The Standard Form Of The Equation Of A Parabola Is $y = 7x^2 + 14x + 4$. What Is The Vertex Form Of The Equation?A. $y = 7(x + 1)^2 + 3$B. $y = 7(x + 2)^2 - 3$C. $y = 7(x + 2)^2 + 3$D. $y = 7(x + 1)^2 - 3$

Introduction

In mathematics, a parabola is a quadratic curve that can be represented in various forms, including the standard form and vertex form. The standard form of a parabola is given by the equation $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. However, the vertex form of a parabola is more useful in certain applications, such as graphing and solving quadratic equations. In this article, we will discuss how to convert the standard form of a parabola to its vertex form.

The Standard Form of a Parabola

The standard form of a parabola is given by the equation $y = ax^2 + bx + c$. This form is useful for solving quadratic equations and graphing parabolas. However, it can be difficult to determine the vertex of the parabola from this form.

Converting to Vertex Form

To convert the standard form of a parabola to its vertex form, we need to complete the square. 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.

Step 1: Factor Out the Coefficient of $x^2$

The first step in converting the standard form of a parabola to its vertex form is to factor out the coefficient of $x^2$. In the given equation $y = 7x^2 + 14x + 4$, the coefficient of $x^2$ is 7. We can factor this out as follows:

$$y = 7(x^2 + 2x) + 4$$

Step 2: Complete the Square

The next step is to complete the square. To do this, we need to add and subtract $(2/2)^2 = 1$ inside the parentheses:

$$y = 7(x^2 + 2x + 1 - 1) + 4$$

Step 3: Simplify the Expression

Now, we can simplify the expression by combining like terms:

$$y = 7(x^2 + 2x + 1) - 7 + 4$$

$$y = 7(x + 1)^2 - 3$$

Conclusion

In this article, we discussed how to convert the standard form of a parabola to its vertex form. We used the given equation $y = 7x^2 + 14x + 4$ as an example and completed the square to obtain the vertex form $y = 7(x + 1)^2 - 3$. This form is more useful in certain applications, such as graphing and solving quadratic equations.

Answer

The correct answer is D. $y = 7(x + 1)^2 - 3$.

Discussion

The vertex form of a parabola is a useful tool in mathematics, particularly in graphing and solving quadratic equations. By converting the standard form of a parabola to its vertex form, we can easily determine the vertex of the parabola and graph the curve. In this article, we used the given equation $y = 7x^2 + 14x + 4$ as an example and completed the square to obtain the vertex form $y = 7(x + 1)^2 - 3$. This form is more useful in certain applications, such as graphing and solving quadratic equations.

Related Topics

• Graphing Parabolas: Graphing parabolas is an important application of the vertex form of a parabola. By using the vertex form, we can easily graph the curve and determine the vertex of the parabola.
• Solving Quadratic Equations: Solving quadratic equations is another important application of the vertex form of a parabola. By using the vertex form, we can easily solve quadratic equations and determine the solutions.
• Completing the Square: Completing the square is a useful technique in mathematics, particularly in converting the standard form of a parabola to its vertex form. By completing the square, we can easily obtain the vertex form of a parabola.

References

• Algebra: Algebra is a branch of mathematics that deals with the study of variables and their relationships. It is an important subject in mathematics, particularly in graphing and solving quadratic equations.
• Geometry: Geometry is a branch of mathematics that deals with the study of shapes and their properties. It is an important subject in mathematics, particularly in graphing and solving quadratic equations.
• Calculus: Calculus is a branch of mathematics that deals with the study of rates of change and accumulation. It is an important subject in mathematics, particularly in graphing and solving quadratic equations.
  The Standard Form of a Parabola: Converting to Vertex Form ===========================================================

Q&A: Converting the Standard Form of a Parabola to its Vertex Form

Q: What is the standard form of a parabola?

A: The standard form of a parabola is given by the equation $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

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 convert the standard form of a parabola to its vertex form?

A: To convert the standard form of a parabola to its vertex form, you need to complete the square. This involves factoring out the coefficient of $x^2$, adding and subtracting $(b/2)^2$ inside the parentheses, and then simplifying the expression.

Q: What is the first step in converting the standard form of a parabola to its vertex form?

A: The first step in converting the standard form of a parabola to its vertex form is to factor out the coefficient of $x^2$. In the given equation $y = 7x^2 + 14x + 4$, the coefficient of $x^2$ is 7. We can factor this out as follows:

$$y = 7(x^2 + 2x) + 4$$

Q: What is the next step in converting the standard form of a parabola to its vertex form?

A: The next step is to complete the square. To do this, we need to add and subtract $(b/2)^2$ inside the parentheses. In the given equation $y = 7(x^2 + 2x) + 4$, we need to add and subtract $(2/2)^2 = 1$ inside the parentheses:

$$y = 7(x^2 + 2x + 1 - 1) + 4$$

Q: How do I simplify the expression after completing the square?

A: After completing the square, we can simplify the expression by combining like terms. In the given equation $y = 7(x^2 + 2x + 1) - 7 + 4$, we can simplify the expression as follows:

$$y = 7(x + 1)^2 - 3$$

Q: What is the final answer for the given equation $y = 7x^2 + 14x + 4$?

A: The final answer for the given equation $y = 7x^2 + 14x + 4$ is $y = 7(x + 1)^2 - 3$.

Q: What are some related topics to the standard form and vertex form of a parabola?

A: Some related topics to the standard form and vertex form of a parabola include graphing parabolas, solving quadratic equations, and completing the square.

Q: Why is the vertex form of a parabola useful?

A: The vertex form of a parabola is useful because it allows us to easily determine the vertex of the parabola and graph the curve. It is also useful in solving quadratic equations.

Q: How do I graph a parabola using the vertex form?

A: To graph a parabola using the vertex form, we need to identify the vertex of the parabola and then use the equation to determine the x-intercepts and y-intercepts of the parabola.

Q: How do I solve a quadratic equation using the vertex form?

A: To solve a quadratic equation using the vertex form, we need to identify the vertex of the parabola and then use the equation to determine the solutions of the quadratic equation.

Q: What are some common mistakes to avoid when converting the standard form of a parabola to its vertex form?

A: Some common mistakes to avoid when converting the standard form of a parabola to its vertex form include forgetting to factor out the coefficient of $x^2$, forgetting to add and subtract $(b/2)^2$ inside the parentheses, and not simplifying the expression after completing the square.