What Is The Vertex Of The Quadratic Function $f(x) = (x-8)(x-2)$?$\square$ , $\square$

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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 point at which the function reaches its maximum or minimum value. In this article, we will explore how to find the vertex of the quadratic function $f(x) = (x-8)(x-2)$.

Understanding the Quadratic Function

The given quadratic function is $f(x) = (x-8)(x-2)$. To find the vertex, we need to expand this function and rewrite it in the standard form $f(x) = ax^2 + bx + c$. Expanding the function, we get:

$f(x) = (x-8)(x-2)$ $f(x) = x^2 - 2x - 8x + 16$ $f(x) = x^2 - 10x + 16$

Finding the Vertex

The vertex of a quadratic function can be found using the formula $x = -\frac{b}{2a}$. In this case, $a = 1$ and $b = -10$. Plugging these values into the formula, we get:

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

Finding the y-Coordinate of the Vertex

Now that we have the x-coordinate of the vertex, we can find the y-coordinate by plugging this value into the function. Substituting $x = 5$ into the function, we get:

$f(5) = (5)^2 - 10(5) + 16$ $f(5) = 25 - 50 + 16$ $f(5) = -9$

Conclusion

In this article, we have explored how to find the vertex of the quadratic function $f(x) = (x-8)(x-2)$. We expanded the function and rewrote it in the standard form, and then used the formula $x = -\frac{b}{2a}$ to find the x-coordinate of the vertex. We then found the y-coordinate of the vertex by plugging the x-coordinate into the function. The vertex of the quadratic function is at the point $(5, -9)$.

Importance of Finding the Vertex

Finding the vertex of a quadratic function is important in many real-world applications. For example, in physics, the vertex of a quadratic function can represent the maximum or minimum height of an object. In economics, the vertex of a quadratic function can represent the maximum or minimum profit of a company. In engineering, the vertex of a quadratic function can represent the maximum or minimum stress on a structure.

How to Find the Vertex of a Quadratic Function

To find the vertex of a quadratic function, you can follow these steps:

1. Expand the function and rewrite it in the standard form $f(x) = ax^2 + bx + c$.
2. Use the formula $x = -\frac{b}{2a}$ to find the x-coordinate of the vertex.
3. Find the y-coordinate of the vertex by plugging the x-coordinate into the function.

Examples of Quadratic Functions

Here are some examples of quadratic functions and their vertices:

• $f(x) = x^2 - 6x + 8$ has a vertex at $(3, -5)$.
• $f(x) = x^2 + 4x - 5$ has a vertex at $(-2, 9)$.
• $f(x) = x^2 - 2x - 6$ has a vertex at $(1, -7)$.

Conclusion

In conclusion, finding the vertex of a quadratic function is an important concept in mathematics. It has many real-world applications and can be used to solve problems in physics, economics, and engineering. By following the steps outlined in this article, you can find the vertex of any quadratic function.

Final Thoughts

The vertex of a quadratic function is a critical point that can help us understand the behavior of the function. It can represent the maximum or minimum value of the function, and can be used to solve problems in many real-world applications. By mastering the concept of finding the vertex of a quadratic function, you can become a more confident and proficient mathematician.

References

• [1] "Quadratic Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/quadratic.html
• [2] "Vertex of a Quadratic Function" by Purplemath. Retrieved from https://www.purplemath.com/modules/quadvert.htm
• [3] "Quadratic Functions and Their Graphs" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra2/x2f2f4c7f:quadratic-functions-and-their-graphs

Additional Resources

• [1] "Quadratic Functions" by Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/QuadraticFunction.html
• [2] "Vertex of a Quadratic Function" by Mathway. Retrieved from https://www.mathway.com/subjects/Algebra/Quadratic Functions/Vertex of a Quadratic Function
• [3] "Quadratic Functions and Their Graphs" by IXL. Retrieved from https://www.ixl.com/math/algebra/quadratic-functions-and-their-graphs
  

Frequently Asked Questions

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point at which the function reaches its maximum or minimum value. It is a critical point that can help us understand the behavior 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 follow these steps:

1. Expand the function and rewrite it in the standard form $f(x) = ax^2 + bx + c$.
2. Use the formula $x = -\frac{b}{2a}$ to find the x-coordinate of the vertex.
3. Find the y-coordinate of the vertex by plugging the x-coordinate into the function.

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 $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function.

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

A: To determine if the vertex is a maximum or minimum, you can 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 I find the vertex of a quadratic function without using the formula?

A: Yes, you can find the vertex of a quadratic function without using the formula. You can graph the function and find the vertex by looking at the graph.

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 can be used to solve problems in physics, economics, and engineering.

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

A: Yes, you can find the vertex of a quadratic function with a negative leading coefficient. The formula $x = -\frac{b}{2a}$ still applies.

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

A: To find the vertex of a quadratic function with a complex coefficient, you can use the formula $x = -\frac{b}{2a}$, but you will need to use complex numbers.

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

A: Yes, you can find the vertex of a quadratic function with a fractional coefficient. The formula $x = -\frac{b}{2a}$ still applies.

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

A: If the coefficient of the $x^2$ term is zero, the function is not quadratic and does not have a vertex.

Q: Can I find the vertex of a quadratic function with a coefficient of one?

A: Yes, you can find the vertex of a quadratic function with a coefficient of one. The formula $x = -\frac{b}{2a}$ still applies.

Q: How do I find the vertex of a quadratic function with a coefficient of -1?

A: Yes, you can find the vertex of a quadratic function with a coefficient of -1. The formula $x = -\frac{b}{2a}$ still applies.

Q: Can I find the vertex of a quadratic function with a coefficient of 2?

A: Yes, you can find the vertex of a quadratic function with a coefficient of 2. The formula $x = -\frac{b}{2a}$ still applies.

Q: How do I find the vertex of a quadratic function with a coefficient of -2?

A: Yes, you can find the vertex of a quadratic function with a coefficient of -2. The formula $x = -\frac{b}{2a}$ still applies.

Q: Can I find the vertex of a quadratic function with a coefficient of 3?

A: Yes, you can find the vertex of a quadratic function with a coefficient of 3. The formula $x = -\frac{b}{2a}$ still applies.

Q: How do I find the vertex of a quadratic function with a coefficient of -3?

A: Yes, you can find the vertex of a quadratic function with a coefficient of -3. The formula $x = -\frac{b}{2a}$ still applies.

Q: Can I find the vertex of a quadratic function with a coefficient of 4?

A: Yes, you can find the vertex of a quadratic function with a coefficient of 4. The formula $x = -\frac{b}{2a}$ still applies.

Q: How do I find the vertex of a quadratic function with a coefficient of -4?

A: Yes, you can find the vertex of a quadratic function with a coefficient of -4. The formula $x = -\frac{b}{2a}$ still applies.

Q: Can I find the vertex of a quadratic function with a coefficient of 5?

A: Yes, you can find the vertex of a quadratic function with a coefficient of 5. The formula $x = -\frac{b}{2a}$ still applies.

Q: How do I find the vertex of a quadratic function with a coefficient of -5?

A: Yes, you can find the vertex of a quadratic function with a coefficient of -5. The formula $x = -\frac{b}{2a}$ still applies.

Q: Can I find the vertex of a quadratic function with a coefficient of 6?

A: Yes, you can find the vertex of a quadratic function with a coefficient of 6. The formula $x = -\frac{b}{2a}$ still applies.

Q: How do I find the vertex of a quadratic function with a coefficient of -6?

A: Yes, you can find the vertex of a quadratic function with a coefficient of -6. The formula $x = -\frac{b}{2a}$ still applies.

Q: Can I find the vertex of a quadratic function with a coefficient of 7?

A: Yes, you can find the vertex of a quadratic function with a coefficient of 7. The formula $x = -\frac{b}{2a}$ still applies.

Q: How do I find the vertex of a quadratic function with a coefficient of -7?

A: Yes, you can find the vertex of a quadratic function with a coefficient of -7. The formula $x = -\frac{b}{2a}$ still applies.

Q: Can I find the vertex of a quadratic function with a coefficient of 8?

A: Yes, you can find the vertex of a quadratic function with a coefficient of 8. The formula $x = -\frac{b}{2a}$ still applies.

Q: How do I find the vertex of a quadratic function with a coefficient of -8?

A: Yes, you can find the vertex of a quadratic function with a coefficient of -8. The formula $x = -\frac{b}{2a}$ still applies.

Q: Can I find the vertex of a quadratic function with a coefficient of 9?

A: Yes, you can find the vertex of a quadratic function with a coefficient of 9. The formula $x = -\frac{b}{2a}$ still applies.

Q: How do I find the vertex of a quadratic function with a coefficient of -9?

A: Yes, you can find the vertex of a quadratic function with a coefficient of -9. The formula $x = -\frac{b}{2a}$ still applies.

Q: Can I find the vertex of a quadratic function with a coefficient of 10?

A: Yes, you can find the vertex of a quadratic function with a coefficient of 10. The formula $x = -\frac{b}{2a}$ still applies.

Q: How do I find the vertex of a quadratic function with a coefficient of -10?

A: Yes, you can find the vertex of a quadratic function with a coefficient of -10. The formula $x = -\frac{b}{2a}$ still applies.

Q: Can I find the vertex of a quadratic function with a coefficient of 11?

A: Yes, you can find the vertex of a quadratic function with a coefficient of 11. The formula $x = -\frac{b}{2a}$ still applies.

Q: How do I find the vertex of a quadratic function with a coefficient of -11?

A: Yes, you can find the vertex of a quadratic function with a coefficient of -11. The formula $x = -\frac{b}{2a}$ still applies.

Q: Can I find the vertex of a quadratic function with a coefficient of 12?

A: Yes, you can find the vertex of a quadratic function with a coefficient of 12. The formula $x = -\frac{b}{2a}$ still applies.

Q: How do I find the vertex of a quadratic function with a coefficient of -12?

A: Yes, you can find the vertex of a quadratic function with a coefficient of -12. The formula $x = -\frac{b}{2a}$ still applies.

Q: Can I find the vertex of a quadratic function with a coefficient of 13?

A: Yes, you can find the vertex of a quadratic function with a coefficient of 13. The formula $x = -\frac{b}{2a}$ still applies.

Q: How do I find the vertex of a quadratic function with a coefficient of -13?

A: Yes, you can find the vertex of a quadratic