Find The Vertex Of The Parabola $y = X^2 + X$.Simplify Both Coordinates And Write Them As Proper Fractions.

Introduction

In mathematics, a parabola is a quadratic curve that can be represented by the equation $y = ax^2 + bx + c$. The vertex of a parabola is the point at which the curve changes direction, and it is a crucial concept in algebra and calculus. In this article, we will focus on finding the vertex of the parabola $y = x^2 + x$ and simplify both coordinates to write them as proper fractions.

What is a Parabola?

A parabola is a quadratic curve that can be represented by the equation $y = ax^2 + bx + c$. The graph of a parabola is a U-shaped curve that opens upwards or downwards. The vertex of a parabola is the point at which the curve changes direction, and it is the minimum or maximum point of the curve.

The Standard Form of a Parabola

The standard form of a parabola is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. To find the vertex of a parabola, we need to rewrite the equation in the standard form.

Finding the Vertex of the Parabola $y = x^2 + x$

To find the vertex of the parabola $y = x^2 + x$, we need to rewrite the equation in the standard form. We can do this by completing the square.

Step 1: Complete the Square

To complete the square, we need to add and subtract $(b/2)^2$ to the equation.

y = x^2 + x
y = x^2 + x + (1/4) - (1/4)
y = (x + 1/2)^2 - (1/4)

Step 2: Rewrite the Equation in the Standard Form

Now that we have completed the square, we can rewrite the equation in the standard form.

y = (x + 1/2)^2 - (1/4)
y = (x + 1/2)^2 - 1/4

Step 3: Identify the Vertex

The vertex of the parabola is the point at which the curve changes direction. In the standard form of a parabola, the vertex is represented by the point $(h, k)$. In this case, the vertex is $(h, k) = (-1/2, -1/4)$.

Simplifying the Coordinates

The coordinates of the vertex are $(-1/2, -1/4)$. We can simplify these coordinates to write them as proper fractions.

Step 1: Simplify the x-Coordinate

The x-coordinate of the vertex is $-1/2$. This is already a proper fraction.

Step 2: Simplify the y-Coordinate

The y-coordinate of the vertex is $-1/4$. This is also already a proper fraction.

Conclusion

In this article, we have found the vertex of the parabola $y = x^2 + x$ and simplified both coordinates to write them as proper fractions. The vertex of the parabola is $(-1/2, -1/4)$. We have also discussed the concept of a parabola and the standard form of a parabola. We hope that this article has provided a clear and concise explanation of how to find the vertex of a parabola.

Frequently Asked Questions

Q: What is a parabola?

A: A parabola is a quadratic curve that can be represented by the equation $y = ax^2 + bx + c$.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point at which the curve changes direction.

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

A: To find the vertex of a parabola, you need to rewrite the equation in the standard form by completing the square.

Q: What is the standard form of a parabola?

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

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for the Nonmathematician" by Morris Kline

Glossary

• Parabola: A quadratic curve that can be represented by the equation $y = ax^2 + bx + c$.
• Vertex: The point at which the curve changes direction.
• Standard form: The standard form of a parabola is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.
• Completing the square: A method of rewriting the equation of a parabola in the standard form by adding and subtracting $(b/2)^2$ to the equation.
  Vertex of a Parabola: Frequently Asked Questions =====================================================

Introduction

In our previous article, we discussed how to find the vertex of a parabola and simplified both coordinates to write them as proper fractions. In this article, we will answer some of the most frequently asked questions about the vertex of a parabola.

Q&A

Q: What is a parabola?

A: A parabola is a quadratic curve that can be represented by the equation $y = ax^2 + bx + c$.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point at which the curve changes direction.

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

A: To find the vertex of a parabola, you need to rewrite the equation in the standard form by completing the square.

Q: What is the standard form of a parabola?

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

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

A: To complete the square, you need to add and subtract $(b/2)^2$ to the equation. This will allow you to rewrite the equation in the standard form.

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

A: The vertex of a parabola is the point at which the curve changes direction. It is also the minimum or maximum point of the curve.

Q: Can the vertex of a parabola be negative?

A: Yes, the vertex of a parabola can be negative. In fact, the vertex of a parabola can be any real number.

Q: Can the vertex of a parabola be a fraction?

A: Yes, the vertex of a parabola can be a fraction. In fact, the vertex of a parabola can be any real number, including fractions.

Q: How do I simplify the coordinates of the vertex of a parabola?

A: To simplify the coordinates of the vertex of a parabola, you need to simplify the x and y coordinates separately. You can do this by dividing both numbers by their greatest common divisor.

Q: What is the greatest common divisor (GCD) of two numbers?

A: The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder.

Q: How do I find the GCD of two numbers?

A: To find the GCD of two numbers, you can use the Euclidean algorithm. This algorithm involves repeatedly dividing the larger number by the smaller number and taking the remainder.

Q: What is the Euclidean algorithm?

A: The Euclidean algorithm is a method for finding the greatest common divisor (GCD) of two numbers. It involves repeatedly dividing the larger number by the smaller number and taking the remainder.

Conclusion

In this article, we have answered some of the most frequently asked questions about the vertex of a parabola. We hope that this article has provided a clear and concise explanation of the concept of the vertex of a parabola and how to find it.

Frequently Asked Questions (FAQs)

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point at which the curve changes direction.

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

A: To find the vertex of a parabola, you need to rewrite the equation in the standard form by completing the square.

Q: What is the standard form of a parabola?

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

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

A: To complete the square, you need to add and subtract $(b/2)^2$ to the equation. This will allow you to rewrite the equation in the standard form.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for the Nonmathematician" by Morris Kline

Glossary

• Parabola: A quadratic curve that can be represented by the equation $y = ax^2 + bx + c$.
• Vertex: The point at which the curve changes direction.
• Standard form: The standard form of a parabola is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.
• Completing the square: A method of rewriting the equation of a parabola in the standard form by adding and subtracting $(b/2)^2$ to the equation.
• Greatest common divisor (GCD): The largest number that divides both numbers without leaving a remainder.
• Euclidean algorithm: A method for finding the greatest common divisor (GCD) of two numbers.