Complete The Square To Rewrite The Quadratic Function In Vertex Form:$ Y = X^2 + 9x + 7 $

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Introduction

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Quadratic functions are a fundamental concept in mathematics, and understanding how to rewrite them in vertex form is crucial for solving various mathematical problems. The vertex form of a quadratic function is a powerful tool for analyzing and graphing quadratic functions. In this article, we will focus on rewriting the quadratic function $y = x^2 + 9x + 7$ in vertex form using the method of completing the square.

What is Completing the Square?

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Completing the square is a mathematical technique used to rewrite a quadratic function in a specific form, known as the vertex form. This technique involves manipulating the quadratic function to create a perfect square trinomial, which can be factored into the square of a binomial. The vertex form of a quadratic function is given by:

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

where $(h, k)$ is the vertex of the parabola.

Step 1: Identify the Coefficients

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To rewrite the quadratic function $y = x^2 + 9x + 7$ in vertex form, we need to identify the coefficients of the quadratic function. The coefficients are the numbers that multiply the variables in the function. In this case, the coefficients are:

• $a = 1$ (coefficient of $x^2$)
• $b = 9$ (coefficient of $x$)
• $c = 7$ (constant term)

Step 2: Calculate the Value of $h$

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The value of $h$ is given by the formula:

$$h = -\frac{b}{2a}$$

Substituting the values of $a$ and $b$, we get:

$$h = -\frac{9}{2(1)} = -\frac{9}{2}$$

Step 3: Calculate the Value of $k$

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The value of $k$ is given by the formula:

$$k = c - \frac{b^2}{4a}$$

Substituting the values of $a$, $b$, and $c$, we get:

$$k = 7 - \frac{(9)^2}{4(1)} = 7 - \frac{81}{4} = -\frac{55}{4}$$

Step 4: Rewrite the Quadratic Function in Vertex Form

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Now that we have the values of $h$ and $k$, we can rewrite the quadratic function in vertex form:

$$y = (x + \frac{9}{2})^2 - \frac{55}{4}$$

Conclusion

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In this article, we have shown how to rewrite the quadratic function $y = x^2 + 9x + 7$ in vertex form using the method of completing the square. We have identified the coefficients of the quadratic function, calculated the values of $h$ and $k$, and rewritten the quadratic function in vertex form. This technique is a powerful tool for analyzing and graphing quadratic functions, and it is essential for solving various mathematical problems.

Example Problems

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Problem 1

Rewrite the quadratic function $y = x^2 + 6x + 2$ in vertex form.

Solution

To rewrite the quadratic function in vertex form, we need to identify the coefficients of the quadratic function. The coefficients are:

• $a = 1$ (coefficient of $x^2$)
• $b = 6$ (coefficient of $x$)
• $c = 2$ (constant term)

The value of $h$ is given by the formula:

$$h = -\frac{b}{2a} = -\frac{6}{2(1)} = -3$$

The value of $k$ is given by the formula:

$$k = c - \frac{b^2}{4a} = 2 - \frac{(6)^2}{4(1)} = 2 - \frac{36}{4} = -7$$

Now that we have the values of $h$ and $k$, we can rewrite the quadratic function in vertex form:

$$y = (x + 3)^2 - 7$$

Problem 2

Rewrite the quadratic function $y = x^2 - 4x + 3$ in vertex form.

Solution

To rewrite the quadratic function in vertex form, we need to identify the coefficients of the quadratic function. The coefficients are:

• $a = 1$ (coefficient of $x^2$)
• $b = -4$ (coefficient of $x$)
• $c = 3$ (constant term)

The value of $h$ is given by the formula:

$$h = -\frac{b}{2a} = -\frac{-4}{2(1)} = 2$$

The value of $k$ is given by the formula:

$$k = c - \frac{b^2}{4a} = 3 - \frac{(-4)^2}{4(1)} = 3 - \frac{16}{4} = -2$$

Now that we have the values of $h$ and $k$, we can rewrite the quadratic function in vertex form:

$$y = (x - 2)^2 - 2$$

Final Thoughts

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In conclusion, rewriting quadratic functions in vertex form using the method of completing the square is a powerful tool for analyzing and graphing quadratic functions. By identifying the coefficients of the quadratic function, calculating the values of $h$ and $k$, and rewriting the quadratic function in vertex form, we can gain a deeper understanding of the properties of quadratic functions. This technique is essential for solving various mathematical problems, and it is a fundamental concept in mathematics.

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Introduction

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In our previous article, we discussed how to rewrite quadratic functions in vertex form using the method of completing the square. In this article, we will provide a Q&A guide to help you better understand the concept of quadratic functions in vertex form.

Q: What is the vertex form of a quadratic function?

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A: The vertex form of a quadratic function is given by:

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

where $(h, k)$ is the vertex of the parabola.

Q: How do I identify the coefficients of a quadratic function?

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A: To identify the coefficients of a quadratic function, you need to look at the equation and identify the numbers that multiply the variables. In the equation $y = x^2 + 9x + 7$, the coefficients are:

• $a = 1$ (coefficient of $x^2$)
• $b = 9$ (coefficient of $x$)
• $c = 7$ (constant term)

Q: How do I calculate the value of $h$?

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A: The value of $h$ is given by the formula:

$$h = -\frac{b}{2a}$$

Substituting the values of $a$ and $b$, you get:

$$h = -\frac{9}{2(1)} = -\frac{9}{2}$$

Q: How do I calculate the value of $k$?

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A: The value of $k$ is given by the formula:

$$k = c - \frac{b^2}{4a}$$

Substituting the values of $a$, $b$, and $c$, you get:

$$k = 7 - \frac{(9)^2}{4(1)} = 7 - \frac{81}{4} = -\frac{55}{4}$$

Q: How do I rewrite a quadratic function in vertex form?

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A: To rewrite a quadratic function in vertex form, you need to follow these steps:

1. Identify the coefficients of the quadratic function.
2. Calculate the value of $h$ using the formula $h = -\frac{b}{2a}$.
3. Calculate the value of $k$ using the formula $k = c - \frac{b^2}{4a}$.
4. Rewrite the quadratic function in vertex form using the formula $y = a(x - h)^2 + k$.

Q: What is the significance of the vertex form of a quadratic function?

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A: The vertex form of a quadratic function is significant because it allows us to easily identify the vertex of the parabola, which is the point where the parabola changes direction. The vertex form also makes it easier to graph the parabola and to find the maximum or minimum value of the function.

Q: Can I use the vertex form of a quadratic function to solve systems of equations?

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A: Yes, you can use the vertex form of a quadratic function to solve systems of equations. By rewriting the quadratic function in vertex form, you can easily identify the vertex of the parabola, which can be used to solve the system of equations.

Q: Are there any other applications of the vertex form of a quadratic function?

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A: Yes, there are many other applications of the vertex form of a quadratic function. Some examples include:

• Modeling real-world situations, such as the motion of an object under the influence of gravity.
• Finding the maximum or minimum value of a function.
• Graphing quadratic functions.
• Solving systems of equations.

Conclusion

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In conclusion, the vertex form of a quadratic function is a powerful tool for analyzing and graphing quadratic functions. By understanding how to rewrite a quadratic function in vertex form, you can gain a deeper understanding of the properties of quadratic functions and apply this knowledge to solve a wide range of mathematical problems.

Example Problems

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Problem 1

Rewrite the quadratic function $y = x^2 + 6x + 2$ in vertex form.

Solution

To rewrite the quadratic function in vertex form, we need to identify the coefficients of the quadratic function. The coefficients are:

• $a = 1$ (coefficient of $x^2$)
• $b = 6$ (coefficient of $x$)
• $c = 2$ (constant term)

The value of $h$ is given by the formula:

$$h = -\frac{b}{2a} = -\frac{6}{2(1)} = -3$$

The value of $k$ is given by the formula:

$$k = c - \frac{b^2}{4a} = 2 - \frac{(6)^2}{4(1)} = 2 - \frac{36}{4} = -7$$

Now that we have the values of $h$ and $k$, we can rewrite the quadratic function in vertex form:

$$y = (x + 3)^2 - 7$$

Problem 2

Rewrite the quadratic function $y = x^2 - 4x + 3$ in vertex form.

Solution

To rewrite the quadratic function in vertex form, we need to identify the coefficients of the quadratic function. The coefficients are:

• $a = 1$ (coefficient of $x^2$)
• $b = -4$ (coefficient of $x$)
• $c = 3$ (constant term)

The value of $h$ is given by the formula:

$$h = -\frac{b}{2a} = -\frac{-4}{2(1)} = 2$$

The value of $k$ is given by the formula:

$$k = c - \frac{b^2}{4a} = 3 - \frac{(-4)^2}{4(1)} = 3 - \frac{16}{4} = -2$$

Now that we have the values of $h$ and $k$, we can rewrite the quadratic function in vertex form:

$$y = (x - 2)^2 - 2$$