1. \( y=x^{2}+6 x+9 \)

Certainly! Let's analyze and work with the quadratic function you've provided: \[ y = x^{2} + 6x + 9 \] ### 1. **Factoring the Quadratic** First, let's see if we can factor the quadratic expression. We look for two numbers that multiply to **9** (the constant term) and add up to **6** (the coefficient of the middle term, \( x \)). - **Numbers**: 3 and 3 - **Verification**: - \( 3 \times 3 = 9 \) - \( 3 + 3 = 6 \) Thus, the quadratic can be factored as: \[ y = (x + 3)(x + 3) \quad \text{or} \quad y = (x + 3)^2 \] ### 2. **Vertex Form** The factored form \( y = (x + 3)^2 \) is also known as the vertex form of a quadratic equation, which is generally expressed as: \[ y = a(x - h)^2 + k \] where \((h, k)\) is the vertex of the parabola. Comparing the two forms: \[ y = (x + 3)^2 = 1(x - (-3))^2 + 0 \] - **Vertex**: \((-3, 0)\) ### 3. **Graphing the Quadratic** Given the vertex and the fact that the coefficient of \( x^2 \) is positive, we know that the parabola opens **upwards**. - **Vertex**: \((-3, 0)\) - **Axis of Symmetry**: \( x = -3 \) - **Y-intercept**: When \( x = 0 \), \[ y = 0^2 + 6(0) + 9 = 9 \] So, the y-intercept is at \((0, 9)\). - **X-intercept(s)**: Setting \( y = 0 \), \[ 0 = (x + 3)^2 \implies x = -3 \] This is a **double root**, meaning the parabola touches the x-axis at \((-3, 0)\) and does not cross it. ### 4. **Summary of Key Features** - **Standard Form**: \( y = x^{2} + 6x + 9 \) - **Factored Form**: \( y = (x + 3)^2 \) - **Vertex**: \((-3, 0)\) - **Axis of Symmetry**: \( x = -3 \) - **Direction**: Opens upwards - **Y-intercept**: \((0, 9)\) - **X-intercept**: \((-3, 0)\) (double root) ### 5. **Graph of the Function** Here's a sketch of the quadratic function: ``` | 9 | * | 6 | | 3 | * | 0 |----------*---------- | (-3,0) | | +------------------------- ``` - The parabola touches the x-axis at \((-3, 0)\) and reaches its minimum point there. - It opens upwards, meaning it extends infinitely in the positive y-direction. ### 6. **Applications and Further Analysis** Understanding the structure of a quadratic function is fundamental in various applications such as physics (projectile motion), engineering, economics (profit modeling), and more. By identifying key features like the vertex, axis of symmetry, and intercepts, one can gain deep insights into the behavior of the function and its real-world implications. If you have a specific aspect of this quadratic equation you'd like to explore further—such as solving equations, applications, or transformations—feel free to ask!