By Completing The Square, Work Out The Coordinates Of The Turning Point Of The Curve $y = X^2 + 12x - 5$.

Introduction

In mathematics, the process of completing the square is a powerful technique used to rewrite quadratic expressions in a specific form. This form allows us to easily identify the vertex of a parabola, which is the turning point of the curve. In this article, we will explore how to use completing the square to find the coordinates of the turning point of the curve $y = x^2 + 12x - 5$.

Understanding the Curve

The given curve is a quadratic function in the form $y = ax^2 + bx + c$, where $a = 1$, $b = 12$, and $c = -5$. The graph of this curve is a parabola that opens upwards, since the coefficient of the $x^2$ term is positive.

Completing the Square

To complete the square, we need to rewrite the quadratic expression in the form $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. The process of completing the square involves the following steps:

1. Take half of the coefficient of the $x$ term: In this case, the coefficient of the $x$ term is $12$, so we take half of this value, which is $6$.
2. Square the result: We square the value obtained in step 1, which gives us $6^2 = 36$.
3. Add and subtract the squared value: We add and subtract the squared value obtained in step 2 to the quadratic expression, which gives us $y = x^2 + 12x + 36 - 36 - 5$.
4. Write the expression as a perfect square: We can now write the expression as a perfect square, which gives us $y = (x + 6)^2 - 41$.

Finding the Coordinates of the Turning Point

Now that we have rewritten the quadratic expression in the form $y = a(x - h)^2 + k$, we can easily identify the vertex of the parabola, which is the turning point of the curve. The vertex is given by the coordinates $(h, k)$, where $h$ is the value that we added to $x$ to complete the square, and $k$ is the constant term.

In this case, we added $6$ to $x$ to complete the square, so $h = -6$. The constant term is $-41$, so $k = -41$. Therefore, the coordinates of the turning point of the curve are $(-6, -41)$.

Conclusion

In this article, we used the process of completing the square to find the coordinates of the turning point of the curve $y = x^2 + 12x - 5$. We rewrote the quadratic expression in the form $y = a(x - h)^2 + k$, which allowed us to easily identify the vertex of the parabola. The coordinates of the turning point are $(-6, -41)$.

Example

Let's consider another example to illustrate the process of completing the square. Suppose we want to find the coordinates of the turning point of the curve $y = x^2 + 8x + 12$. We can complete the square by following the same steps as before:

1. Take half of the coefficient of the $x$ term: $8/2 = 4$
2. Square the result: $4^2 = 16$
3. Add and subtract the squared value: $y = x^2 + 8x + 16 - 16 + 12$
4. Write the expression as a perfect square: $y = (x + 4)^2 - 4$

Now that we have rewritten the quadratic expression in the form $y = a(x - h)^2 + k$, we can easily identify the vertex of the parabola, which is the turning point of the curve. The vertex is given by the coordinates $(h, k)$, where $h$ is the value that we added to $x$ to complete the square, and $k$ is the constant term.

In this case, we added $4$ to $x$ to complete the square, so $h = -4$. The constant term is $-4$, so $k = -4$. Therefore, the coordinates of the turning point of the curve are $(-4, -4)$.

Applications of Completing the Square

Completing the square has many applications in mathematics and other fields. Some of the applications include:

• Graphing quadratic functions: Completing the square allows us to easily identify the vertex of a parabola, which is the turning point of the curve.
• Finding the maximum or minimum value of a quadratic function: The vertex of a parabola represents the maximum or minimum value of the quadratic function.
• Solving quadratic equations: Completing the square can be used to solve quadratic equations by rewriting them in the form $y = a(x - h)^2 + k$.
• Optimization problems: Completing the square can be used to solve optimization problems by finding the maximum or minimum value of a quadratic function.

Conclusion

In conclusion, completing the square is a powerful technique used to rewrite quadratic expressions in a specific form. This form allows us to easily identify the vertex of a parabola, which is the turning point of the curve. We used this technique to find the coordinates of the turning point of the curve $y = x^2 + 12x - 5$. We also discussed some of the applications of completing the square, including graphing quadratic functions, finding the maximum or minimum value of a quadratic function, solving quadratic equations, and optimization problems.

Introduction

Completing the square is a powerful technique used to rewrite quadratic expressions in a specific form. This form allows us to easily identify the vertex of a parabola, which is the turning point of the curve. In this article, we will answer some of the most frequently asked questions about completing the square.

Q: What is completing the square?

A: Completing the square is a technique used to rewrite quadratic expressions in the form $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Q: Why is completing the square useful?

A: Completing the square is useful because it allows us to easily identify the vertex of a parabola, which is the turning point of the curve. This is useful in graphing quadratic functions, finding the maximum or minimum value of a quadratic function, solving quadratic equations, and optimization problems.

Q: How do I complete the square?

A: To complete the square, follow these steps:

1. Take half of the coefficient of the $x$ term.
2. Square the result.
3. Add and subtract the squared value to the quadratic expression.
4. Write the expression as a perfect square.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the turning point of the curve. It is given by the coordinates $(h, k)$, where $h$ is the value that we added to $x$ to complete the square, and $k$ is the constant term.

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

A: To find the coordinates of the vertex of a parabola, complete the square and identify the values of $h$ and $k$.

Q: Can I use completing the square to solve quadratic equations?

A: Yes, completing the square can be used to solve quadratic equations by rewriting them in the form $y = a(x - h)^2 + k$.

Q: What are some common mistakes to avoid when completing the square?

A: Some common mistakes to avoid when completing the square include:

• Not taking half of the coefficient of the $x$ term.
• Not squaring the result.
• Not adding and subtracting the squared value to the quadratic expression.
• Not writing the expression as a perfect square.

Q: Can I use completing the square to graph quadratic functions?

A: Yes, completing the square can be used to graph quadratic functions by identifying the vertex of the parabola.

Q: What are some real-world applications of completing the square?

A: Some real-world applications of completing the square include:

• Graphing quadratic functions to model real-world situations.
• Finding the maximum or minimum value of a quadratic function to optimize a process.
• Solving quadratic equations to find the solution to a problem.
• Using completing the square to solve optimization problems.

Q: Can I use completing the square to solve systems of equations?

A: Yes, completing the square can be used to solve systems of equations by rewriting the equations in the form $y = a(x - h)^2 + k$.

Q: What are some tips for mastering completing the square?

A: Some tips for mastering completing the square include:

• Practicing, practicing, practicing!
• Understanding the concept of completing the square.
• Using visual aids to help you understand the process.
• Breaking down the process into smaller steps.

Conclusion

In conclusion, completing the square is a powerful technique used to rewrite quadratic expressions in a specific form. This form allows us to easily identify the vertex of a parabola, which is the turning point of the curve. We answered some of the most frequently asked questions about completing the square, including how to complete the square, what is the vertex of a parabola, and how to find the coordinates of the vertex of a parabola. We also discussed some common mistakes to avoid when completing the square and some real-world applications of completing the square.