Determine The Shape And Direction Of The Parabola Formed By The Given Function.Given Function: Y = − 1 6 X 2 Y = -\frac{1}{6} X^2 Y = − 6 1 ​ X 2 Solution:- Because A A A Is Negative, The Parabola Opens Downward.- The Parabola Is Symmetric About Its Line Of

Introduction

In mathematics, a parabola is a type of quadratic curve that can be represented by a quadratic equation in the form of $y = ax^2 + bx + c$. The shape and direction of a parabola are determined by the value of the coefficient $a$. In this article, we will determine the shape and direction of the parabola formed by the given function $y = -\frac{1}{6} x^2$.

Understanding the Coefficient $a$

The coefficient $a$ plays a crucial role in determining the shape and direction of a parabola. If $a$ is positive, the parabola opens upward, and if $a$ is negative, the parabola opens downward. In the given function $y = -\frac{1}{6} x^2$, the coefficient $a$ is negative, which means the parabola will open downward.

Symmetry of the Parabola

A parabola is symmetric about its line of symmetry, which is also known as the axis of symmetry. The line of symmetry is a vertical line that passes through the vertex of the parabola. In the case of the given function $y = -\frac{1}{6} x^2$, the line of symmetry is the y-axis, which is represented by the equation $x = 0$.

Vertex of the Parabola

The vertex of a parabola is the point where the parabola changes direction. In the case of the given function $y = -\frac{1}{6} x^2$, the vertex is the point where the parabola intersects the y-axis. To find the vertex, we can use the formula $x = -\frac{b}{2a}$. In this case, $a = -\frac{1}{6}$ and $b = 0$, so the vertex is at the point $(0, 0)$.

Graph of the Parabola

The graph of the parabola formed by the given function $y = -\frac{1}{6} x^2$ is a downward-facing parabola that is symmetric about the y-axis. The vertex of the parabola is at the point $(0, 0)$, and the parabola opens downward.

Conclusion

In conclusion, the shape and direction of the parabola formed by the given function $y = -\frac{1}{6} x^2$ are determined by the value of the coefficient $a$. Since $a$ is negative, the parabola opens downward, and the line of symmetry is the y-axis. The vertex of the parabola is at the point $(0, 0)$, and the parabola is symmetric about the y-axis.

Mathematical Representation

The given function $y = -\frac{1}{6} x^2$ can be represented mathematically as a quadratic equation in the form of $y = ax^2 + bx + c$. In this case, $a = -\frac{1}{6}$, $b = 0$, and $c = 0$. The graph of the parabola formed by this equation is a downward-facing parabola that is symmetric about the y-axis.

Graphical Representation

The graph of the parabola formed by the given function $y = -\frac{1}{6} x^2$ can be represented graphically as a downward-facing parabola that is symmetric about the y-axis. The vertex of the parabola is at the point $(0, 0)$, and the parabola opens downward.

Real-World Applications

The parabola formed by the given function $y = -\frac{1}{6} x^2$ has several real-world applications. For example, the trajectory of a projectile under the influence of gravity can be represented by a parabola. The parabola can also be used to model the motion of a pendulum or a spring-mass system.

Conclusion

In conclusion, the shape and direction of the parabola formed by the given function $y = -\frac{1}{6} x^2$ are determined by the value of the coefficient $a$. Since $a$ is negative, the parabola opens downward, and the line of symmetry is the y-axis. The vertex of the parabola is at the point $(0, 0)$, and the parabola is symmetric about the y-axis. The parabola has several real-world applications, including modeling the motion of a projectile or a pendulum.

References

• [1] "Parabola" by Math Open Reference. Retrieved from https://www.mathopenref.com/parabola.html
• [2] "Quadratic Equations" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f-quadratic-equations/x2f-quadratic-equations-article
• [3] "Graphing Quadratic Equations" by Purplemath. Retrieved from https://www.purplemath.com/modules/graphquad.htm

Further Reading

• "Introduction to Algebra" by Michael Artin
• "Calculus" by Michael Spivak
• "Differential Equations" by Morris Tenenbaum
  Determine the Shape and Direction of the Parabola Formed by the Given Function: Q&A ====================================================================================

Introduction

In our previous article, we discussed how to determine the shape and direction of a parabola formed by a given function. In this article, we will provide a Q&A section to help you better understand the concepts and solve problems related to parabolas.

Q: What is a parabola?

A: A parabola is a type of quadratic curve that can be represented by a quadratic equation in the form of $y = ax^2 + bx + c$. The shape and direction of a parabola are determined by the value of the coefficient $a$.

Q: How do I determine the shape and direction of a parabola?

A: To determine the shape and direction of a parabola, you need to look at the value of the coefficient $a$. If $a$ is positive, the parabola opens upward, and if $a$ is negative, the parabola opens downward.

Q: What is the line of symmetry of a parabola?

A: The line of symmetry of a parabola is a vertical line that passes through the vertex of the parabola. It is also known as the axis of symmetry.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you can use the formula $x = -\frac{b}{2a}$. This will give you the x-coordinate of the vertex. To find the y-coordinate, you can plug the x-coordinate into the equation of the parabola.

Q: What is the difference between a parabola and a circle?

A: A parabola is a type of quadratic curve that opens upward or downward, while a circle is a type of conic section that is symmetrical about its center.

Q: Can a parabola have a negative coefficient $a$?

A: Yes, a parabola can have a negative coefficient $a$. In this case, the parabola will open downward.

Q: Can a parabola have a zero coefficient $a$?

A: No, a parabola cannot have a zero coefficient $a$. If $a$ is zero, the equation is not a quadratic equation, and it does not represent a parabola.

Q: How do I graph a parabola?

A: To graph a parabola, you can use the equation of the parabola and plot points on a coordinate plane. You can also use a graphing calculator or a computer program to graph the parabola.

Q: What are some real-world applications of parabolas?

A: Parabolas have several real-world applications, including modeling the motion of a projectile, a pendulum, or a spring-mass system. They are also used in engineering to design curves and surfaces.

Conclusion

In conclusion, parabolas are an important concept in mathematics and have several real-world applications. By understanding the shape and direction of a parabola, you can solve problems and model real-world situations. We hope this Q&A section has helped you better understand the concepts and solve problems related to parabolas.

References

• [1] "Parabola" by Math Open Reference. Retrieved from https://www.mathopenref.com/parabola.html
• [2] "Quadratic Equations" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f-quadratic-equations/x2f-quadratic-equations-article
• [3] "Graphing Quadratic Equations" by Purplemath. Retrieved from https://www.purplemath.com/modules/graphquad.htm

Further Reading

• "Introduction to Algebra" by Michael Artin
• "Calculus" by Michael Spivak
• "Differential Equations" by Morris Tenenbaum