Finding the Vertex of the Given Function

Oct 13, 2025 by miodeya.com 88 views

Understanding the Concept of Vertex

The vertex of a quadratic function is the highest or lowest point on the graph of the function. It is a crucial concept in mathematics, particularly in algebra and calculus. The vertex form of a quadratic function is given by $f(x) = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola.

Finding the Vertex of the Given Function

To find the vertex of the function $f(x) = x^2 + 8x - 2$ , we can use the method of completing the square. This method involves rewriting the quadratic function in the form $f(x) = a(x - h)^2 + k$ . The vertex of the parabola is then given by the point $(h, k)$ .

Completing the Square

To complete the square, we start by taking half of the coefficient of the $x$ term, which is $8$ , and squaring it. This gives us $(8/2)^2 = 16$ . We then add and subtract $16$ inside the parentheses to maintain the value of the expression.

f(x) = x^2 + 8x - 2
= (x^2 + 8x + 16) - 16 - 2
= (x + 4)^2 - 18

Identifying the Vertex

Comparing the completed square form with the vertex form, we can see that the vertex of the parabola is given by the point $(-4, -18)$ .

Conclusion

In conclusion, the vertex of the graph of the function $f(x) = x^2 + 8x - 2$ is the point $(-4, -18)$ . This is the highest point on the graph of the function, and it is a crucial concept in mathematics, particularly in algebra and calculus.

Answer

The correct answer is D. $(-4, -18)$ .

Why is the Vertex Important?

The vertex of a quadratic function is important because it helps us understand the behavior of the function. The vertex is the highest or lowest point on the graph of the function, and it is a crucial concept in mathematics, particularly in algebra and calculus.

How to Find the Vertex of a Quadratic Function

To find the vertex of a quadratic function, we can use the method of completing the square. This method involves rewriting the quadratic function in the form $f(x) = a(x - h)^2 + k$ . The vertex of the parabola is then given by the point $(h, k)$ .

Examples of Finding the Vertex

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

$f(x) = x^2 + 6x + 8$
$f(x) = x^2 - 4x - 3$
$f(x) = x^2 + 2x - 6$

Tips and Tricks

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

Use the method of completing the square to rewrite the quadratic function in the form $f(x) = a(x - h)^2 + k$ .
Identify the vertex of the parabola by finding the point $(h, k)$ .
Use the vertex form to graph the quadratic function.

Real-World Applications

The vertex of a quadratic function has many real-world applications, including:

Optimization: The vertex of a quadratic function can be used to find the maximum or minimum value of a function.
Physics: The vertex of a quadratic function can be used to model the motion of an object under the influence of gravity.
Economics: The vertex of a quadratic function can be used to model the demand or supply of a product.

Conclusion

In conclusion, the vertex of a quadratic function is an important concept in mathematics, particularly in algebra and calculus. It helps us understand the behavior of the function and has many real-world applications. By using the method of completing the square, we can find the vertex of a quadratic function and use it to graph the function.

Final Thoughts

The vertex of a quadratic function is a powerful tool that can be used to solve many problems in mathematics and real-world applications. By understanding the concept of the vertex, we can gain a deeper understanding of the behavior of quadratic functions and use it to make informed decisions.

References

Algebra and Calculus: These are the two main branches of mathematics that deal with the concept of the vertex.
Quadratic Functions: These are functions that can be written in the form $f(x) = ax^2 + bx + c$ .
Completing the Square: This is a method used to rewrite a quadratic function in the form $f(x) = a(x - h)^2 + k$ .

Glossary

Vertex: The highest or lowest point on the graph of a quadratic function.
Quadratic Function: A function that can be written in the form $f(x) = ax^2 + bx + c$ .
Completing the Square: A method used to rewrite a quadratic function in the form $f(x) = a(x - h)^2 + k$ .

Conclusion

Understanding the Concept of Vertex

The vertex of a quadratic function is the highest or lowest point on the graph of the function. It is a crucial concept in mathematics, particularly in algebra and calculus.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the highest or lowest 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 method of completing the square. This method involves rewriting the quadratic function in the form $f(x) = a(x - h)^2 + k$ . The vertex of the parabola is then given by the point $(h, k)$ .

Q: What is the method of completing the square?

A: The method of completing the square is a technique used to rewrite a quadratic function in the form $f(x) = a(x - h)^2 + k$ . This involves taking half of the coefficient of the $x$ term, squaring it, and adding and subtracting it inside the parentheses.

Q: How do I use the method of completing the square to find the vertex of a quadratic function?

A: To use the method of completing the square, follow these steps:

Take half of the coefficient of the $x$ term and square it.
Add and subtract the result inside the parentheses.
Rewrite the quadratic function in the form $f(x) = a(x - h)^2 + k$ .
Identify the vertex of the parabola by finding the point $(h, k)$ .

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

A: The vertex of a quadratic function has many real-world applications, including:

Optimization: The vertex of a quadratic function can be used to find the maximum or minimum value of a function.
Physics: The vertex of a quadratic function can be used to model the motion of an object under the influence of gravity.
Economics: The vertex of a quadratic function can be used to model the demand or supply of a product.

Q: How do I graph a quadratic function using the vertex?

A: To graph a quadratic function using the vertex, follow these steps:

Identify the vertex of the parabola by finding the point $(h, k)$ .
Use the vertex form to write the quadratic function as $f(x) = a(x - h)^2 + k$ .
Plot the vertex on the graph.
Use the vertex as a reference point to plot the rest of the graph.

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 completing the square correctly: Make sure to take half of the coefficient of the $x$ term and square it correctly.
Not identifying the vertex correctly: Make sure to identify the vertex of the parabola by finding the point $(h, k)$ .
Not using the vertex form correctly: Make sure to use the vertex form to write the quadratic function as $f(x) = a(x - h)^2 + k$ .

Q: How do I check my work when finding the vertex of a quadratic function?

A: To check your work when finding the vertex of a quadratic function, follow these steps:

Plug the vertex into the original quadratic function to make sure it is true.
Use the vertex form to graph the quadratic function and make sure it matches the original graph.
Check your work by using a calculator or graphing software to verify the vertex.