To find the vertex of the quadratic function \( g(x) = x^2 + 10x + 9 \), we can either complete the square or use the vertex formula. The vertex formula for a quadratic function \( g(x) = ax^2 + bx + c \) is:

\[
x_{\text{vertex}} = \frac{-b}{2a}
\]

Here:
- \( a = 1 \) (coefficient of \( x^2 \)),
- \( b = 10 \) (coefficient of \( x \)),
- \( c = 9 \) (constant).

### Step 1: Find the x-coordinate of the vertex
\[
x_{\text{vertex}} = \frac{-10}{2(1)} = \frac{-10}{2} = -5
\]

### Step 2: Find the y-coordinate of the vertex
Now, substitute \( x = -5 \) into the original function \( g(x) = x^2 + 10x + 9 \):

\[
g(-5) = (-5)^2 + 10(-5) + 9 = 25 - 50 + 9 = -16
\]

### Vertex:
The vertex of the function \( g(x) = x^2 + 10x + 9 \) is at \( (-5, -16) \).

Would you like further details on how this relates to the graph?

Here are 5 related questions to explore this topic:
1. What is the axis of symmetry for this quadratic function?
2. How do you determine if the vertex is a maximum or minimum point?
3. Can you rewrite \( g(x) = x^2 + 10x + 9 \) in vertex form?
4. How does the discriminant of a quadratic equation relate to the graph?
5. What is the domain and range of the function \( g(x) \)?

**Tip:** The vertex of a parabola can always be found using the vertex formula or by completing the square!

Learn how to find the vertex of the quadratic function g(x)=x^2+10x+9 using the vertex formula. This problem covers key algebraic concepts, including how to calculate the vertex of a parabola and understand its graph. Suitable for students in Grades 8-10.

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