The given quadratic equation is:

\[
4x^2 + 24x + 9 = 0
\]

### Steps to Graph the Equation:

1. **Rewrite the Equation** in standard form:  
   The equation is already in standard form \(ax^2 + bx + c = 0\), where:
   - \(a = 4\)
   - \(b = 24\)
   - \(c = 9\)

2. **Find the Vertex:**  
   The x-coordinate of the vertex of a parabola is given by:

   \[
   x = \frac{-b}{2a} = \frac{-24}{2(4)} = -3
   \]

   Now, substitute \(x = -3\) back into the equation to find the y-coordinate of the vertex:

   \[
   y = 4(-3)^2 + 24(-3) + 9 = 4(9) - 72 + 9 = 36 - 72 + 9 = -27
   \]

   So, the vertex is at \((-3, -27)\).

3. **Determine the Axis of Symmetry:**  
   The axis of symmetry is the vertical line that passes through the vertex. The equation for the axis of symmetry is:

   \[
   x = -3
   \]

4. **Find the Y-Intercept:**  
   The y-intercept occurs when \(x = 0\). Substitute \(x = 0\) into the equation:

   \[
   y = 4(0)^2 + 24(0) + 9 = 9
   \]

   So, the y-intercept is at \((0, 9)\).

5. **Determine the Shape of the Parabola:**  
   Since \(a = 4\) is positive, the parabola opens **upward**.

### Plot Summary:
- Vertex: \((-3, -27)\)
- Y-intercept: \((0, 9)\)
- Axis of symmetry: \(x = -3\)
- Parabola opens upward.

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### Related Questions:
1. How do you find the roots of the quadratic equation?
2. What effect does changing the coefficient \(a\) have on the graph's shape?
3. Can this quadratic equation be factored easily?
4. How is the discriminant of a quadratic equation calculated, and what does it tell you?
5. What is the general form of a quadratic equation?

**Tip:** The vertex form of a quadratic equation is useful for identifying the vertex quickly.

Learn how to graph the quadratic equation 4x^2 + 24x + 9 with a detailed step-by-step solution. This problem covers key algebraic concepts, such as finding the vertex, axis of symmetry, y-intercept, and determining the shape of the parabola. Suitable for students in Grades 9-10.

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