Consider The Function P ( X ) = − 2 X 2 − 8 X − 5 P(x) = -2x^2 - 8x - 5 P ( X ) = − 2 X 2 − 8 X − 5 .1. What Is The Vertex Of P P P ?2. What Is The Equation Of The Line Of Symmetry Of P P P ? □ \square □ Graph P ( X P(x P ( X ].

In mathematics, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is $p(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this case, we are given the quadratic function $p(x) = -2x^2 - 8x - 5$. Our goal is to find the vertex of this function and the equation of the line of symmetry.

Finding the Vertex

The vertex of a quadratic function is the maximum or minimum point of the function. To find the vertex, we can use the formula $x = -\frac{b}{2a}$. In this case, $a = -2$ and $b = -8$. Plugging these values into the formula, we get:

$x = -\frac{-8}{2(-2)}$ $x = -\frac{-8}{-4}$ $x = 2$

Now that we have the x-coordinate of the vertex, we can find the y-coordinate by plugging this value back into the original function:

$p(2) = -2(2)^2 - 8(2) - 5$ $p(2) = -2(4) - 16 - 5$ $p(2) = -8 - 16 - 5$ $p(2) = -29$

So, the vertex of the function $p(x) = -2x^2 - 8x - 5$ is at the point $(2, -29)$.

Finding the Equation of the Line of Symmetry

The line of symmetry of a quadratic function is a vertical line that passes through the vertex of the function. The equation of this line is given by $x = h$, where $h$ is the x-coordinate of the vertex. In this case, we found that the x-coordinate of the vertex is $2$. Therefore, the equation of the line of symmetry is $x = 2$.

Graphing the Function

To graph the function $p(x) = -2x^2 - 8x - 5$, we can start by plotting the vertex at the point $(2, -29)$. We can then use the equation of the line of symmetry to draw a vertical line through this point. Next, we can use the fact that the function is a quadratic function to determine the direction of the parabola. Since the coefficient of the $x^2$ term is negative, the parabola opens downward.

To complete the graph, we can plot a few more points on the function. We can start by plugging in some values of $x$ into the function and calculating the corresponding values of $y$. For example, we can plug in $x = 0$ to get:

$p(0) = -2(0)^2 - 8(0) - 5$ $p(0) = -5$

So, the point $(0, -5)$ is on the graph. We can also plug in $x = 1$ to get:

$p(1) = -2(1)^2 - 8(1) - 5$ $p(1) = -2 - 8 - 5$ $p(1) = -15$

So, the point $(1, -15)$ is also on the graph. We can continue this process to plot more points on the graph.

Conclusion

In this article, we have learned how to find the vertex and the equation of the line of symmetry of a quadratic function. We have also learned how to graph a quadratic function. The vertex of a quadratic function is the maximum or minimum point of the function, and the equation of the line of symmetry is a vertical line that passes through the vertex. We have used the formula $x = -\frac{b}{2a}$ to find the x-coordinate of the vertex, and we have used the equation of the line of symmetry to draw a vertical line through the vertex. We have also learned how to graph a quadratic function by plotting the vertex and using the equation of the line of symmetry to draw a vertical line through the vertex.

Graph of the Function

Here is the graph of the function $p(x) = -2x^2 - 8x - 5$:

### Graph of p(x) = -2x^2 - 8x - 5
Vertex: (2, -29)
Line of Symmetry: x = 2
Graph:

-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30

-30 | * -25 | * -20 | * -15 | * -10 | * -5 | * 0 | * 5 | * 10 | * 15 | * 20 | * 25 | * 30 | *

Note: The graph is a parabola that opens downward, with the vertex at the point (2, -29) and the line of symmetry at x = 2.<br/>
**Quadratic Function Q&A**
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In this article, we will answer some common questions about quadratic functions. Whether you are a student, a teacher, or just someone who wants to learn more about quadratic functions, this article is for you.
Q: What is a quadratic function?
A: A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is $p(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.
Q: What is the vertex of a quadratic function?
A: The vertex of a quadratic function is the maximum or minimum point of the function. It is the point where the function changes from increasing to decreasing or from decreasing to increasing.
Q: How do I find the vertex of a quadratic function?
A: To find the vertex of a quadratic function, you can use the formula $x = -\frac{b}{2a}$. This will give you the x-coordinate of the vertex. To find the y-coordinate of the vertex, you can plug this value back into the original function.
Q: What is the line of symmetry of a quadratic function?
A: The line of symmetry of a quadratic function is a vertical line that passes through the vertex of the function. The equation of this line is given by $x = h$, where $h$ is the x-coordinate of the vertex.
Q: How do I graph a quadratic function?
A: To graph a quadratic function, you can start by plotting the vertex at the point $(h, k)$. You can then use the equation of the line of symmetry to draw a vertical line through this point. Next, you can use the fact that the function is a quadratic function to determine the direction of the parabola. Since the coefficient of the $x^2$ term is negative, the parabola opens downward.
Q: What is the difference between a quadratic function and a linear function?
A: A quadratic function is a polynomial function of degree two, while a linear function is a polynomial function of degree one. In other words, a quadratic function has a highest power of two, while a linear function has a highest power of one.
Q: Can a quadratic function have more than one vertex?
A: No, a quadratic function can only have one vertex. The vertex is the point where the function changes from increasing to decreasing or from decreasing to increasing.
Q: Can a quadratic function be a perfect square?
A: Yes, a quadratic function can be a perfect square. For example, the function $p(x) = x^2$ is a perfect square.
Q: Can a quadratic function be a perfect cube?
A: No, a quadratic function cannot be a perfect cube. A perfect cube is a polynomial function of degree three, while a quadratic function is a polynomial function of degree two.
Q: Can a quadratic function be a rational function?
A: Yes, a quadratic function can be a rational function. For example, the function $p(x) = \frac{x^2}{x+1}$ is a rational function.
Conclusion
In this article, we have answered some common questions about quadratic functions. We have learned about the vertex, the line of symmetry, and how to graph a quadratic function. We have also learned about the differences between quadratic functions and linear functions, and about the properties of quadratic functions.
Additional Resources
If you want to learn more about quadratic functions, here are some additional resources:

Khan Academy: Quadratic Functions
Mathway: Quadratic Functions
Wolfram Alpha: Quadratic Functions

Note: These resources are online and can be accessed for free.