vertex form.

Learn how to express the quadratic function y = 4x^2 - 8x + 3 in vertex form by completing the square. The solution explains each step in detail and simplifies the function to vertex form, where the vertex is (1, -1) and the value of 'a' is 4. Ideal for students in Grades 9-12.

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