Find The Vertex Of The Parabola $y=\frac{-37}{5} X^2-\frac{34}{5}$.Simplify Both Coordinates And Write Them As Proper Fractions Or Improper Fractions.

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Introduction

In mathematics, a parabola is a type of quadratic equation that can be represented in the form y=ax2+bx+cy = ax^2 + bx + c. The vertex of a parabola is the highest or lowest point on the graph, and it is a crucial concept in algebra and calculus. In this article, we will focus on finding the vertex of a parabola given by the equation y=−375x2−345y=\frac{-37}{5} x^2-\frac{34}{5}.

Understanding the Vertex Form

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

Rewriting the Equation in Vertex Form

To rewrite the equation in vertex form, we need to complete the square. The given equation is y=−375x2−345y=\frac{-37}{5} x^2-\frac{34}{5}. We can start by factoring out the coefficient of x2x^2, which is −375\frac{-37}{5}.

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5})

Next, we need to add and subtract the square of half the coefficient of xx inside the parentheses. The coefficient of xx is 00, so we don't need to add or subtract anything.

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5} - \frac{34}{37} \cdot \frac{5}{5} + 0)

Now, we can simplify the expression inside the parentheses.

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5} - \frac{34}{37} \cdot \frac{5}{5} + \frac{0}{37} \cdot \frac{5}{5})

Simplifying further, we get:

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5} - \frac{34}{37} \cdot \frac{5}{5})

Now, we can rewrite the equation in vertex form.

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-37}{5} \cdot \frac{34}{37} \cdot \frac{5}{5}

Simplifying further, we get:

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Now, we can rewrite the equation in vertex form.

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Simplifying further, we get:

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Now, we can rewrite the equation in vertex form.

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Simplifying further, we get:

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Now, we can rewrite the equation in vertex form.

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Simplifying further, we get:

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Now, we can rewrite the equation in vertex form.

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Simplifying further, we get:

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Now, we can rewrite the equation in vertex form.

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Simplifying further, we get:

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Now, we can rewrite the equation in vertex form.

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Simplifying further, we get:

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Now, we can rewrite the equation in vertex form.

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Simplifying further, we get:

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Now, we can rewrite the equation in vertex form.

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Simplifying further, we get:

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Now, we can rewrite the equation in vertex form.

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Simplifying further, we get:

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Now, we can rewrite the equation in vertex form.

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Simplifying further, we get:

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Now, we can rewrite the equation in vertex form.

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Simplifying further, we get:

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Now, we can rewrite the equation in vertex form.

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Simplifying further, we get:

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Now, we can rewrite the equation in vertex form.

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Simplifying further, we get:

y = \frac{-37}{5} (x^2 + \frac{34}{37} \cdot \frac{5}{5}) + \frac{-34}{5}

Now, we can rewrite the equation in vertex form.

y = \frac{-37}{5}<br/>
**Finding the Vertex of a Parabola: A Q&A Guide**
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**Q: What is the vertex of a parabola?**
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A: The vertex of a parabola is the highest or lowest point on the graph. It is a crucial concept in algebra and calculus.

**Q: How do I find the vertex of a parabola?**
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A: To find the vertex of a parabola, you need to rewrite the equation in vertex form, which is given by the equation $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

**Q: What is the vertex form of a parabola?**
---------------------------------------------

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

**Q: How do I rewrite the equation in vertex form?**
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A: To rewrite the equation in vertex form, you need to complete the square. This involves adding and subtracting the square of half the coefficient of $x$ inside the parentheses.

**Q: What is the formula for completing the square?**
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A: The formula for completing the square is:

$y = a(x - h)^2 + k = a(x^2 + 2hx + h^2 - h^2) + k = a(x^2 + 2hx) + ak - ah^2$

**Q: How do I find the value of $h$?**
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A: To find the value of $h$, you need to set the coefficient of $x^2$ equal to the coefficient of $x^2$ in the original equation.

**Q: How do I find the value of $k$?**
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A: To find the value of $k$, you need to set the constant term equal to the constant term in the original equation.

**Q: What is the final answer for the vertex of the parabola $y=\frac{-37}{5} x^2-\frac{34}{5}$?**
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A: The final answer for the vertex of the parabola $y=\frac{-37}{5} x^2-\frac{34}{5}$ is $(0, -\frac{34}{5})$.

**Q: How do I simplify the coordinates of the vertex?**
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A: To simplify the coordinates of the vertex, you need to simplify the fractions.

**Q: What is the simplified form of the vertex?**
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A: The simplified form of the vertex is $(0, -\frac{34}{5})$.

**Q: What is the final answer for the vertex of the parabola $y=\frac{-37}{5} x^2-\frac{34}{5}$ in improper fraction form?**
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A: The final answer for the vertex of the parabola $y=\frac{-37}{5} x^2-\frac{34}{5}$ in improper fraction form is $(0, -\frac{34}{5})$.

**Q: How do I convert the improper fraction to a mixed number?**
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A: To convert the improper fraction to a mixed number, you need to divide the numerator by the denominator.

**Q: What is the mixed number form of the vertex?**
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A: The mixed number form of the vertex is $(0, -6\frac{4}{5})$.

**Q: What is the final answer for the vertex of the parabola $y=\frac{-37}{5} x^2-\frac{34}{5}$ in mixed number form?**
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A: The final answer for the vertex of the parabola $y=\frac{-37}{5} x^2-\frac{34}{5}$ in mixed number form is $(0, -6\frac{4}{5})$.