What is the first step in rewriting the expression y = 4x^2 + 2x + 7 in the form y = a(x – h)^2 + k?

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To rewrite the expression y = 4x2 + 2x + 7 in the form y = a(x – h)2 + k, we need to complete the square. This process allows us to express the quadratic in vertex form, which makes it easier to analyze its properties.

The first step in this process is to factor out the coefficient of x2 from the first two terms of the quadratic. In our expression, the coefficient of x2 is 4. We can factor this out from the first two terms:

y = 4(x2 + (2/4)x) + 7

Now, this simplifies to:

y = 4(x2 + 0.5x) + 7

Next, we will complete the square inside the parentheses. To do this, take the coefficient of x (which is 0.5), divide it by 2 (getting 0.25), and then square it (resulting in 0.0625). We will add and subtract this value inside the parentheses:

y = 4(x2 + 0.5x + 0.0625 - 0.0625) + 7

This can be rearranged to group the completed square together:

y = 4((x + 0.25)2 - 0.0625) + 7

Now distribute the 4:

y = 4(x + 0.25)2 - 4(0.0625) + 7

Calculating the constants gives us:

- 0.25 + 7 = 6.75

So our expression becomes:

y = 4(x + 0.25)2 + 6.75

In conclusion, the first step to rewriting y = 4x2 + 2x + 7 in vertex form y = a(x – h)2 + k involves factoring out the leading coefficient and completing the square. By following these steps, we have successfully transformed the equation and identified the vertex of the parabola represented by the equation.

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