Which Of The Following Describes The Parabola With The Equation Y = X 2 + 2 X + 1 Y = X^2 + 2x + 1 Y = X 2 + 2 X + 1 ?A. The Axis Of Symmetry Is X = 1 X = 1 X = 1 And The Vertex Is ( 1 , 4 (1, 4 ( 1 , 4 ].B. The Axis Of Symmetry Is X = − 1 X = -1 X = − 1 And The Vertex Is $(-1,

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Introduction

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Parabolas are a fundamental concept in mathematics, particularly in algebra and geometry. They are a type of quadratic equation that can be represented in various forms, including standard, vertex, and factored forms. In this article, we will delve into the world of parabolas, focusing on the equation $y = x^2 + 2x + 1$. We will explore the characteristics of this parabola, including its axis of symmetry and vertex.

The Equation $y = x^2 + 2x + 1$

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The given equation is a quadratic equation in the form of $y = ax^2 + bx + c$, where $a = 1$, $b = 2$, and $c = 1$. To understand the characteristics of this parabola, we need to rewrite the equation in vertex form, which is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Rewriting the Equation in Vertex Form

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To rewrite the equation in vertex form, we need to complete the square. We start by factoring out the coefficient of $x^2$, which is 1.

$$y = x^2 + 2x + 1$$

Next, we add and subtract the square of half the coefficient of $x$ inside the parentheses.

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

Now, we can rewrite the equation as:

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

Comparing this with the vertex form, we can see that the vertex is $(-1, -1)$.

Axis of Symmetry and Vertex

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The axis of symmetry is the vertical line that passes through the vertex of the parabola. In this case, the axis of symmetry is $x = -1$. The vertex is the point where the parabola changes direction, and it is represented by the coordinates $(h, k)$ in the vertex form.

Conclusion

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In conclusion, the parabola with the equation $y = x^2 + 2x + 1$ has an axis of symmetry of $x = -1$ and a vertex of $(-1, -1)$. This is a fundamental concept in mathematics, and understanding parabolas is essential for solving various problems in algebra and geometry.

Example Problems

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Problem 1

Find the axis of symmetry and vertex of the parabola with the equation $y = x^2 - 4x + 4$.

Solution

To find the axis of symmetry and vertex, we need to rewrite the equation in vertex form. We start by factoring out the coefficient of $x^2$, which is 1.

$$y = x^2 - 4x + 4$$

Next, we add and subtract the square of half the coefficient of $x$ inside the parentheses.

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

Now, we can rewrite the equation as:

$$y = (x - 2)^2 - 4$$

Comparing this with the vertex form, we can see that the vertex is $(2, -4)$.

The axis of symmetry is the vertical line that passes through the vertex of the parabola. In this case, the axis of symmetry is $x = 2$.

Problem 2

Find the axis of symmetry and vertex of the parabola with the equation $y = x^2 + 6x + 9$.

Solution

To find the axis of symmetry and vertex, we need to rewrite the equation in vertex form. We start by factoring out the coefficient of $x^2$, which is 1.

$$y = x^2 + 6x + 9$$

Next, we add and subtract the square of half the coefficient of $x$ inside the parentheses.

$$y = (x^2 + 6x + 9) - 9$$

Now, we can rewrite the equation as:

$$y = (x + 3)^2 - 9$$

Comparing this with the vertex form, we can see that the vertex is $(-3, -9)$.

The axis of symmetry is the vertical line that passes through the vertex of the parabola. In this case, the axis of symmetry is $x = -3$.

Final Thoughts

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In conclusion, understanding parabolas is essential for solving various problems in algebra and geometry. The axis of symmetry and vertex are fundamental concepts that can be used to analyze and solve problems involving parabolas. By following the steps outlined in this article, you can find the axis of symmetry and vertex of any parabola with a given equation.

References

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• [1] "Algebra and Geometry" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Glossary

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• Axis of Symmetry: The vertical line that passes through the vertex of a parabola.
• Vertex: The point where the parabola changes direction.
• Parabola: A type of quadratic equation that can be represented in various forms, including standard, vertex, and factored forms.
  

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Introduction

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Parabolas are a fundamental concept in mathematics, particularly in algebra and geometry. In our previous article, we explored the characteristics of parabolas, including their axis of symmetry and vertex. In this article, we will answer some frequently asked questions about parabolas, providing a comprehensive guide to help you understand these fascinating curves.

Q&A

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Q1: What is a parabola?

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A parabola is a type of quadratic equation that can be represented in various forms, including standard, vertex, and factored forms.

A1:

A parabola is a U-shaped curve that can be represented by the equation $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q2: What is the axis of symmetry?

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The axis of symmetry is the vertical line that passes through the vertex of a parabola.

A2:

The axis of symmetry is the vertical line that passes through the vertex of a parabola. It is a fundamental concept in mathematics, and understanding it is essential for solving problems involving parabolas.

Q3: How do I find the axis of symmetry?

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To find the axis of symmetry, you need to rewrite the equation in vertex form, which is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

A3:

To find the axis of symmetry, you need to rewrite the equation in vertex form. You can do this by completing the square, which involves adding and subtracting the square of half the coefficient of $x$ inside the parentheses.

Q4: What is the vertex of a parabola?

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The vertex of a parabola is the point where the parabola changes direction.

A4:

The vertex of a parabola is the point where the parabola changes direction. It is represented by the coordinates $(h, k)$ in the vertex form of the equation.

Q5: How do I find the vertex of a parabola?

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To find the vertex of a parabola, you need to rewrite the equation in vertex form, which is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

A5:

To find the vertex of a parabola, you need to rewrite the equation in vertex form. You can do this by completing the square, which involves adding and subtracting the square of half the coefficient of $x$ inside the parentheses.

Q6: What is the difference between a parabola and a circle?

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A parabola is a U-shaped curve, while a circle is a round shape.

A6:

A parabola is a U-shaped curve, while a circle is a round shape. The key difference between the two is that a parabola has a single axis of symmetry, while a circle has no axis of symmetry.

Q7: Can a parabola have more than one axis of symmetry?

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No, a parabola can only have one axis of symmetry.

A7:

No, a parabola can only have one axis of symmetry. This is because the axis of symmetry is a fundamental property of a parabola, and it is defined by the equation of the parabola.

Q8: How do I graph a parabola?

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To graph a parabola, you need to use a graphing calculator or a computer program to plot the points on the graph.

A8:

To graph a parabola, you need to use a graphing calculator or a computer program to plot the points on the graph. You can also use a table of values to find the points on the graph.

Conclusion

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In conclusion, parabolas are a fundamental concept in mathematics, and understanding them is essential for solving problems in algebra and geometry. By answering these frequently asked questions, we hope to have provided a comprehensive guide to help you understand parabolas and their characteristics.

References

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• [1] "Algebra and Geometry" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Glossary

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• Axis of Symmetry: The vertical line that passes through the vertex of a parabola.
• Vertex: The point where the parabola changes direction.
• Parabola: A type of quadratic equation that can be represented in various forms, including standard, vertex, and factored forms.