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The equation given is \( y = x^2 + 6x + 8 \), which is a quadratic equation, representing a parabola. To help you graph it, let's go through the steps to better understand its properties.

### 1. Rewrite the equation in vertex form:
You can either graph it directly or by completing the square, but I'll simplify it without changing form.

### 2. Identify key features:
- **Y-intercept:** Set \( x = 0 \) to find \( y = 0^2 + 6(0) + 8 = 8 \). So, the y-intercept is \( (0, 8) \).
- **X-intercepts (Roots):** Solve \( x^2 + 6x + 8 = 0 \). Factoring:
  \[
  (x + 2)(x + 4) = 0 \quad \Rightarrow \quad x = -2 \text{ or } x = -4
  \]
  The x-intercepts are \( (-2, 0) \) and \( (-4, 0) \).

### 3. Find the vertex:
The vertex can be found using the formula \( x = -\frac{b}{2a} \), where \( a = 1 \) and \( b = 6 \):
\[
x = -\frac{6}{2(1)} = -3
\]
Plug \( x = -3 \) into the original equation:
\[
y = (-3)^2 + 6(-3) + 8 = 9 - 18 + 8 = -1
\]
So, the vertex is \( (-3, -1) \).

### 4. Graphing steps:
- Plot the vertex at \( (-3, -1) \).
- Plot the y-intercept at \( (0, 8) \).
- Plot the x-intercepts at \( (-2, 0) \) and \( (-4, 0) \).
- The parabola opens upward because the coefficient of \( x^2 \) is positive.
  
Once you plot these points, sketch the curve.

Would you like more details or have any questions? Here are some relative questions:

1. How do you complete the square to convert to vertex form?
2. What is the axis of symmetry for the parabola?
3. How can you determine if the parabola opens upwards or downwards?
4. What role does the discriminant play in determining the number of roots?
5. How would you solve this quadratic equation using the quadratic formula?

**Tip:** To check your graph's accuracy, ensure symmetry on either side of the vertex!

Graph the parabola whose equation is y = x^2 + 6x + 8.

Learn how to graph the quadratic equation y = x^2 + 6x + 8 step-by-step. Discover key concepts such as vertex, intercepts, and graphing techniques for parabolas. Suitable for students in Grades 8-10.

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