Graph The Equation: $y = 2(x-4)^2 + 5$

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Introduction

Graphing equations is a fundamental concept in mathematics, particularly in algebra and geometry. It involves visualizing the relationship between variables in an equation, which can be used to solve problems, model real-world situations, and make predictions. In this article, we will focus on graphing the equation y=2(x−4)2+5y = 2(x-4)^2 + 5, which is a quadratic equation in the form of y=ax2+bx+cy = ax^2 + bx + c. We will explore the properties of this equation, its graph, and how to interpret the results.

Understanding the Equation

The given equation is y=2(x−4)2+5y = 2(x-4)^2 + 5. To understand this equation, let's break it down into its components:

  • Quadratic term: The equation contains a quadratic term, which is (x−4)2(x-4)^2. This term represents a parabola that opens upwards or downwards.
  • Coefficient: The coefficient of the quadratic term is 2, which indicates the direction and steepness of the parabola.
  • Constant term: The constant term is 5, which represents the vertical shift of the parabola.

Graphing the Equation

To graph the equation, we need to identify the key features of the parabola, such as its vertex, axis of symmetry, and intercepts.

Vertex

The vertex of a parabola is the point where the parabola changes direction. In the equation y=2(x−4)2+5y = 2(x-4)^2 + 5, the vertex is at the point (4, 5). This is because the equation is in the form of y=a(x−h)2+ky = a(x-h)^2 + k, where (h, k) represents the vertex.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola. In this case, the axis of symmetry is the line x = 4.

Intercepts

The x-intercepts are the points where the parabola intersects the x-axis. To find the x-intercepts, we need to set y = 0 and solve for x.

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the equation
equation = 2*(x-4)**2 + 5

# Solve for x
solutions = sp.solve(equation, x)

print(solutions)

The x-intercepts are at the points (2, 0) and (6, 0).

Graphing the Parabola

Now that we have identified the key features of the parabola, we can graph it using a graphing tool or software.

import matplotlib.pyplot as plt
import numpy as np

# Define the x values
x = np.linspace(-10, 10, 400)

# Define the y values
y = 2*(x-4)**2 + 5

# Create the plot
plt.plot(x, y)
plt.xlabel('x')
plt.ylabel('y')
plt.title('Graph of the Equation $y = 2(x-4)^2 + 5{{content}}#39;)
plt.grid(True)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.show()

Interpreting the Graph

The graph of the equation y=2(x−4)2+5y = 2(x-4)^2 + 5 is a parabola that opens upwards. The vertex of the parabola is at the point (4, 5), and the axis of symmetry is the line x = 4. The x-intercepts are at the points (2, 0) and (6, 0).

Conclusion

Graphing equations is a powerful tool for visualizing the relationship between variables in an equation. In this article, we graphed the equation y=2(x−4)2+5y = 2(x-4)^2 + 5 and identified its key features, such as the vertex, axis of symmetry, and intercepts. We also created a graph of the parabola using a graphing tool and software. By understanding and interpreting the graph, we can gain insights into the behavior of the equation and make predictions about its behavior.

Applications of Graphing Equations

Graphing equations has numerous applications in various fields, including:

  • Science: Graphing equations is used to model real-world phenomena, such as the motion of objects, the growth of populations, and the behavior of physical systems.
  • Engineering: Graphing equations is used to design and optimize systems, such as bridges, buildings, and electronic circuits.
  • Economics: Graphing equations is used to model economic systems, such as supply and demand curves, and to make predictions about economic trends.
  • Computer Science: Graphing equations is used to visualize data, such as the behavior of algorithms and the performance of computer systems.

Conclusion

Q&A: Graphing the Equation y=2(x−4)2+5y = 2(x-4)^2 + 5

Q: What is the vertex of the parabola?

A: The vertex of the parabola is at the point (4, 5). This is because the equation is in the form of y=a(x−h)2+ky = a(x-h)^2 + k, where (h, k) represents the vertex.

Q: What is the axis of symmetry?

A: The axis of symmetry is a vertical line that passes through the vertex of the parabola. In this case, the axis of symmetry is the line x = 4.

Q: What are the x-intercepts?

A: The x-intercepts are the points where the parabola intersects the x-axis. To find the x-intercepts, we need to set y = 0 and solve for x. The x-intercepts are at the points (2, 0) and (6, 0).

Q: How do I graph the parabola?

A: To graph the parabola, we can use a graphing tool or software. We can also use a graphing calculator or a computer program to graph the parabola.

Q: What is the significance of the graph?

A: The graph of the equation y=2(x−4)2+5y = 2(x-4)^2 + 5 is a parabola that opens upwards. The vertex of the parabola is at the point (4, 5), and the axis of symmetry is the line x = 4. The x-intercepts are at the points (2, 0) and (6, 0). By understanding and interpreting the graph, we can gain insights into the behavior of the equation and make predictions about its behavior.

Q: What are some real-world applications of graphing equations?

A: Graphing equations has numerous applications in various fields, including:

  • Science: Graphing equations is used to model real-world phenomena, such as the motion of objects, the growth of populations, and the behavior of physical systems.
  • Engineering: Graphing equations is used to design and optimize systems, such as bridges, buildings, and electronic circuits.
  • Economics: Graphing equations is used to model economic systems, such as supply and demand curves, and to make predictions about economic trends.
  • Computer Science: Graphing equations is used to visualize data, such as the behavior of algorithms and the performance of computer systems.

Q: How do I use graphing equations in my daily life?

A: Graphing equations can be used in various ways in your daily life, such as:

  • Modeling real-world phenomena: Graphing equations can be used to model real-world phenomena, such as the motion of objects, the growth of populations, and the behavior of physical systems.
  • Designing and optimizing systems: Graphing equations can be used to design and optimize systems, such as bridges, buildings, and electronic circuits.
  • Making predictions: Graphing equations can be used to make predictions about economic trends, population growth, and other real-world phenomena.
  • Visualizing data: Graphing equations can be used to visualize data, such as the behavior of algorithms and the performance of computer systems.

Conclusion

In conclusion, graphing equations is a fundamental concept in mathematics that has numerous applications in various fields. By understanding and interpreting the graph of an equation, we can gain insights into the behavior of the equation and make predictions about its behavior.