Expansion and Simplification of a Quadratic Function

Oct 13, 2025 by miodeya.com 125 views

Introduction

Quadratic functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. In this article, we will delve into the world of quadratic functions, focusing on the expansion and simplification of a given quadratic function, and identifying its key characteristics.

Expansion and Simplification of a Quadratic Function

Given the quadratic function $f(x) = (x-6)(x-4)$ , our goal is to expand and simplify it to obtain the standard form of a quadratic function. To do this, we will use the distributive property of multiplication over addition.

$f(x) = (x-6)(x-4)$

Using the distributive property, we can expand the expression as follows:

$f(x) = x(x-4) - 6(x-4)$

$f(x) = x^2 - 4x - 6x + 24$

$f(x) = x^2 - 10x + 24$

Therefore, the expanded and simplified form of the given quadratic function is $f(x) = x^2 - 10x + 24$ .

Identifying Key Characteristics

Now that we have the expanded and simplified form of the quadratic function, we can identify its key characteristics.

Vertex

The vertex of a quadratic function is the point at which the function changes from decreasing to increasing or vice versa. To find the vertex, we can use the formula:

$x = \frac{-b}{2a}$

where $a$ and $b$ are the coefficients of the quadratic function.

In this case, $a = 1$ and $b = -10$ . Plugging these values into the formula, we get:

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

$x = \frac{10}{2}$

$x = 5$

To find the y-coordinate of the vertex, we can plug the x-coordinate into the quadratic function:

$f(5) = (5)^2 - 10(5) + 24$

$f(5) = 25 - 50 + 24$

$f(5) = -1$

Therefore, the vertex of the quadratic function is $(5, -1)$ .

Minimum/Maximum

Since the vertex is at $(5, -1)$ , we can conclude that the quadratic function has a minimum value at this point.

Y-intercept

The y-intercept of a quadratic function is the point at which the function intersects the y-axis. To find the y-intercept, we can plug $x = 0$ into the quadratic function:

$f(0) = (0)^2 - 10(0) + 24$

$f(0) = 24$

Therefore, the y-intercept of the quadratic function is $(0, 24)$ .

Axis of Symmetry

The axis of symmetry of a quadratic function is the vertical line that passes through the vertex. To find the equation of the axis of symmetry, we can use the formula:

$x = \frac{-b}{2a}$

In this case, $a = 1$ and $b = -10$ . Plugging these values into the formula, we get:

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

$x = \frac{10}{2}$

$x = 5$

Therefore, the equation of the axis of symmetry is $x = 5$ .

Conclusion

In this article, we expanded and simplified a given quadratic function, and identified its key characteristics, including the vertex, minimum/maximum, y-intercept, and axis of symmetry. We hope that this comprehensive analysis has provided a deeper understanding of quadratic functions and their properties.

Real-World Applications

Quadratic functions have numerous real-world applications, including:

Physics: Quadratic functions are used to model the motion of objects under the influence of gravity, friction, and other forces.
Engineering: Quadratic functions are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
Economics: Quadratic functions are used to model the behavior of economic systems, including supply and demand curves.
Computer Science: Quadratic functions are used in algorithms and data structures, such as sorting and searching.

Future Research Directions

There are many areas of research that involve quadratic functions, including:

Quadratic optimization: Developing algorithms and techniques for optimizing quadratic functions.
Quadratic approximation: Developing methods for approximating quadratic functions using simpler functions.
Quadratic interpolation: Developing methods for interpolating quadratic functions using a set of given points.

References

Algebra: A comprehensive textbook on algebra, including quadratic functions.
Calculus: A comprehensive textbook on calculus, including quadratic functions.
Mathematics: A comprehensive textbook on mathematics, including quadratic functions.

Glossary

Quadratic function: A polynomial function of degree two, in the form $f(x) = ax^2 + bx + c$ .
Vertex: The point at which the function changes from decreasing to increasing or vice versa.
Minimum/maximum: The value of the function at the vertex.
Y-intercept: The point at which the function intersects the y-axis.
Axis of symmetry: The vertical line that passes through the vertex.
Quadratic Function Q&A =========================

Frequently Asked Questions

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, in the form $f(x) = ax^2 + bx + c$ . It is a fundamental concept in mathematics and has numerous real-world applications.

Q: How do I expand and simplify a quadratic function?

A: To expand and simplify a quadratic function, you can use the distributive property of multiplication over addition. For example, if you have the quadratic function $f(x) = (x-6)(x-4)$ , you can expand it as follows:

$f(x) = x(x-4) - 6(x-4)$

$f(x) = x^2 - 4x - 6x + 24$

$f(x) = x^2 - 10x + 24$

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:

$x = \frac{-b}{2a}$

where $a$ and $b$ are the coefficients of the quadratic function. For example, if you have the quadratic function $f(x) = x^2 - 10x + 24$ , you can find the vertex as follows:

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

$x = \frac{10}{2}$

$x = 5$

Q: How do I find the minimum/maximum of a quadratic function?

A: To find the minimum/maximum of a quadratic function, you can look at the vertex. If the vertex is at a minimum, the function is decreasing on one side and increasing on the other. If the vertex is at a maximum, the function is increasing on one side and decreasing on the other.

Q: How do I find the y-intercept of a quadratic function?

A: To find the y-intercept of a quadratic function, you can plug $x = 0$ into the function. For example, if you have the quadratic function $f(x) = x^2 - 10x + 24$ , you can find the y-intercept as follows:

$f(0) = (0)^2 - 10(0) + 24$

$f(0) = 24$

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

A: To find the axis of symmetry of a quadratic function, you can use the formula:

$x = \frac{-b}{2a}$

where $a$ and $b$ are the coefficients of the quadratic function. For example, if you have the quadratic function $f(x) = x^2 - 10x + 24$ , you can find the axis of symmetry as follows:

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

$x = \frac{10}{2}$

$x = 5$

Q: What are some real-world applications of quadratic functions?

A: Quadratic functions have numerous real-world applications, including:

Physics: Quadratic functions are used to model the motion of objects under the influence of gravity, friction, and other forces.
Engineering: Quadratic functions are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
Economics: Quadratic functions are used to model the behavior of economic systems, including supply and demand curves.
Computer Science: Quadratic functions are used in algorithms and data structures, such as sorting and searching.

Q: What are some areas of research that involve quadratic functions?

A: There are many areas of research that involve quadratic functions, including:

Quadratic optimization: Developing algorithms and techniques for optimizing quadratic functions.
Quadratic approximation: Developing methods for approximating quadratic functions using simpler functions.
Quadratic interpolation: Developing methods for interpolating quadratic functions using a set of given points.

Q: What are some common mistakes to avoid when working with quadratic functions?

A: Some common mistakes to avoid when working with quadratic functions include:

Not expanding and simplifying the function: Failing to expand and simplify the function can lead to incorrect results.
Not finding the vertex: Failing to find the vertex can lead to incorrect conclusions about the function's behavior.
Not finding the minimum/maximum: Failing to find the minimum/maximum can lead to incorrect conclusions about the function's behavior.
Not finding the y-intercept: Failing to find the y-intercept can lead to incorrect conclusions about the function's behavior.

Q: What are some tips for working with quadratic functions?

A: Some tips for working with quadratic functions include:

Use the distributive property: Use the distributive property to expand and simplify the function.
Find the vertex: Find the vertex to understand the function's behavior.
Find the minimum/maximum: Find the minimum/maximum to understand the function's behavior.
Find the y-intercept: Find the y-intercept to understand the function's behavior.
Use the axis of symmetry: Use the axis of symmetry to understand the function's behavior.