Find The Vertex Of The Function.$\[ F(x) = 5x^2 + 2x - 1 \\]$\[ \left(-\frac{[?]}{\square}, -\square\right) \\]

Introduction

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, and ${ a \neq 0 }$. The vertex of a quadratic function is the maximum or minimum point on the graph of the function. In this article, we will learn how to find the vertex of a quadratic function.

The Formula for the Vertex

The vertex of a quadratic function can be found using the formula:

${ \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) }$

This formula gives us the x-coordinate and the y-coordinate of the vertex.

Example

Let's consider the quadratic function:

${ f(x) = 5x^2 + 2x - 1 }$

To find the vertex of this function, we need to identify the values of ${ a }$, ${ b }$, and ${ c }$. In this case, ${ a = 5 }$, ${ b = 2 }$, and ${ c = -1 }$.

Step 1: Identify the values of a, b, and c

Variable	Value
${ a }$	5
${ b }$	2
${ c }$	-1

Step 2: Plug the values into the formula

Now that we have the values of ${ a }$, ${ b }$, and ${ c }$, we can plug them into the formula:

${ \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) }$

${ \left(-\frac{2}{2(5)}, f\left(-\frac{2}{2(5)}\right)\right) }$

${ \left(-\frac{2}{10}, f\left(-\frac{2}{10}\right)\right) }$

${ \left(-\frac{1}{5}, f\left(-\frac{1}{5}\right)\right) }$

Step 3: Simplify the expression

To simplify the expression, we need to find the value of ${ f\left(-\frac{1}{5}\right) }$. We can do this by plugging ${ x = -\frac{1}{5} }$ into the function:

${ f\left(-\frac{1}{5}\right) = 5\left(-\frac{1}{5}\right)^2 + 2\left(-\frac{1}{5}\right) - 1 }$

${ f\left(-\frac{1}{5}\right) = 5\left(\frac{1}{25}\right) - \frac{2}{5} - 1 }$

${ f\left(-\frac{1}{5}\right) = \frac{1}{5} - \frac{2}{5} - 1 }$

${ f\left(-\frac{1}{5}\right) = -\frac{1}{5} - 1 }$

${ f\left(-\frac{1}{5}\right) = -\frac{6}{5} }$

Step 4: Write the vertex in the correct format

Now that we have the x-coordinate and the y-coordinate of the vertex, we can write it in the correct format:

${ \left(-\frac{1}{5}, -\frac{6}{5}\right) }$

Conclusion

In this article, we learned how to find the vertex of a quadratic function using the formula:

${ \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) }$

We applied this formula to the quadratic function ${ f(x) = 5x^2 + 2x - 1 }$ and found the vertex to be:

${ \left(-\frac{1}{5}, -\frac{6}{5}\right) }$

This formula is a powerful tool for finding the vertex of a quadratic function, and it can be applied to any quadratic function in the form ${ f(x) = ax^2 + bx + c }$.

Tips and Tricks

• Make sure to identify the values of ${ a }$, ${ b }$, and ${ c }$ before plugging them into the formula.
• Simplify the expression as much as possible before writing the vertex in the correct format.
• Practice finding the vertex of different quadratic functions to become more comfortable with the formula.

Common Mistakes

• Failing to identify the values of ${ a }$, ${ b }$, and ${ c }$ before plugging them into the formula.
• Not simplifying the expression before writing the vertex in the correct format.
• Making mistakes when plugging the values into the formula.

Real-World Applications

• Finding the vertex of a quadratic function can be used to determine the maximum or minimum point on a graph.
• This can be useful in a variety of real-world applications, such as:
  • Determining the maximum or minimum cost of a product.
  • Finding the maximum or minimum profit of a business.
  • Determining the maximum or minimum height of a projectile.

Conclusion

In conclusion, finding the vertex of a quadratic function is a powerful tool that can be used to determine the maximum or minimum point on a graph. By using the formula:

${ \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) }$

$$

Introduction

In our previous article, we learned how to find the vertex of a quadratic function using the formula:

${ \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) }$

In this article, we will answer some common 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 on the graph of the function.

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:

${ \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) }$

This formula gives you the x-coordinate and the y-coordinate of the vertex.

Q: What are the values of a, b, and c in the formula?

A: The values of ${ a }$, ${ b }$, and ${ c }$ are the coefficients of the quadratic function. For example, if the quadratic function is:

${ f(x) = 5x^2 + 2x - 1 }$

Then, ${ a = 5 }$, ${ b = 2 }$, and ${ c = -1 }$.

Q: How do I plug the values into the formula?

A: To plug the values into the formula, you need to follow these steps:

1. Identify the values of ${ a }$, ${ b }$, and ${ c }$.
2. Plug the values into the formula:

${ \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) }$ 3. Simplify the expression as much as possible. 4. Write the vertex in the correct format.

Q: What if the quadratic function is in the form f(x) = ax^2 + bx?

A: If the quadratic function is in the form ${ f(x) = ax^2 + bx }$, then the value of ${ c }$ is 0. In this case, the formula becomes:

${ \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) }$

Q: Can I use the formula to find the vertex of a quadratic function with a negative leading coefficient?

A: Yes, you can use the formula to find the vertex of a quadratic function with a negative leading coefficient. The formula will still work, but the vertex will be a maximum point instead of a minimum point.

Q: What if the quadratic function has a complex number as its vertex?

A: If the quadratic function has a complex number as its vertex, then the vertex will be a complex conjugate pair. In this case, the vertex will be of the form:

${ \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) }$

Where the x-coordinate and the y-coordinate are complex numbers.

Q: Can I use the formula to find the vertex of a quadratic function with a fractional coefficient?

A: Yes, you can use the formula to find the vertex of a quadratic function with a fractional coefficient. The formula will still work, but you may need to simplify the expression to get the vertex in the correct format.

Conclusion

In conclusion, finding the vertex of a quadratic function is a powerful tool that can be used to determine the maximum or minimum point on a graph. By using the formula:

${ \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) }$

we can find the vertex of any quadratic function in the form ${ f(x) = ax^2 + bx + c }$. With practice and patience, you can become more comfortable with this formula and apply it to a variety of real-world applications.

Tips and Tricks

• Make sure to identify the values of ${ a }$, ${ b }$, and ${ c }$ before plugging them into the formula.
• Simplify the expression as much as possible before writing the vertex in the correct format.
• Practice finding the vertex of different quadratic functions to become more comfortable with the formula.

Common Mistakes

• Failing to identify the values of ${ a }$, ${ b }$, and ${ c }$ before plugging them into the formula.
• Not simplifying the expression before writing the vertex in the correct format.
• Making mistakes when plugging the values into the formula.

Real-World Applications

• Finding the vertex of a quadratic function can be used to determine the maximum or minimum point on a graph.
• This can be useful in a variety of real-world applications, such as:
  • Determining the maximum or minimum cost of a product.
  • Finding the maximum or minimum profit of a business.
  • Determining the maximum or minimum height of a projectile.