The Graph Of A Quadratic Function Has A Vertex At The Point ( 6 , 4 (6,4 ( 6 , 4 ]. It Passes Through The Point ( − 4 , − 5 (-4,-5 ( − 4 , − 5 ]. Which Of The Following Equations Represents This Function?A. Y = − 9 100 ( X − 6 ) 2 − 4 Y=-\frac{9}{100}(x-6)^2-4 Y = − 100 9 ​ ( X − 6 ) 2 − 4 B.

Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding their graphs is crucial for various applications in science, engineering, and economics. In this article, we will explore the graph of a quadratic function with a vertex at the point $(6,4)$ and passing through the point $(-4,-5)$. Our goal is to determine which of the given equations represents this function.

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, which can be written in the form $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The graph of a quadratic function is a parabola, which can be either upward-facing or downward-facing, depending on the value of $a$. If $a > 0$, the parabola opens upward, and if $a < 0$, it opens downward.

The Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. In this case, we are given that the vertex is at the point $(6, 4)$, so the equation can be written as $y = a(x - 6)^2 + 4$.

Using the Given Point to Find the Value of $a$

We are also given that the graph passes through the point $(-4, -5)$. We can substitute this point into the equation to find the value of $a$. Plugging in $x = -4$ and $y = -5$, we get:

$$-5 = a(-4 - 6)^2 + 4$$

Simplifying the equation, we get:

$$-5 = a(-10)^2 + 4$$

$$-5 = 100a + 4$$

Subtracting 4 from both sides, we get:

$$-9 = 100a$$

Dividing both sides by 100, we get:

$$a = -\frac{9}{100}$$

The Final Equation

Now that we have found the value of $a$, we can write the final equation of the quadratic function:

$$y = -\frac{9}{100}(x - 6)^2 + 4$$

This is the equation that represents the graph of the quadratic function with a vertex at the point $(6, 4)$ and passing through the point $(-4, -5)$.

Conclusion

In this article, we have explored the graph of a quadratic function with a vertex at the point $(6, 4)$ and passing through the point $(-4, -5)$. We have used the vertex form of a quadratic function and the given point to find the value of $a$ and write the final equation of the function. The equation that represents this function is $y = -\frac{9}{100}(x - 6)^2 + 4$.

The Importance of Quadratic Functions

Quadratic functions are used in various applications in science, engineering, and economics. They are used to model real-world phenomena, such as the motion of objects, the growth of populations, and the behavior of electrical circuits. Understanding quadratic functions is essential for solving problems in these fields.

Real-World Applications of Quadratic Functions

Quadratic functions have many real-world applications. Some examples include:

• Projectile Motion: Quadratic functions are used to model the motion of projectiles, such as the trajectory of a thrown ball or the path of a rocket.
• Population Growth: Quadratic functions are used to model the growth of populations, such as the growth of a city or the spread of a disease.
• Electrical Circuits: Quadratic functions are used to model the behavior of electrical circuits, such as the voltage and current in a circuit.

Conclusion

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Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding their properties and applications is essential for various fields of study. In this article, we will provide a Q&A guide to help you better understand quadratic functions and their uses.

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which can be written in the form $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: What is the vertex form of a quadratic function?

A: The vertex form of a quadratic function is given by $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

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 $h = -\frac{b}{2a}$ and $k = c - \frac{b^2}{4a}$.

Q: What is the axis of symmetry of a quadratic function?

A: The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the parabola. It can be found using the formula $x = -\frac{b}{2a}$.

Q: How do I determine the direction of the parabola?

A: To determine the direction of the parabola, you can look at the value of $a$. If $a > 0$, the parabola opens upward, and if $a < 0$, it opens downward.

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

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

• Projectile Motion: Quadratic functions are used to model the motion of projectiles, such as the trajectory of a thrown ball or the path of a rocket.
• Population Growth: Quadratic functions are used to model the growth of populations, such as the growth of a city or the spread of a disease.
• Electrical Circuits: Quadratic functions are used to model the behavior of electrical circuits, such as the voltage and current in a circuit.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: What is the discriminant of a quadratic equation?

A: The discriminant of a quadratic equation is the expression $b^2 - 4ac$ under the square root in the quadratic formula. It can be used to determine the nature of the solutions to the equation.

Q: What are the different types of solutions to a quadratic equation?

A: The different types of solutions to a quadratic equation are:

• Real and distinct solutions: The equation has two distinct real solutions.
• Real and repeated solutions: The equation has one real solution that is repeated.
• Complex solutions: The equation has two complex solutions.

Conclusion

In conclusion, quadratic functions are a fundamental concept in mathematics, and understanding their properties and applications is essential for various fields of study. This Q&A guide has provided you with a better understanding of quadratic functions and their uses. Whether you are a student or a professional, understanding quadratic functions is crucial for solving problems in science, engineering, and economics.

Frequently Asked Questions

• Q: What is the difference between a quadratic function and a quadratic equation? A: A quadratic function is a polynomial function of degree two, while a quadratic equation is an equation that can be written in the form $ax^2 + bx + c = 0$.
• Q: How do I graph a quadratic function? A: To graph a quadratic function, you can use the vertex form of the function and plot the vertex and the axis of symmetry.
• Q: What is the relationship between the vertex and the axis of symmetry of a quadratic function? A: The vertex and the axis of symmetry of a quadratic function are related by the formula $x = -\frac{b}{2a}$.

Additional Resources

• Quadratic Function Calculator: A calculator that can be used to solve quadratic equations and graph quadratic functions.
• Quadratic Function Grapher: A grapher that can be used to graph quadratic functions.
• Quadratic Function Solver: A solver that can be used to solve quadratic equations.

Conclusion

In conclusion, quadratic functions are a fundamental concept in mathematics, and understanding their properties and applications is essential for various fields of study. This Q&A guide has provided you with a better understanding of quadratic functions and their uses. Whether you are a student or a professional, understanding quadratic functions is crucial for solving problems in science, engineering, and economics.