Graph The Parabola:$\[ Y = X^2 - 4x + 8 \\]

Introduction

In mathematics, a parabola is a quadratic curve that is U-shaped and can be represented by a quadratic equation. The standard form of a quadratic equation is $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this article, we will focus on graphing the parabola represented by the equation $y = x^2 - 4x + 8$. We will explore the properties of the parabola, including its vertex, axis of symmetry, and x-intercepts.

Understanding the Equation

The given equation is $y = x^2 - 4x + 8$. To graph this parabola, we need to understand the properties of the equation. The coefficient of the $x^2$ term is $1$, which means that the parabola opens upwards. The coefficient of the $x$ term is $-4$, which means that the parabola is shifted to the right. The constant term is $8$, which means that the parabola is shifted upwards.

Vertex Form

The vertex form of a quadratic equation is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. To convert the given equation to vertex form, we need to complete the square.

Completing the Square

To complete the square, we need to add and subtract $(b/2)^2$ to the equation.

import sympy as sp

x = sp.symbols('x')

# Define the equation
equation = x**2 - 4*x + 8

# Complete the square
completed_square = sp.expand((x - 2)**2) + 4

print(completed_square)

The completed square is $(x - 2)^2 + 4$. Therefore, the vertex form of the equation is $y = (x - 2)^2 + 4$.

Vertex

The vertex of the parabola is $(h, k) = (2, 4)$. This means that the parabola has a minimum point at $(2, 4)$.

Axis of Symmetry

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

X-Intercepts

To find the x-intercepts of the parabola, we need to set $y = 0$ and solve for $x$.

import sympy as sp

x = sp.symbols('x')

# Define the equation
equation = x**2 - 4*x + 8

# Set y = 0 and solve for x
solutions = sp.solve(equation, x)

print(solutions)

The x-intercepts of the parabola are $x = 2 \pm 2\sqrt{3}$.

Graphing the Parabola

To graph the parabola, we can use the properties we have discussed. We can start by plotting the vertex at $(2, 4)$. Then, we can plot the axis of symmetry at $x = 2$. Finally, we can plot the x-intercepts at $x = 2 \pm 2\sqrt{3}$.

Graph

Here is a graph of the parabola:

import matplotlib.pyplot as plt
import numpy as np

x = np.linspace(-5, 5, 400)
y = x**2 - 4*x + 8

plt.plot(x, y)
plt.axvline(x=2, color='k', linestyle='--')
plt.scatter(2, 4, color='r')
plt.scatter(2 - 2*np.sqrt(3), 0, color='r')
plt.scatter(2 + 2*np.sqrt(3), 0, color='r')
plt.show()

Conclusion

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Introduction

In our previous article, we discussed graphing the parabola represented by the equation $y = x^2 - 4x + 8$. We explored the properties of the parabola, including its vertex, axis of symmetry, and x-intercepts. In this article, we will answer some frequently asked questions about graphing the parabola.

Q&A

Q: What is the vertex of the parabola?

A: The vertex of the parabola is $(h, k) = (2, 4)$. This means that the parabola has a minimum point at $(2, 4)$.

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

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

Q: How do I find the x-intercepts of the parabola?

A: To find the x-intercepts of the parabola, you need to set $y = 0$ and solve for $x$. In this case, the x-intercepts are $x = 2 \pm 2\sqrt{3}$.

Q: How do I graph the parabola?

A: To graph the parabola, you can use the properties we have discussed. You can start by plotting the vertex at $(2, 4)$. Then, you can plot the axis of symmetry at $x = 2$. Finally, you can plot the x-intercepts at $x = 2 \pm 2\sqrt{3}$.

Q: What is the equation of the parabola in vertex form?

A: The equation of the parabola in vertex form is $y = (x - 2)^2 + 4$.

Q: How do I convert the equation of the parabola to vertex form?

A: To convert the equation of the parabola to vertex form, you need to complete the square. You can do this by adding and subtracting $(b/2)^2$ to the equation.

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

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

Q: How do I find the y-intercept of the parabola?

A: To find the y-intercept of the parabola, you need to set $x = 0$ and solve for $y$. In this case, the y-intercept is $y = 8$.

Q: What is the equation of the parabola in standard form?

A: The equation of the parabola in standard form is $y = x^2 - 4x + 8$.

Q: How do I graph the parabola using a graphing calculator?

A: To graph the parabola using a graphing calculator, you can enter the equation of the parabola and use the graphing function. You can also use the vertex form of the equation to graph the parabola.

Conclusion

In this article, we have answered some frequently asked questions about graphing the parabola. We have discussed the properties of the parabola, including its vertex, axis of symmetry, and x-intercepts. We have also provided examples of how to graph the parabola using different methods.