Examples of Finding the Vertex of a Quadratic Function

Oct 13, 2025 by wbrum.org 34 views

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. The vertex of a quadratic function is the maximum or minimum point of the function, and it is an important concept in algebra and calculus.

What is the Vertex of a Quadratic Function?

The vertex of a quadratic function is the point at which the function changes from decreasing to increasing or vice versa. It is the lowest or highest point on the graph of the function. The vertex can be found using the formula $x = -\frac{b}{2a}$ , which gives the x-coordinate of the vertex. The y-coordinate of the vertex can be found by substituting the x-coordinate into the function.

Finding the Vertex of the Function $f(x) = 2(x-7)^2 + 12$

To find the vertex of the function $f(x) = 2(x-7)^2 + 12$ , we need to use the formula $x = -\frac{b}{2a}$ . However, the function is not in the standard form $f(x) = ax^2 + bx + c$ . Instead, it is in the form $f(x) = a(x-h)^2 + k$ , where $(h,k)$ is the vertex of the function.

Step 1: Identify the Values of $a$ , $h$ , and $k$

In the function $f(x) = 2(x-7)^2 + 12$ , we can see that $a = 2$ , $h = 7$ , and $k = 12$ . The value of $a$ is the coefficient of the squared term, the value of $h$ is the value inside the parentheses, and the value of $k$ is the constant term.

Step 2: Find the x-Coordinate of the Vertex

The x-coordinate of the vertex is given by the formula $x = -\frac{b}{2a}$ . However, in this case, we don't need to use this formula because the function is already in the form $f(x) = a(x-h)^2 + k$ . The x-coordinate of the vertex is simply the value of $h$ , which is $7$ .

Step 3: Find the y-Coordinate of the Vertex

The y-coordinate of the vertex is given by the formula $y = f(h)$ . We can substitute the value of $h$ into the function to find the y-coordinate of the vertex.

$f(7) = 2(7-7)^2 + 12$ $f(7) = 2(0)^2 + 12$ $f(7) = 0 + 12$ $f(7) = 12$

Conclusion

The vertex of the function $f(x) = 2(x-7)^2 + 12$ is the point $(7,12)$ . This means that the function has a minimum value of $12$ at $x = 7$ .

Why is Finding the Vertex of a Quadratic Function 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, such as finding the maximum profit or minimum cost.
Physics: The vertex of a quadratic function can represent the position of an object at a given time, and the slope of the function can represent the velocity of the object.
Engineering: The vertex of a quadratic function can represent the maximum or minimum value of a system, such as the maximum or minimum stress on a beam.

Examples of Finding the Vertex of a Quadratic Function

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

Example 1: Find the vertex of the function $f(x) = x^2 - 6x + 8$ .
Example 2: Find the vertex of the function $f(x) = 3x^2 + 12x - 7$ .
Example 3: Find the vertex of the function $f(x) = 2x^2 - 4x + 1$ .

Tips and Tricks for Finding the Vertex of a Quadratic Function

Here are some tips and tricks for finding the vertex of a quadratic function:

Use the formula $x = -\frac{b}{2a}$ : This formula can be used to find the x-coordinate of the vertex.
Use the formula $y = f(h)$ : This formula can be used to find the y-coordinate of the vertex.
Graph the function: Graphing the function can help you visualize the vertex and find its coordinates.

Conclusion

$In conclusion, finding the vertex of a quadratic function is an important concept in algebra and calculus. The vertex of a quadratic function can be found using the formula $x = -\frac{b}{2a}$ or by using the formula $y = f(h)$. The vertex of a quadratic function can represent the maximum or minimum value of a function, and it is an important concept in many real-world applications.<br/> **Finding the Vertex of a Quadratic Function: Q&A**$

Introduction

In our previous article, we discussed how to find the vertex of a quadratic function. 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 point at which the function changes from decreasing to increasing or vice versa. It is the lowest or highest 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 $x = -\frac{b}{2a}$ or by using the formula $y = f(h)$ . If the function is in the form $f(x) = a(x-h)^2 + k$ , the vertex is the point $(h,k)$ .

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

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

Q: What is the formula for finding the y-coordinate of the vertex?

A: The formula for finding the y-coordinate of the vertex is $y = f(h)$ .

Q: How do I know if the vertex is a maximum or minimum point?

A: If the coefficient of the squared term is positive, the vertex is a minimum point. If the coefficient of the squared term is negative, the vertex is a maximum point.

Q: Can I find the vertex of a quadratic function if it is not in the standard form?

A: Yes, you can find the vertex of a quadratic function even if it is not in the standard form. You can use the formula $x = -\frac{b}{2a}$ or by using the formula $y = f(h)$ .

Q: How do I graph a quadratic function to find its vertex?

A: To graph a quadratic function, you can use a graphing calculator or a computer program. You can also use a table of values to find the x and y coordinates of the vertex.

Q: What are some real-world applications of finding the vertex of a quadratic function?

A: Finding the vertex of a quadratic function has many real-world applications, such as:

Optimization: Finding the maximum or minimum value of a function is crucial in optimization problems, such as finding the maximum profit or minimum cost.
Physics: The vertex of a quadratic function can represent the position of an object at a given time, and the slope of the function can represent the velocity of the object.
Engineering: The vertex of a quadratic function can represent the maximum or minimum value of a system, such as the maximum or minimum stress on a beam.

Q: Can I use technology to find the vertex of a quadratic function?

A: Yes, you can use technology to find the vertex of a quadratic function. Graphing calculators and computer programs can help you find the vertex of a quadratic function.

Q: What are some common mistakes to avoid when finding the vertex of a quadratic function?

A: Some common mistakes to avoid when finding the vertex of a quadratic function include:

Not using the correct formula: Make sure to use the correct formula for finding the x and y coordinates of the vertex.
Not checking the sign of the coefficient: Make sure to check the sign of the coefficient of the squared term to determine if the vertex is a maximum or minimum point.
Not using a graphing calculator or computer program: Using a graphing calculator or computer program can help you visualize the vertex and find its coordinates.

Conclusion

In conclusion, finding the vertex of a quadratic function is an important concept in algebra and calculus. By understanding the formulas and techniques for finding the vertex, you can apply this knowledge to real-world problems and make informed decisions.