Complete the square to re-write the quadratic function in vertex form:

y, equals, x, squared, plus, 9, x, plus, 7

*y*=*x*2+9*x*+7

To rewrite the quadratic function \( y = x^2 + 9x + 7 \) in **vertex form** by completing the square, follow these steps:

1. **Start with the given quadratic function**:
   \[
   y = x^2 + 9x + 7
   \]

2. **Isolate the constant term**:
   We focus on the quadratic and linear terms first:
   \[
   y = (x^2 + 9x) + 7
   \]

3. **Complete the square**:
   To complete the square, take half of the coefficient of \( x \) (which is 9), square it, and add and subtract this value inside the parentheses.
   Half of 9 is \( \frac{9}{2} = 4.5 \), and squaring it gives \( \left( \frac{9}{2} \right)^2 = 20.25 \).

   Now, add and subtract 20.25 inside the parentheses:
   \[
   y = (x^2 + 9x + 20.25 - 20.25) + 7
   \]
   Simplify the expression inside the parentheses:
   \[
   y = ((x + 4.5)^2 - 20.25) + 7
   \]

4. **Simplify the equation**:
   Combine the constants:
   \[
   y = (x + 4.5)^2 - 20.25 + 7
   \]
   \[
   y = (x + 4.5)^2 - 13.25
   \]

Thus, the quadratic function in **vertex form** is:
\[
y = (x + 4.5)^2 - 13.25
\]

Here, the vertex of the parabola is at \( (-4.5, -13.25) \).

---

Would you like more details on this process or have any questions?

Here are 5 related questions:
1. What is the vertex of the quadratic function \( y = x^2 + 9x + 7 \)?
2. How does completing the square help in identifying the vertex of a parabola?
3. How do you find the axis of symmetry from the vertex form?
4. What is the difference between standard form and vertex form of a quadratic function?
5. How does the vertex form make it easier to graph a quadratic function?

**Tip:** In vertex form \( y = a(x - h)^2 + k \), the vertex is always \( (h, k) \), and the sign of \( a \) determines if the parabola opens up or down.

Complete the square to re-write the quadratic function in vertex form: y = x^2 + 9x + 7

Learn how to rewrite the quadratic equation y = x^2 + 9x + 7 in vertex form by completing the square. This detailed solution explains the steps to find the vertex, simplify the equation, and understand the properties of the parabola. Ideal for students in Grades 9-11.

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