find the vertex and line of symmetry of y=2x squared plus 16x plus 24

To find the vertex and the line of symmetry for the quadratic function \( y = 2x^2 + 16x + 24 \), we can follow the same process as before:

### 1. Identify the coefficients
The quadratic equation is in the form:
\[
y = ax^2 + bx + c
\]
For this equation:
- \( a = 2 \)
- \( b = 16 \)
- \( c = 24 \)

### 2. Calculate the x-coordinate of the vertex
The x-coordinate of the vertex is given by the formula:
\[
x = -\frac{b}{2a}
\]
Substitute the values of \( a \) and \( b \):
\[
x = -\frac{16}{2(2)} = -\frac{16}{4} = -4
\]

### 3. Calculate the y-coordinate of the vertex
Substitute \( x = -4 \) back into the original equation to find the y-coordinate:
\[
y = 2(-4)^2 + 16(-4) + 24
\]
\[
y = 2(16) - 64 + 24 = 32 - 64 + 24 = -8
\]
So, the vertex is at \( (-4, -8) \).

### 4. Find the equation of the line of symmetry
The line of symmetry is a vertical line that passes through the vertex. The equation of the line of symmetry is:
\[
x = -4
\]

### Final Answer:
- **Vertex**: \( (-4, -8) \)
- **Line of Symmetry**: \( x = -4 \)

Would you like further details or have any questions?

Here are some related questions:
1. How does changing the coefficient \( a \) affect the width of the parabola?
2. What are the roots (x-intercepts) of this quadratic equation?
3. How can you rewrite this quadratic equation in vertex form?
4. What is the significance of the y-intercept in the context of this function?
5. How would you graph this quadratic function using the vertex and the line of symmetry?

**Tip:** The x-coordinate of the vertex can also be interpreted as the value of \( x \) that makes the function's derivative equal to zero, indicating a maximum or minimum point.

Learn how to find the vertex and line of symmetry of the quadratic function y = 2x^2 + 16x + 24 step-by-step. This problem involves calculating the vertex coordinates using the vertex formula and determining the line of symmetry. Suitable for high school students.

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