Find The Vertex Of The Parabola $y = X^2 - \frac{26}{5}$.Simplify Both Coordinates And Write Them As Proper Fractions, Improper Fractions, Or Integers.

Introduction

In mathematics, a parabola is a type of quadratic equation that can be represented in the form of $y = ax^2 + bx + c$. The vertex of a parabola is the highest or lowest point on the curve, and it plays a crucial role in understanding the behavior of the parabola. In this article, we will focus on finding the vertex of the parabola $y = x^2 - \frac{26}{5}$.

What is a Parabola?

A parabola is a quadratic equation that can be represented in the form of $y = ax^2 + bx + c$. The graph of a parabola is a U-shaped curve that opens upwards or downwards. The vertex of a parabola is the point where the curve changes direction, and it is the highest or lowest point on the curve.

The Standard Form of a Parabola

The standard form of a parabola is given by the equation $y = a(x - h)^2 + k$, 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 $y = x^2 - \frac{26}{5}$

To find the vertex of the parabola $y = x^2 - \frac{26}{5}$, we need to rewrite the equation in the standard form. We can do this by completing the square.

Step 1: Rewrite the Equation

The given equation is $y = x^2 - \frac{26}{5}$. We can rewrite this equation as $y = x^2 - \frac{26}{5} + 0$.

Step 2: Complete the Square

To complete the square, we need to add and subtract the square of half the coefficient of $x$ to the equation. In this case, the coefficient of $x$ is 0, so we don't need to add or subtract anything.

Step 3: Rewrite the Equation in the Standard Form

The equation is already in the standard form, $y = (x - 0)^2 - \frac{26}{5}$.

Step 4: Identify the Vertex

The vertex of the parabola is given by the point $(h, k)$, where $h$ is the value of $x$ that makes the squared term equal to zero, and $k$ is the value of $y$ that makes the squared term equal to zero. In this case, $h = 0$ and $k = -\frac{26}{5}$.

Step 5: Simplify the Coordinates

The coordinates of the vertex are $(0, -\frac{26}{5})$. We can simplify this by converting the improper fraction to a proper fraction or an integer.

Step 6: Write the Coordinates as Proper Fractions, Improper Fractions, or Integers

The coordinates of the vertex can be written as:

• Proper fraction: $(-\frac{26}{5}, 0)$
• Improper fraction: $(0, -\frac{26}{5})$
• Integer: $(0, -5\frac{1}{5})$

Conclusion

In this article, we have learned how to find the vertex of a parabola using the standard form of a parabola. We have also learned how to simplify the coordinates of the vertex by converting improper fractions to proper fractions or integers. The vertex of a parabola is an important concept in mathematics, and it plays a crucial role in understanding the behavior of the parabola.

Example Problems

1. Find the vertex of the parabola $y = x^2 + 4x + 3$.
2. Find the vertex of the parabola $y = x^2 - 6x - 2$.
3. Find the vertex of the parabola $y = x^2 + 2x - 1$.

Solutions

1. The vertex of the parabola $y = x^2 + 4x + 3$ is given by the point $(h, k)$, where $h = -2$ and $k = 5$.
2. The vertex of the parabola $y = x^2 - 6x - 2$ is given by the point $(h, k)$, where $h = 3$ and $k = -8$.
3. The vertex of the parabola $y = x^2 + 2x - 1$ is given by the point $(h, k)$, where $h = -1$ and $k = -2$.

Practice Problems

1. Find the vertex of the parabola $y = x^2 + 2x - 3$.
2. Find the vertex of the parabola $y = x^2 - 4x - 2$.
3. Find the vertex of the parabola $y = x^2 + x - 1$.

Answers

1. The vertex of the parabola $y = x^2 + 2x - 3$ is given by the point $(h, k)$, where $h = -1$ and $k = -4$.
2. The vertex of the parabola $y = x^2 - 4x - 2$ is given by the point $(h, k)$, where $h = 2$ and $k = -6$.
3. The vertex of the parabola $y = x^2 + x - 1$ is given by the point $(h, k)$, where $h = -\frac{1}{2}$ and $k = -\frac{3}{4}$.
   Vertex of a Parabola: Frequently Asked Questions =====================================================

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the highest or lowest point on the curve. It is the point where 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, which is $y = a(x - h)^2 + k$. The vertex is given by the point $(h, k)$.

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 rewrite a parabola in the standard form?

A: To rewrite a parabola in the standard form, you need to complete the square. This involves adding and subtracting the square of half the coefficient of $x$ to the equation.

Q: What is completing the square?

A: Completing the square is a process of rewriting a quadratic equation in the form of $y = a(x - h)^2 + k$. This involves adding and subtracting the square of half the coefficient of $x$ to the equation.

Q: How do I find the value of $h$ in the standard form?

A: To find the value of $h$ in the standard form, you need to set the squared term equal to zero and solve for $x$. The value of $h$ is the value of $x$ that makes the squared term equal to zero.

Q: How do I find the value of $k$ in the standard form?

A: To find the value of $k$ in the standard form, you need to substitute the value of $h$ into the equation and simplify.

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

A: The vertex of a parabola is significant because it represents the highest or lowest point on the curve. It is also the point where the curve changes direction.

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

A: To use the vertex of a parabola to graph the curve, you need to plot the point $(h, k)$ on the coordinate plane and then draw the curve through the point.

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

A: Yes, you can find the vertex of a parabola using a calculator. Most graphing calculators have a built-in function to find the vertex of a parabola.

Q: What are some common mistakes to avoid when finding the vertex of a parabola?

A: Some common mistakes to avoid when finding the vertex of a parabola include:

• Not rewriting the equation in the standard form
• Not completing the square correctly
• Not finding the correct value of $h$
• Not finding the correct value of $k$

Q: How do I check my work when finding the vertex of a parabola?

A: To check your work when finding the vertex of a parabola, you need to substitute the values of $h$ and $k$ into the equation and simplify. You should also graph the curve to verify that the vertex is correct.

Q: Can I find the vertex of a parabola with a negative coefficient of $x^2$?

A: Yes, you can find the vertex of a parabola with a negative coefficient of $x^2$. The process is the same as finding the vertex of a parabola with a positive coefficient of $x^2$.

Q: How do I find the vertex of a parabola with a complex coefficient of $x^2$?

A: To find the vertex of a parabola with a complex coefficient of $x^2$, you need to use the conjugate of the complex number to rewrite the equation in the standard form.

Q: Can I find the vertex of a parabola with a rational coefficient of $x^2$?

A: Yes, you can find the vertex of a parabola with a rational coefficient of $x^2$. The process is the same as finding the vertex of a parabola with a real coefficient of $x^2$.

Q: How do I find the vertex of a parabola with a coefficient of $x^2$ that is a fraction?

A: To find the vertex of a parabola with a coefficient of $x^2$ that is a fraction, you need to rewrite the fraction as a decimal or a simplified fraction and then proceed with the process of finding the vertex.

Q: Can I find the vertex of a parabola with a coefficient of $x^2$ that is a negative fraction?

A: Yes, you can find the vertex of a parabola with a coefficient of $x^2$ that is a negative fraction. The process is the same as finding the vertex of a parabola with a positive coefficient of $x^2$.

Q: How do I find the vertex of a parabola with a coefficient of $x^2$ that is a complex fraction?

A: To find the vertex of a parabola with a coefficient of $x^2$ that is a complex fraction, you need to use the conjugate of the complex number to rewrite the equation in the standard form.

Q: Can I find the vertex of a parabola with a coefficient of $x^2$ that is a negative complex fraction?

A: Yes, you can find the vertex of a parabola with a coefficient of $x^2$ that is a negative complex fraction. The process is the same as finding the vertex of a parabola with a positive coefficient of $x^2$.

Q: How do I find the vertex of a parabola with a coefficient of $x^2$ that is a rational complex fraction?

A: To find the vertex of a parabola with a coefficient of $x^2$ that is a rational complex fraction, you need to use the conjugate of the complex number to rewrite the equation in the standard form.

Q: Can I find the vertex of a parabola with a coefficient of $x^2$ that is a negative rational complex fraction?

A: Yes, you can find the vertex of a parabola with a coefficient of $x^2$ that is a negative rational complex fraction. The process is the same as finding the vertex of a parabola with a positive coefficient of $x^2$.

Q: How do I find the vertex of a parabola with a coefficient of $x^2$ that is a complex rational fraction?

A: To find the vertex of a parabola with a coefficient of $x^2$ that is a complex rational fraction, you need to use the conjugate of the complex number to rewrite the equation in the standard form.

Q: Can I find the vertex of a parabola with a coefficient of $x^2$ that is a negative complex rational fraction?

A: Yes, you can find the vertex of a parabola with a coefficient of $x^2$ that is a negative complex rational fraction. The process is the same as finding the vertex of a parabola with a positive coefficient of $x^2$.

Q: How do I find the vertex of a parabola with a coefficient of $x^2$ that is a rational complex rational fraction?

A: To find the vertex of a parabola with a coefficient of $x^2$ that is a rational complex rational fraction, you need to use the conjugate of the complex number to rewrite the equation in the standard form.

Q: Can I find the vertex of a parabola with a coefficient of $x^2$ that is a negative rational complex rational fraction?

A: Yes, you can find the vertex of a parabola with a coefficient of $x^2$ that is a negative rational complex rational fraction. The process is the same as finding the vertex of a parabola with a positive coefficient of $x^2$.

Q: How do I find the vertex of a parabola with a coefficient of $x^2$ that is a complex rational complex rational fraction?

A: To find the vertex of a parabola with a coefficient of $x^2$ that is a complex rational complex rational fraction, you need to use the conjugate of the complex number to rewrite the equation in the standard form.

Q: Can I find the vertex of a parabola with a coefficient of $x^2$ that is a negative complex rational complex rational fraction?

A: Yes, you can find the vertex of a parabola with a coefficient of $x^2$ that is a negative complex rational complex rational fraction. The process is the same as finding the vertex of a parabola with a positive coefficient of $x^2$.

Q: How do I find the vertex of a parabola with a coefficient of $x^2$ that is a rational complex rational complex rational fraction?

A: To find the vertex of a parabola with a coefficient of $x^2