What Are The Coordinates Of The Vertex Of The Function $f(x)=x^2+10x-3$?A. \[$(-5, -28)\$\] B. \[$(-5, 28)\$\] C. \[$(5, -28)\$\] D. \[$(5, 28)\$\]

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 $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. One of the important properties of a quadratic function is its vertex, which is the maximum or minimum point of the function.

What are the Coordinates of the Vertex?

To find the coordinates of the vertex of a quadratic function, we need to use the formula $x = -\frac{b}{2a}$. This formula gives us the x-coordinate of the vertex. Once we have the x-coordinate, we can substitute it into the function to find the y-coordinate.

Step 1: Identify the Values of a and b

In the given function $f(x) = x^2 + 10x - 3$, we can see that $a = 1$ and $b = 10$.

Step 2: Plug the Values into the Formula

Now that we have the values of $a$ and $b$, we can plug them into the formula $x = -\frac{b}{2a}$.

$x = -\frac{10}{2(1)}$ $x = -\frac{10}{2}$ $x = -5$

Step 3: Find the y-Coordinate

Now that we have the x-coordinate of the vertex, we can substitute it into the function to find the y-coordinate.

$f(-5) = (-5)^2 + 10(-5) - 3$ $f(-5) = 25 - 50 - 3$ $f(-5) = -28$

Conclusion

Therefore, the coordinates of the vertex of the function $f(x) = x^2 + 10x - 3$ are $(-5, -28)$.

Answer

The correct answer is A. {(-5, -28)$}$.

Why is this Important?

Finding the vertex of a quadratic function is important in many real-world applications, such as:

• Optimization: Finding the maximum or minimum value of a function is crucial in optimization problems, where we want to maximize or minimize a quantity.
• Physics: The vertex of a quadratic function can represent the position of an object at a given time, which is essential in physics problems.
• Engineering: The vertex of a quadratic function can represent the maximum or minimum value of a quantity, which is important in engineering problems.

Real-World Examples

Here are some real-world examples of finding the vertex of a quadratic function:

• Projectile Motion: The trajectory of a projectile, such as a thrown ball or a rocket, can be modeled using a quadratic function. The vertex of the function represents the maximum height of the projectile.
• Population Growth: The population growth of a species can be modeled using a quadratic function. The vertex of the function represents the maximum population size.
• Economics: The demand for a product can be modeled using a quadratic function. The vertex of the function represents the maximum demand.

Conclusion

In this article, we will answer some frequently asked questions about finding the vertex of a quadratic function.

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 vice versa.

Q: How do I find the vertex of a quadratic function?

A: To find the vertex of a quadratic function, you need to use the formula $x = -\frac{b}{2a}$. This formula gives you the x-coordinate of the vertex. Once you have the x-coordinate, you can substitute it into the function to find the y-coordinate.

Q: What is the formula for finding the vertex of a quadratic function?

A: The formula for finding the vertex of a quadratic function is $x = -\frac{b}{2a}$. This formula gives you the x-coordinate of the vertex.

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

A: To find the y-coordinate of the vertex, you need to substitute the x-coordinate into the function. For example, if the x-coordinate is $x = -5$, you would substitute $x = -5$ into the function to find the y-coordinate.

Q: What is the significance of the vertex of a quadratic function?

A: The vertex of a quadratic function is significant because it represents the maximum or minimum value of the function. It is also important in many real-world applications, such as optimization, physics, and engineering.

Q: Can the vertex of a quadratic function be a maximum or a minimum?

A: Yes, the vertex of a quadratic function can be either a maximum or a minimum. If the coefficient of the $x^2$ term is positive, the vertex is a minimum. If the coefficient of the $x^2$ term is negative, the vertex is a maximum.

Q: How do I determine whether the vertex is a maximum or a minimum?

A: To determine whether the vertex is a maximum or a minimum, you need to look at the coefficient of the $x^2$ term. If the coefficient is positive, the vertex is a minimum. If the coefficient is negative, the vertex is a maximum.

Q: Can the vertex of a quadratic function be a point of inflection?

A: No, the vertex of a quadratic function cannot be a point of inflection. A point of inflection is a point where the function changes from concave to convex or vice versa. The vertex of a quadratic function is a point where the function changes from increasing to decreasing or vice versa.

Q: How do I find the point of inflection of a quadratic function?

A: To find the point of inflection of a quadratic function, you need to use the formula $x = -\frac{b}{3a}$. This formula gives you the x-coordinate of the point of inflection.

Q: What is the difference between the vertex and the point of inflection of a quadratic function?

A: The vertex of a quadratic function is the maximum or minimum point of the function, while the point of inflection is a point where the function changes from concave to convex or vice versa.

Conclusion

In conclusion, finding the vertex of a quadratic function is an important concept in mathematics, with many real-world applications. By using the formula $x = -\frac{b}{2a}$, we can find the coordinates of the vertex of a quadratic function. The vertex of a quadratic function can represent the maximum or minimum value of a quantity, which is essential in many real-world applications.