Consider The Quadratic Function $f(x) = X^2 - 2x - 24$.- Its Vertex Is $( \square, \square $\].- Its Largest $x$-intercept Is $x = \square$.- Its $y$-intercept Is $(0, \square $\].

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 given function $f(x) = x^2 - 2x - 24$. We will explore its vertex, largest $x$-intercept, and $y$-intercept, providing a comprehensive understanding of this mathematical concept.

The Quadratic Function

The given quadratic function is $f(x) = x^2 - 2x - 24$. This function can be written in the standard form of a quadratic function, which is $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this case, $a = 1$, $b = -2$, and $c = -24$.

Vertex of the Quadratic Function

The vertex of a quadratic function is the point at which the function changes direction. It is the minimum or maximum point of the function, depending on the value of $a$. If $a$ is positive, the vertex is the minimum point, and if $a$ is negative, the vertex is the maximum point.

To find the vertex of the given quadratic function, we can use the formula $x = -\frac{b}{2a}$. Plugging in the values of $a$ and $b$, we get $x = -\frac{-2}{2(1)} = 1$. To find the $y$-coordinate of the vertex, we substitute $x = 1$ into the function: $f(1) = (1)^2 - 2(1) - 24 = -25$.

Therefore, the vertex of the quadratic function is $(1, -25)$.

Largest $x$-Intercept

The $x$-intercepts of a quadratic function are the points at which the function crosses the $x$-axis. To find the $x$-intercepts, we set the function equal to zero and solve for $x$.

Setting $f(x) = 0$, we get $x^2 - 2x - 24 = 0$. We can factor this quadratic equation as $(x - 6)(x + 4) = 0$. This gives us two possible values for $x$: $x = 6$ and $x = -4$.

Since $x = 6$ is the larger value, the largest $x$-intercept is $x = 6$.

$y$-Intercept

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

Plugging in $x = 0$, we get $f(0) = (0)^2 - 2(0) - 24 = -24$. Therefore, the $y$-intercept is $(0, -24)$.

Conclusion

In this article, we have explored the quadratic function $f(x) = x^2 - 2x - 24$. We have found its vertex, largest $x$-intercept, and $y$-intercept, providing a comprehensive understanding of this mathematical concept. The vertex is $(1, -25)$, the largest $x$-intercept is $x = 6$, and the $y$-intercept is $(0, -24)$. This analysis demonstrates the importance of quadratic functions in mathematics and their applications in various fields.

Applications of Quadratic Functions

Quadratic functions have numerous applications in various fields, 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.

Real-World Examples

Quadratic functions have numerous real-world applications, including:

• Projectile Motion: The trajectory of a projectile, such as a thrown ball or a rocket, can be modeled using a quadratic function.
• Optimization: Quadratic functions are used to optimize systems, such as finding the minimum or maximum value of a function.
• Signal Processing: Quadratic functions are used in signal processing, such as filtering and modulation.

Conclusion

In conclusion, quadratic functions are a fundamental concept in mathematics, and they have numerous applications in various fields. The given quadratic function $f(x) = x^2 - 2x - 24$ has a vertex of $(1, -25)$, a largest $x$-intercept of $x = 6$, and a $y$-intercept of $(0, -24)$. This analysis demonstrates the importance of quadratic functions in mathematics and their applications in various fields.

Introduction

Quadratic functions are a fundamental concept in mathematics, and they have numerous applications in various fields. In this article, we will answer some of the most frequently asked questions about quadratic functions, providing a comprehensive understanding of this mathematical concept.

Q1: What is a quadratic function?

A quadratic function is a polynomial function of degree two, which means that the highest power of the variable is two. It can be written in the standard form of a quadratic function, which is $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q2: What is the vertex of a quadratic function?

The vertex of a quadratic function is the point at which the function changes direction. It is the minimum or maximum point of the function, depending on the value of $a$. If $a$ is positive, the vertex is the minimum point, and if $a$ is negative, the vertex is the maximum point.

Q3: How do I find the vertex of a quadratic function?

To find the vertex of a quadratic function, you can use the formula $x = -\frac{b}{2a}$. This will give you the $x$-coordinate of the vertex. To find the $y$-coordinate of the vertex, you can substitute $x = -\frac{b}{2a}$ into the function.

Q4: What is the largest $x$-intercept of a quadratic function?

The largest $x$-intercept of a quadratic function is the point at which the function crosses the $x$-axis. To find the largest $x$-intercept, you can set the function equal to zero and solve for $x$.

Q5: How do I find the $y$-intercept of a quadratic function?

To find the $y$-intercept of a quadratic function, you can substitute $x = 0$ into the function. This will give you the $y$-coordinate of the $y$-intercept.

Q6: What is the difference between a quadratic function and a linear function?

A quadratic function is a polynomial function of degree two, while a linear function is a polynomial function of degree one. This means that a quadratic function has a higher degree than a linear function, and it can have a minimum or maximum point.

Q7: Can a quadratic function have more than one $x$-intercept?

Yes, a quadratic function can have more than one $x$-intercept. This occurs when the function crosses the $x$-axis at more than one point.

Q8: How do I graph a quadratic function?

To graph a quadratic function, you can use the following steps:

1. Find the vertex of the function.
2. Find the $x$-intercepts of the function.
3. Plot the vertex and the $x$-intercepts on a coordinate plane.
4. Draw a smooth curve through the points to graph the function.

Q9: What are some real-world applications of quadratic functions?

Quadratic functions have numerous real-world applications, including:

• Projectile Motion: The trajectory of a projectile, such as a thrown ball or a rocket, can be modeled using a quadratic function.
• Optimization: Quadratic functions are used to optimize systems, such as finding the minimum or maximum value of a function.
• Signal Processing: Quadratic functions are used in signal processing, such as filtering and modulation.

Q10: Can I use a calculator to solve quadratic equations?

Yes, you can use a calculator to solve quadratic equations. Most calculators have a built-in quadratic equation solver that can be used to find the solutions to a quadratic equation.

Conclusion

In conclusion, quadratic functions are a fundamental concept in mathematics, and they have numerous applications in various fields. By understanding the properties and behavior of quadratic functions, you can solve a wide range of problems and model real-world phenomena. We hope that this Q&A article has provided you with a comprehensive understanding of quadratic functions and their applications.