Identifying the Vertex of the Parabola

Oct 13, 2025 by miodeya.com 131 views

Introduction

In mathematics, a parabola is a fundamental concept that plays a crucial role in various fields, including algebra, geometry, and calculus. The given function, ${$ h(x) = 2(x-3)^2 - 9 $}$, represents a parabola in the form of a quadratic equation. In this article, we will delve into the properties of this parabola, including identifying the vertex, determining the direction of opening, and calculating the y-intercept.

Identifying the Vertex of the Parabola

The vertex of a parabola is the highest or lowest point on the curve, which is also known as the turning point. To identify the vertex of the given parabola, we need to rewrite the function in the standard form, ${$ h(x) = a(x-h)^2 + k $}$, where ${$ (h,k) $}$ represents the coordinates of the vertex.

Given the function ${$ h(x) = 2(x-3)^2 - 9 $}$, we can rewrite it as:

Q&A: Frequently Asked Questions about Parabolas

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the highest or lowest point on the curve, which is also known as the turning point.

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

A: To find the vertex of a parabola, you need to rewrite the function in the standard form, [$ h(x) = a(x-h)^2 + k $}$, where ${$ (h,k) $}$ represents the coordinates of the vertex.

Q: What is the standard form of a parabola?

A: The standard form of a parabola is ${$ h(x) = a(x-h)^2 + k $}$, where ${$ (h,k) $}$ represents the coordinates of the vertex.

Q: How do I determine if a parabola opens upwards or downwards?

A: To determine if a parabola opens upwards or downwards, you need to look at the coefficient of the squared term. If the coefficient is positive, the parabola opens upwards. If the coefficient is negative, the parabola opens downwards.

Q: What is the y-intercept of a parabola?

A: The y-intercept of a parabola is the point where the parabola intersects the y-axis. To find the y-intercept, you need to set x = 0 and solve for y.

Q: How do I calculate the y-intercept of a parabola?

A: To calculate the y-intercept of a parabola, you need to set x = 0 and solve for y. This can be done by substituting x = 0 into the equation of the parabola and solving for y.

Q: What is the significance of the vertex of a parabola?

A: The vertex of a parabola is significant because it represents the maximum or minimum point of the parabola. This point is also known as the turning point.

Q: How do I use the vertex of a parabola to graph the parabola?

A: To use the vertex of a parabola to graph the parabola, you need to plot the vertex and then use the equation of the parabola to find other points on the curve.

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

A: Parabolas have many real-world applications, including the design of satellite dishes, the trajectory of projectiles, and the shape of mirrors and lenses.

Conclusion

In conclusion, parabolas are an important concept in mathematics that have many real-world applications. By understanding the properties of parabolas, including the vertex, direction of opening, and y-intercept, you can use them to model and solve problems in a variety of fields.

Glossary of Terms

Vertex: The highest or lowest point on a parabola.
Standard form: The form of a parabola's equation, ${$ h(x) = a(x-h)^2 + k $}$.
Y-intercept: The point where a parabola intersects the y-axis.
Turning point: The point on a parabola where the curve changes direction.

References

[1] "Parabolas" by Math Open Reference. Retrieved from https://www.mathopenref.com/parabola.html
[2] "Vertex Form of a Parabola" by Purplemath. Retrieved from https://www.purplemath.com/modules/parabola/vertexform.htm
[3] "Y-Intercept of a Parabola" by Math Is Fun. Retrieved from https://www.mathisfun.com/algebra/parabolas-y-intercept.html