Find The Coordinates Of The Vertex For The Parabola Defined By The Given Quadratic Function: F ( X ) = − X 2 − 4 X − 7 F(x) = -x^2 - 4x - 7 F ( X ) = − X 2 − 4 X − 7 A. { (-2, -3)$}$ B. { (2, -11)$}$ C. { (-4, -7)$}$ D. { (2, -19)$}$

Introduction

In mathematics, a parabola is a type of quadratic function that can be represented in the form of $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The vertex of a parabola is the highest or lowest point on the curve, and it is represented by the coordinates $(h, k)$. In this article, we will focus on finding the coordinates of the vertex for a parabola defined by the given quadratic function $f(x) = -x^2 - 4x - 7$.

Understanding the Quadratic Function

The given quadratic function is $f(x) = -x^2 - 4x - 7$. To find the coordinates of the vertex, we need to use the formula for the x-coordinate of the vertex, which is given by $h = -\frac{b}{2a}$. In this case, $a = -1$ and $b = -4$. Plugging these values into the formula, we get:

$h = -\frac{-4}{2(-1)}$ $h = -\frac{-4}{-2}$ $h = -2$

Finding the y-Coordinate of the Vertex

Now that we have found the x-coordinate of the vertex, we can find the y-coordinate by plugging the value of $h$ into the original quadratic function:

$f(-2) = -(-2)^2 - 4(-2) - 7$ $f(-2) = -4 + 8 - 7$ $f(-2) = -3$

Conclusion

Therefore, the coordinates of the vertex for the parabola defined by the given quadratic function $f(x) = -x^2 - 4x - 7$ are $(-2, -3)$.

Why is it Important to Find the Coordinates of the Vertex?

Finding the coordinates of the vertex is an important concept in mathematics, particularly in algebra and calculus. The vertex of a parabola can be used to determine the maximum or minimum value of a quadratic function, which is essential in various real-world applications such as physics, engineering, and economics.

Real-World Applications of Finding the Coordinates of the Vertex

In physics, the vertex of a parabola can be used to determine the maximum height of a projectile, such as a thrown ball or a rocket. In engineering, the vertex of a parabola can be used to design the shape of a bridge or a building. In economics, the vertex of a parabola can be used to determine the maximum profit or minimum cost of a business.

How to Find the Coordinates of the Vertex for Other Quadratic Functions

To find the coordinates of the vertex for other quadratic functions, you can use the formula $h = -\frac{b}{2a}$ to find the x-coordinate of the vertex, and then plug the value of $h$ into the original quadratic function to find the y-coordinate of the vertex.

Common Mistakes to Avoid When Finding the Coordinates of the Vertex

When finding the coordinates of the vertex, it is essential to avoid common mistakes such as:

• Not using the correct formula: Make sure to use the correct formula for finding the x-coordinate of the vertex, which is $h = -\frac{b}{2a}$.
• Not plugging in the correct values: Make sure to plug in the correct values of $a$ and $b$ into the formula.
• Not simplifying the expression: Make sure to simplify the expression for the y-coordinate of the vertex.

Conclusion

In conclusion, finding the coordinates of the vertex for a parabola defined by a quadratic function is an essential concept in mathematics. By using the formula $h = -\frac{b}{2a}$ and plugging in the correct values, you can find the coordinates of the vertex for any quadratic function. Remember to avoid common mistakes and to simplify the expression for the y-coordinate of the vertex.

Final Answer

The final answer is: $\boxed{A. (-2, -3)}$

Q: What is the formula for finding the x-coordinate of the vertex of a parabola?

A: The formula for finding the x-coordinate of the vertex of a parabola is $h = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function.

Q: How do I find the y-coordinate of the vertex of a parabola?

A: To find the y-coordinate of the vertex of a parabola, plug the value of $h$ into the original quadratic function. This will give you the y-coordinate of the vertex.

Q: What is the significance of the vertex of a parabola?

A: The vertex of a parabola represents the highest or lowest point on the curve. It is essential in various real-world applications such as physics, engineering, and economics.

Q: How do I determine if a parabola opens upward or downward?

A: To determine if a parabola opens upward or downward, look at the coefficient of the $x^2$ term. If the coefficient is positive, the parabola opens upward. If the coefficient is negative, the parabola opens downward.

Q: Can I find the coordinates of the vertex for a quadratic function with a negative leading coefficient?

A: Yes, you can find the coordinates of the vertex for a quadratic function with a negative leading coefficient. Simply use the formula $h = -\frac{b}{2a}$ to find the x-coordinate of the vertex, and then plug the value of $h$ into the original quadratic function to find the y-coordinate of the vertex.

Q: What is the difference between the x-coordinate and the y-coordinate of the vertex?

A: The x-coordinate of the vertex represents the horizontal position of the vertex, while the y-coordinate of the vertex represents the vertical position of the vertex.

Q: Can I use the coordinates of the vertex to graph a parabola?

A: Yes, you can use the coordinates of the vertex to graph a parabola. The vertex is the highest or lowest point on the curve, and it can be used as a reference point to graph the parabola.

Q: How do I find the coordinates of the vertex for a parabola with a horizontal axis of symmetry?

A: To find the coordinates of the vertex for a parabola with a horizontal axis of symmetry, use the formula $h = -\frac{b}{2a}$ to find the x-coordinate of the vertex. The y-coordinate of the vertex can be found by plugging the value of $h$ into the original quadratic function.

Q: Can I use the coordinates of the vertex to find the maximum or minimum value of a quadratic function?

A: Yes, you can use the coordinates of the vertex to find the maximum or minimum value of a quadratic function. The y-coordinate of the vertex represents the maximum or minimum value of the function.

Q: How do I determine if a parabola has a maximum or minimum value?

A: To determine if a parabola has a maximum or minimum value, look at the coefficient of the $x^2$ term. If the coefficient is positive, the parabola has a minimum value. If the coefficient is negative, the parabola has a maximum value.

Q: Can I use the coordinates of the vertex to find the equation of a parabola?

A: Yes, you can use the coordinates of the vertex to find the equation of a parabola. The vertex is the highest or lowest point on the curve, and it can be used as a reference point to find the equation of the parabola.

Q: How do I find the equation of a parabola given its vertex?

A: To find the equation of a parabola given its vertex, use the formula $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Q: Can I use the coordinates of the vertex to find the axis of symmetry of a parabola?

A: Yes, you can use the coordinates of the vertex to find the axis of symmetry of a parabola. The axis of symmetry is a vertical line that passes through the vertex of the parabola.

Q: How do I find the axis of symmetry of a parabola given its vertex?

A: To find the axis of symmetry of a parabola given its vertex, use the formula $x = h$, where $(h, k)$ is the vertex of the parabola.

Q: Can I use the coordinates of the vertex to find the x-intercepts of a parabola?

A: Yes, you can use the coordinates of the vertex to find the x-intercepts of a parabola. The x-intercepts are the points where the parabola intersects the x-axis.

Q: How do I find the x-intercepts of a parabola given its vertex?

A: To find the x-intercepts of a parabola given its vertex, use the formula $x = h \pm \sqrt{\frac{-b}{a}}$, where $(h, k)$ is the vertex of the parabola.

Q: Can I use the coordinates of the vertex to find the y-intercept of a parabola?

A: Yes, you can use the coordinates of the vertex to find the y-intercept of a parabola. The y-intercept is the point where the parabola intersects the y-axis.

Q: How do I find the y-intercept of a parabola given its vertex?

A: To find the y-intercept of a parabola given its vertex, use the formula $y = k$, where $(h, k)$ is the vertex of the parabola.

Q: Can I use the coordinates of the vertex to find the equation of the axis of symmetry of a parabola?

A: Yes, you can use the coordinates of the vertex to find the equation of the axis of symmetry of a parabola. The axis of symmetry is a vertical line that passes through the vertex of the parabola.

Q: How do I find the equation of the axis of symmetry of a parabola given its vertex?

A: To find the equation of the axis of symmetry of a parabola given its vertex, use the formula $x = h$, where $(h, k)$ is the vertex of the parabola.

Q: Can I use the coordinates of the vertex to find the equation of the tangent line to a parabola?

A: Yes, you can use the coordinates of the vertex to find the equation of the tangent line to a parabola. The tangent line is a line that touches the parabola at a single point.

Q: How do I find the equation of the tangent line to a parabola given its vertex?

A: To find the equation of the tangent line to a parabola given its vertex, use the formula $y = mx + b$, where $m$ is the slope of the tangent line and $b$ is the y-intercept.

Q: Can I use the coordinates of the vertex to find the equation of the normal line to a parabola?

A: Yes, you can use the coordinates of the vertex to find the equation of the normal line to a parabola. The normal line is a line that is perpendicular to the tangent line at a single point.

Q: How do I find the equation of the normal line to a parabola given its vertex?

A: To find the equation of the normal line to a parabola given its vertex, use the formula $y = mx + b$, where $m$ is the slope of the normal line and $b$ is the y-intercept.

Q: Can I use the coordinates of the vertex to find the equation of the asymptote of a parabola?

A: Yes, you can use the coordinates of the vertex to find the equation of the asymptote of a parabola. The asymptote is a line that the parabola approaches as it goes to infinity.

Q: How do I find the equation of the asymptote of a parabola given its vertex?

A: To find the equation of the asymptote of a parabola given its vertex, use the formula $y = mx + b$, where $m$ is the slope of the asymptote and $b$ is the y-intercept.

Q: Can I use the coordinates of the vertex to find the equation of the parabola in polar coordinates?

A: Yes, you can use the coordinates of the vertex to find the equation of the parabola in polar coordinates. The polar coordinates are a way of representing the position of a point in a two-dimensional space using a distance and an angle.

Q: How do I find the equation of the parabola in polar coordinates given its vertex?

A: To find the equation of the parabola in polar coordinates given its vertex, use the formula $r = a \sin^2 \theta + b \cos^2 \theta + c$, where $a$, $b$, and $c$ are constants.

Q: Can I use the coordinates of the vertex to find the equation of the parabola in parametric form?

A: Yes, you can use the coordinates of the vertex