Find The Vertex Of The Parabola Y = X 2 − 33 4 Y = X^2 - \frac{33}{4} Y = X 2 − 4 33 ​ .Simplify Both Coordinates And Write Them As Proper Fractions, Improper Fractions, Or Integers. □ \square □ , □ \square □

Introduction

In mathematics, a parabola is a type of quadratic equation that can be represented in the form $y = ax^2 + bx + c$. The vertex of a parabola is the point at which the parabola 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 $y = x^2 - \frac{33}{4}$.

What is the Vertex of a Parabola?

The vertex of a parabola is the point at which the parabola changes direction. It is the minimum or maximum point of the parabola, depending on the direction of the parabola. The vertex is represented by the coordinates $(h, k)$, where $h$ is the x-coordinate and $k$ is the y-coordinate.

How to Find the Vertex of a Parabola

To find the vertex of a parabola, we can use the formula $h = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic equation. In the case of the parabola $y = x^2 - \frac{33}{4}$, we have $a = 1$ and $b = 0$. Plugging these values into the formula, we get:

$h = -\frac{0}{2(1)} = 0$

Finding the Y-Coordinate of the Vertex

Now that we have the x-coordinate of the vertex, we can find the y-coordinate by plugging the value of $h$ into the original equation. In this case, we have:

$y = (0)^2 - \frac{33}{4} = -\frac{33}{4}$

Simplifying the Coordinates

The coordinates of the vertex are $(0, -\frac{33}{4})$. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 1. Therefore, the simplified coordinates are still $(0, -\frac{33}{4})$.

Writing the Coordinates as Improper Fractions

We can also write the coordinates as improper fractions by dividing the numerator by the denominator. In this case, we have:

$-\frac{33}{4} = -\frac{33 \div 4}{1 \div 4} = -\frac{33}{4}$

Writing the Coordinates as Integers

Finally, we can write the coordinates as integers by dividing the numerator and the denominator by their greatest common divisor. In this case, we have:

$-\frac{33}{4} = -\frac{33 \div 1}{4 \div 1} = -\frac{33}{4}$

Conclusion

In conclusion, the vertex of the parabola $y = x^2 - \frac{33}{4}$ is the point $(0, -\frac{33}{4})$. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, and we can also write the coordinates as improper fractions or integers.

Key Takeaways

• The vertex of a parabola is the point at which the parabola changes direction.
• The vertex is represented by the coordinates $(h, k)$, where $h$ is the x-coordinate and $k$ is the y-coordinate.
• To find the vertex of a parabola, we can use the formula $h = -\frac{b}{2a}$.
• We can simplify the coordinates of the vertex by dividing both the numerator and the denominator by their greatest common divisor.
• We can also write the coordinates as improper fractions or integers.

Frequently Asked Questions

• Q: What is the vertex of the parabola $y = x^2 - \frac{33}{4}$? A: The vertex of the parabola $y = x^2 - \frac{33}{4}$ is the point $(0, -\frac{33}{4})$.
• Q: How do I find the vertex of a parabola? A: To find the vertex of a parabola, we can use the formula $h = -\frac{b}{2a}$.
• Q: Can I simplify the coordinates of the vertex? A: Yes, we can simplify the coordinates of the vertex by dividing both the numerator and the denominator by their greatest common divisor.

References

• [1] Khan Academy. (n.d.). Vertex Form of a Parabola. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f7c/x2f6f7d/x2f6f7e
• [2] Math Open Reference. (n.d.). Parabola. Retrieved from https://www.mathopenref.com/parabola.html

Additional Resources

• [1] Algebra.com. (n.d.). Vertex of a Parabola. Retrieved from https://www.algebra.com/algebra/homework/vertex/vertex.faq.html
• [2] Purplemath. (n.d.). Vertex Form of a Parabola. Retrieved from https://www.purplemath.com/modules/parabola.htm
  Vertex of a Parabola: Frequently Asked Questions =====================================================

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point at which the parabola changes direction. It is the minimum or maximum point of the parabola, depending on the direction of the parabola.

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

A: To find the vertex of a parabola, you can use the formula $h = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic equation.

Q: What is the formula for finding the vertex of a parabola?

A: The formula for finding the vertex of a parabola is $h = -\frac{b}{2a}$.

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

A: To use the formula, you need to identify the values of $a$ and $b$ in the quadratic equation. Then, plug these values into the formula and simplify to find the x-coordinate of the vertex.

Q: Can I find the y-coordinate of the vertex using the formula?

A: No, the formula only gives you the x-coordinate of the vertex. To find the y-coordinate, you need to plug the x-coordinate into the original equation.

Q: How do I find the y-coordinate of the vertex?

A: To find the y-coordinate of the vertex, plug the x-coordinate into the original equation and simplify.

Q: Can I simplify the coordinates of the vertex?

A: Yes, you can simplify the coordinates of the vertex by dividing both the numerator and the denominator by their greatest common divisor.

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

A: To simplify the coordinates of the vertex, divide both the numerator and the denominator by their greatest common divisor.

Q: Can I write the coordinates of the vertex as improper fractions?

A: Yes, you can write the coordinates of the vertex as improper fractions by dividing the numerator by the denominator.

Q: How do I write the coordinates of the vertex as improper fractions?

A: To write the coordinates of the vertex as improper fractions, divide the numerator by the denominator.

Q: Can I write the coordinates of the vertex as integers?

A: Yes, you can write the coordinates of the vertex as integers by dividing the numerator and the denominator by their greatest common divisor.

Q: How do I write the coordinates of the vertex as integers?

A: To write the coordinates of the vertex as integers, divide the numerator and the denominator by their greatest common divisor.

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

A: The vertex of a parabola is significant because it represents the minimum or maximum point of the parabola, depending on the direction of the parabola.

Q: How do I use the vertex of a parabola in real-world applications?

A: The vertex of a parabola can be used in real-world applications such as designing roller coasters, calculating the trajectory of a projectile, and modeling population growth.

Q: Can I find the vertex of a parabola using a graphing calculator?

A: Yes, you can find the vertex of a parabola using a graphing calculator by graphing the equation and using the calculator's built-in features to find the vertex.

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

A: To find the vertex of a parabola using a graphing calculator, graph the equation and use the calculator's built-in features to find the vertex.

Q: Can I find the vertex of a parabola using a computer algebra system?

A: Yes, you can find the vertex of a parabola using a computer algebra system by entering the equation and using the system's built-in features to find the vertex.

Q: How do I find the vertex of a parabola using a computer algebra system?

A: To find the vertex of a parabola using a computer algebra system, enter the equation and use the system's built-in features to find the vertex.

Conclusion

In conclusion, the vertex of a parabola is a crucial concept in algebra and calculus. It represents the minimum or maximum point of the parabola, depending on the direction of the parabola. We have discussed how to find the vertex of a parabola using the formula $h = -\frac{b}{2a}$, and how to simplify the coordinates of the vertex. We have also discussed how to use the vertex of a parabola in real-world applications and how to find the vertex using a graphing calculator and a computer algebra system.

Key Takeaways

• The vertex of a parabola is the point at which the parabola changes direction.
• The vertex is represented by the coordinates $(h, k)$, where $h$ is the x-coordinate and $k$ is the y-coordinate.
• To find the vertex of a parabola, use the formula $h = -\frac{b}{2a}$.
• You can simplify the coordinates of the vertex by dividing both the numerator and the denominator by their greatest common divisor.
• You can write the coordinates of the vertex as improper fractions or integers.
• The vertex of a parabola can be used in real-world applications such as designing roller coasters, calculating the trajectory of a projectile, and modeling population growth.

Frequently Asked Questions

• Q: What is the vertex of a parabola? A: The vertex of a parabola is the point at which the parabola changes direction.
• Q: How do I find the vertex of a parabola? A: To find the vertex of a parabola, use the formula $h = -\frac{b}{2a}$.
• Q: Can I simplify the coordinates of the vertex? A: Yes, you can simplify the coordinates of the vertex by dividing both the numerator and the denominator by their greatest common divisor.

References

• [1] Khan Academy. (n.d.). Vertex Form of a Parabola. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f7c/x2f6f7d/x2f6f7e
• [2] Math Open Reference. (n.d.). Parabola. Retrieved from https://www.mathopenref.com/parabola.html

Additional Resources

• [1] Algebra.com. (n.d.). Vertex of a Parabola. Retrieved from https://www.algebra.com/algebra/homework/vertex/vertex.faq.html
• [2] Purplemath. (n.d.). Vertex Form of a Parabola. Retrieved from https://www.purplemath.com/modules/parabola.htm