find the vertex of $$ p\left(x\right)=\text{ }3x^2+9x-2 $$

To find the vertex of the quadratic function $ p(x) = 3x^2 + 9x - 2 $, we can use the vertex formula:

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

For the quadratic equation in the form $ ax^2 + bx + c $, we have:
- $ a = 3 $
- $ b = 9 $

Now, substitute $ a $ and $ b $ into the formula:

\[
x_{\text{vertex}} = \frac{-9}{2(3)} = \frac{-9}{6} = -\frac{3}{2}
\]

To find the $ y $-coordinate of the vertex, substitute $ x = -\frac{3}{2} $ back into the equation $ p(x) $:

\[
p\left(-\frac{3}{2}\right) = 3\left(-\frac{3}{2}\right)^2 + 9\left(-\frac{3}{2}\right) - 2
\]
\[
= 3\left(\frac{9}{4}\right) + 9\left(-\frac{3}{2}\right) - 2
\]
\[
= \frac{27}{4} - \frac{27}{2} - 2
\]
\[
= \frac{27}{4} - \frac{54}{4} - \frac{8}{4}
\]
\[
= \frac{27 - 54 - 8}{4} = \frac{-35}{4}
\]

Thus, the vertex is:

\[
\left( -\frac{3}{2}, -\frac{35}{4} \right)
\]

Would you like any more details or explanations?

Here are 5 questions to further explore:
1. How do you determine the axis of symmetry from a quadratic equation?
2. What is the significance of the vertex in the graph of a quadratic function?
3. Can you solve for the roots of this quadratic equation using the quadratic formula?
4. What are the maximum or minimum values in terms of the vertex for different quadratic functions?
5. How does changing the coefficient $ a $ affect the shape and orientation of the parabola?

**Tip:** For any quadratic equation, the vertex provides crucial information about the maximum or minimum value of the function.

find the vertex of $$ p(x) = 3x^2 + 9x - 2 $$

This problem demonstrates how to find the vertex of the quadratic function p(x) = 3x^2 + 9x - 2 using the vertex formula. The solution includes a detailed step-by-step explanation for calculating both the x and y coordinates of the vertex. Suitable for students in Grades 9-11.

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