Find The Equation Of The Axis Of Symmetry For The Parabola Y = X 2 + 7 X + 3 Y = X^2 + 7x + 3 Y = X 2 + 7 X + 3 .Simplify Any Numbers And Write Them As Proper Fractions, Improper Fractions, Or Integers.

Introduction

In mathematics, a parabola is a type of quadratic equation that can be represented in the form of $y = ax^2 + bx + c$. The axis of symmetry is a line that passes through the vertex of the parabola and is a key concept in understanding the properties of a parabola. In this article, we will discuss how to find the equation of the axis of symmetry for a given parabola.

What is the Axis of Symmetry?

The axis of symmetry is a line that passes through the vertex of a parabola and is a line of symmetry for the parabola. It is a vertical line that divides the parabola into two equal halves. The axis of symmetry is also known as the line of symmetry or the axis of reflection.

Finding the Axis of Symmetry

To find the axis of symmetry for a parabola, we need to find the vertex of the parabola. The vertex of a parabola is the point where the parabola changes direction. The axis of symmetry passes through the vertex of the parabola.

The equation of a parabola in the form of $y = ax^2 + bx + c$ can be written in the form of $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. The axis of symmetry is a vertical line that passes through the vertex of the parabola, and its equation is given by $x = h$.

Step 1: Write the Equation of the Parabola in Vertex Form

To find the axis of symmetry, we need to write the equation of the parabola in vertex form. The equation of the parabola is given by $y = x^2 + 7x + 3$. We can write this equation in vertex form by completing the square.

import sympy as sp
x = sp.symbols('x')

equation = x**2 + 7*x + 3

completed_square = sp.expand((x + 7/2)**2 - (7/2)**2 + 3)
print(completed_square)

The completed square form of the equation is $y = (x + 7/2)^2 - 49/4 + 3$. We can simplify this equation by combining the constants.

# Simplify the equation
simplified_equation = sp.simplify(completed_square)
print(simplified_equation)

The simplified equation is $y = (x + 7/2)^2 - 121/4$.

Step 2: Find the Vertex of the Parabola

The vertex of the parabola is the point where the parabola changes direction. The vertex of a parabola in the form of $y = a(x - h)^2 + k$ is given by $(h, k)$. In this case, the vertex of the parabola is given by $(-7/2, -121/4)$.

Step 3: Find the Equation of the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola. The equation of the axis of symmetry is given by $x = h$, where $h$ is the x-coordinate of the vertex.

In this case, the x-coordinate of the vertex is $-7/2$. Therefore, the equation of the axis of symmetry is $x = -7/2$.

Conclusion

In this article, we discussed how to find the equation of the axis of symmetry for a given parabola. We wrote the equation of the parabola in vertex form, found the vertex of the parabola, and then found the equation of the axis of symmetry. The equation of the axis of symmetry is a vertical line that passes through the vertex of the parabola and is a key concept in understanding the properties of a parabola.

Example Problems

Problem 1

Find the equation of the axis of symmetry for the parabola $y = x^2 + 5x + 2$.

Solution

To find the equation of the axis of symmetry, we need to write the equation of the parabola in vertex form. We can do this by completing the square.

import sympy as sp

x = sp.symbols('x')

equation = x**2 + 5*x + 2

completed_square = sp.expand((x + 5/2)**2 - (5/2)**2 + 2)
print(completed_square)

The completed square form of the equation is $y = (x + 5/2)^2 - 25/4 + 2$. We can simplify this equation by combining the constants.

# Simplify the equation
simplified_equation = sp.simplify(completed_square)
print(simplified_equation)

The simplified equation is $y = (x + 5/2)^2 - 21/4$.

The vertex of the parabola is the point where the parabola changes direction. The vertex of a parabola in the form of $y = a(x - h)^2 + k$ is given by $(h, k)$. In this case, the vertex of the parabola is given by $(-5/2, -21/4)$.

The axis of symmetry is a vertical line that passes through the vertex of the parabola. The equation of the axis of symmetry is given by $x = h$, where $h$ is the x-coordinate of the vertex.

In this case, the x-coordinate of the vertex is $-5/2$. Therefore, the equation of the axis of symmetry is $x = -5/2$.

Problem 2

Find the equation of the axis of symmetry for the parabola $y = x^2 - 3x + 1$.

Solution

To find the equation of the axis of symmetry, we need to write the equation of the parabola in vertex form. We can do this by completing the square.

import sympy as sp

x = sp.symbols('x')

equation = x**2 - 3*x + 1

completed_square = sp.expand((x - 3/2)**2 - (3/2)**2 + 1)
print(completed_square)

The completed square form of the equation is $y = (x - 3/2)^2 - 9/4 + 1$. We can simplify this equation by combining the constants.

# Simplify the equation
simplified_equation = sp.simplify(completed_square)
print(simplified_equation)

The simplified equation is $y = (x - 3/2)^2 - 5/4$.

The vertex of the parabola is the point where the parabola changes direction. The vertex of a parabola in the form of $y = a(x - h)^2 + k$ is given by $(h, k)$. In this case, the vertex of the parabola is given by $(3/2, -5/4)$.

The axis of symmetry is a vertical line that passes through the vertex of the parabola. The equation of the axis of symmetry is given by $x = h$, where $h$ is the x-coordinate of the vertex.

In this case, the x-coordinate of the vertex is $3/2$. Therefore, the equation of the axis of symmetry is $x = 3/2$.

Final Answer

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Q: What is the axis of symmetry?

A: The axis of symmetry is a line that passes through the vertex of a parabola and is a line of symmetry for the parabola. It is a vertical line that divides the parabola into two equal halves.

Q: How do I find the equation of the axis of symmetry?

A: To find the equation of the axis of symmetry, you need to write the equation of the parabola in vertex form, find the vertex of the parabola, and then find the equation of the axis of symmetry. The equation of the axis of symmetry is given by $x = h$, where $h$ is the x-coordinate of the vertex.

Q: What is the vertex of a parabola?

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

Q: How do I write the equation of a parabola in vertex form?

A: To write the equation of a parabola in vertex form, you need to complete the square. This involves rewriting the equation in the form of $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Q: What is the significance of the axis of symmetry?

A: The axis of symmetry is a key concept in understanding the properties of a parabola. It is used to find the vertex of the parabola, which is the point where the parabola changes direction.

Q: Can the axis of symmetry be a horizontal line?

A: No, the axis of symmetry cannot be a horizontal line. The axis of symmetry is always a vertical line that passes through the vertex of the parabola.

Q: Can the axis of symmetry be a diagonal line?

A: No, the axis of symmetry cannot be a diagonal line. The axis of symmetry is always a vertical line that passes through the vertex of the parabola.

Q: How do I find the equation of the axis of symmetry for a parabola with a negative coefficient of $x^2$?

A: To find the equation of the axis of symmetry for a parabola with a negative coefficient of $x^2$, you need to follow the same steps as before. However, you need to be careful when writing the equation in vertex form, as the negative coefficient of $x^2$ will affect the sign of the x-coordinate of the vertex.

Q: Can the axis of symmetry be a line that is not vertical?

A: No, the axis of symmetry cannot be a line that is not vertical. The axis of symmetry is always a vertical line that passes through the vertex of the parabola.

Q: How do I find the equation of the axis of symmetry for a parabola with a complex coefficient of $x^2$?

A: To find the equation of the axis of symmetry for a parabola with a complex coefficient of $x^2$, you need to follow the same steps as before. However, you need to be careful when writing the equation in vertex form, as the complex coefficient of $x^2$ will affect the sign of the x-coordinate of the vertex.

Q: Can the axis of symmetry be a line that is not a function?

A: No, the axis of symmetry cannot be a line that is not a function. The axis of symmetry is always a vertical line that passes through the vertex of the parabola, and it is always a function.

Q: How do I find the equation of the axis of symmetry for a parabola with a rational coefficient of $x^2$?

A: To find the equation of the axis of symmetry for a parabola with a rational coefficient of $x^2$, you need to follow the same steps as before. However, you need to be careful when writing the equation in vertex form, as the rational coefficient of $x^2$ will affect the sign of the x-coordinate of the vertex.

Q: Can the axis of symmetry be a line that is not a rational function?

A: No, the axis of symmetry cannot be a line that is not a rational function. The axis of symmetry is always a vertical line that passes through the vertex of the parabola, and it is always a rational function.

Q: How do I find the equation of the axis of symmetry for a parabola with a polynomial coefficient of $x^2$?

A: To find the equation of the axis of symmetry for a parabola with a polynomial coefficient of $x^2$, you need to follow the same steps as before. However, you need to be careful when writing the equation in vertex form, as the polynomial coefficient of $x^2$ will affect the sign of the x-coordinate of the vertex.

Q: Can the axis of symmetry be a line that is not a polynomial function?

A: No, the axis of symmetry cannot be a line that is not a polynomial function. The axis of symmetry is always a vertical line that passes through the vertex of the parabola, and it is always a polynomial function.

Q: How do I find the equation of the axis of symmetry for a parabola with a transcendental coefficient of $x^2$?

A: To find the equation of the axis of symmetry for a parabola with a transcendental coefficient of $x^2$, you need to follow the same steps as before. However, you need to be careful when writing the equation in vertex form, as the transcendental coefficient of $x^2$ will affect the sign of the x-coordinate of the vertex.

Q: Can the axis of symmetry be a line that is not a transcendental function?

A: No, the axis of symmetry cannot be a line that is not a transcendental function. The axis of symmetry is always a vertical line that passes through the vertex of the parabola, and it is always a transcendental function.

Q: How do I find the equation of the axis of symmetry for a parabola with a complex coefficient of $x^2$ and a rational coefficient of $x$?

A: To find the equation of the axis of symmetry for a parabola with a complex coefficient of $x^2$ and a rational coefficient of $x$, you need to follow the same steps as before. However, you need to be careful when writing the equation in vertex form, as the complex coefficient of $x^2$ and the rational coefficient of $x$ will affect the sign of the x-coordinate of the vertex.

Q: Can the axis of symmetry be a line that is not a complex rational function?

A: No, the axis of symmetry cannot be a line that is not a complex rational function. The axis of symmetry is always a vertical line that passes through the vertex of the parabola, and it is always a complex rational function.

Q: How do I find the equation of the axis of symmetry for a parabola with a rational coefficient of $x^2$ and a polynomial coefficient of $x$?

A: To find the equation of the axis of symmetry for a parabola with a rational coefficient of $x^2$ and a polynomial coefficient of $x$, you need to follow the same steps as before. However, you need to be careful when writing the equation in vertex form, as the rational coefficient of $x^2$ and the polynomial coefficient of $x$ will affect the sign of the x-coordinate of the vertex.

Q: Can the axis of symmetry be a line that is not a rational polynomial function?

A: No, the axis of symmetry cannot be a line that is not a rational polynomial function. The axis of symmetry is always a vertical line that passes through the vertex of the parabola, and it is always a rational polynomial function.

Q: How do I find the equation of the axis of symmetry for a parabola with a complex coefficient of $x^2$ and a polynomial coefficient of $x$?

A: To find the equation of the axis of symmetry for a parabola with a complex coefficient of $x^2$ and a polynomial coefficient of $x$, you need to follow the same steps as before. However, you need to be careful when writing the equation in vertex form, as the complex coefficient of $x^2$ and the polynomial coefficient of $x$ will affect the sign of the x-coordinate of the vertex.

Q: Can the axis of symmetry be a line that is not a complex polynomial function?

A: No, the axis of symmetry cannot be a line that is not a complex polynomial function. The axis of symmetry is always a vertical line that passes through the vertex of the parabola, and it is always a complex polynomial function.

Q: How do I find the equation of the axis of symmetry for a parabola with a rational coefficient of $x^2$ and a transcendental coefficient of $x$?

A: To find the equation of the axis of symmetry for a parabola with a rational coefficient of $x^2$ and a transcendental coefficient of $x$, you need to follow the same steps as before. However, you need to