How Is The Graph Of The Parent Quadratic Function Transformed To Produce The Graph Of Y = − ( 2 X + 6 ) 2 + 3 Y=-(2x+6)^2+3 Y = − ( 2 X + 6 ) 2 + 3 ?A. The Graph Is Compressed Horizontally By A Factor Of 2, Shifted Left 3 Units, Reflected Over The X X X -axis, And Translated Up 3

Mar 10, 2025 by ADMIN 281 views

**Understanding the Transformation of Quadratic Functions**

Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding how they are transformed is crucial for graphing and solving equations. In this article, we will explore how the graph of the parent quadratic function is transformed to produce the graph of $y=-(2x+6)^2+3$ . We will break down the transformation process step by step, using mathematical notation and visual aids to illustrate each step.

The Parent Quadratic Function

The parent quadratic function is a quadratic function in the form of $y=ax^2$ , where $a$ is a constant. In this case, the parent quadratic function is $y=x^2$ . The graph of this function is a parabola that opens upwards, with its vertex at the origin (0,0).

The Given Quadratic Function

The given quadratic function is $y=-(2x+6)^2+3$ . To understand how the graph of this function is transformed from the parent quadratic function, we need to analyze the equation and identify the transformations that have been applied.

Horizontal Compression

The first transformation that we can identify is the horizontal compression. The equation $y=-(2x+6)^2+3$ can be rewritten as $y=-(2(x+3))^2+3$ . This indicates that the graph of the parent quadratic function has been compressed horizontally by a factor of 2. The factor of 2 inside the parentheses indicates that the graph has been compressed by a factor of 2 in the x-direction.

Shift Left 3 Units

The equation $y=-(2(x+3))^2+3$ also indicates that the graph of the parent quadratic function has been shifted left 3 units. The term $(x+3)$ inside the parentheses indicates that the graph has been shifted 3 units to the left.

Reflection Over the x-Axis

The negative sign in front of the squared term $-(2x+6)^2$ indicates that the graph of the parent quadratic function has been reflected over the x-axis. This means that the graph has been flipped upside down.

Translation Up 3 Units

Finally, the constant term $+3$ at the end of the equation indicates that the graph of the parent quadratic function has been translated up 3 units. This means that the graph has been shifted 3 units upwards.

Conclusion

In conclusion, the graph of the parent quadratic function is transformed to produce the graph of $y=-(2x+6)^2+3$ by applying the following transformations:

Horizontal compression by a factor of 2
Shift left 3 units
Reflection over the x-axis
Translation up 3 units

By understanding these transformations, we can visualize the graph of the given quadratic function and solve equations involving quadratic functions.

Visualizing the Transformation

To visualize the transformation, we can start with the graph of the parent quadratic function $y=x^2$ . We can then apply the transformations one by one to see how the graph changes.

Step 1: Horizontal Compression

The first transformation that we apply is the horizontal compression by a factor of 2. This means that the graph of the parent quadratic function is compressed in the x-direction by a factor of 2.

import matplotlib.pyplot as plt
import numpy as np
def parent_quadratic(x):
return x**2

x = np.linspace(-10, 10, 400)

y = parent_quadratic(x)

plt.plot(x, y)
plt.title('Parent Quadratic Function')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.show()

Step 2: Shift Left 3 Units

The next transformation that we apply is the shift left 3 units. This means that the graph of the parent quadratic function is shifted 3 units to the left.

# Define the quadratic function with horizontal compression and shift left 3 units
def quadratic(x):
    return -(2*(x+3))**2 + 3

x = np.linspace(-10, 10, 400)

y = quadratic(x)

plt.plot(x, y)
plt.title('Quadratic Function with Horizontal Compression and Shift Left 3 Units')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.show()

Step 3: Reflection Over the x-Axis

The next transformation that we apply is the reflection over the x-axis. This means that the graph of the parent quadratic function is flipped upside down.

# Define the quadratic function with horizontal compression, shift left 3 units, and reflection over the x-axis
def quadratic(x):
    return -(2*(x+3))**2 + 3

x = np.linspace(-10, 10, 400)

y = -quadratic(x)

plt.plot(x, y)
plt.title('Quadratic Function with Horizontal Compression, Shift Left 3 Units, and Reflection Over the x-Axis')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.show()

Step 4: Translation Up 3 Units

The final transformation that we apply is the translation up 3 units. This means that the graph of the parent quadratic function is shifted 3 units upwards.

# Define the quadratic function with horizontal compression, shift left 3 units, reflection over the x-axis, and translation up 3 units
def quadratic(x):
    return -(2*(x+3))**2 + 3

x = np.linspace(-10, 10, 400)

y = quadratic(x) + 3

plt.plot(x, y)
plt.title('Quadratic Function with Horizontal Compression, Shift Left 3 Units, Reflection Over the x-Axis, and Translation Up 3 Units')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.show()

Q: What is the parent quadratic function?

A: The parent quadratic function is a quadratic function in the form of $y=ax^2$ , where $a$ is a constant. In this case, the parent quadratic function is $y=x^2$ . The graph of this function is a parabola that opens upwards, with its vertex at the origin (0,0).

Q: What is the given quadratic function?

A: The given quadratic function is $y=-(2x+6)^2+3$ . This function is a transformation of the parent quadratic function, and it has been compressed horizontally by a factor of 2, shifted left 3 units, reflected over the x-axis, and translated up 3 units.

Q: What is the effect of the horizontal compression on the graph of the parent quadratic function?

A: The horizontal compression by a factor of 2 causes the graph of the parent quadratic function to be compressed in the x-direction. This means that the graph becomes narrower and taller.

Q: What is the effect of the shift left 3 units on the graph of the parent quadratic function?

A: The shift left 3 units causes the graph of the parent quadratic function to be shifted 3 units to the left. This means that the graph is moved to the left by 3 units.

Q: What is the effect of the reflection over the x-axis on the graph of the parent quadratic function?

A: The reflection over the x-axis causes the graph of the parent quadratic function to be flipped upside down. This means that the graph is reflected over the x-axis, and the y-values are negated.

Q: What is the effect of the translation up 3 units on the graph of the parent quadratic function?

A: The translation up 3 units causes the graph of the parent quadratic function to be shifted 3 units upwards. This means that the graph is moved upwards by 3 units.

Q: How can I visualize the transformation of the parent quadratic function to the given quadratic function?

A: You can visualize the transformation by applying the transformations one by one. First, apply the horizontal compression by a factor of 2. Then, apply the shift left 3 units. Next, apply the reflection over the x-axis. Finally, apply the translation up 3 units.

Q: What is the significance of the given quadratic function?

A: The given quadratic function is a transformation of the parent quadratic function, and it has been used to illustrate the concept of transformations in mathematics. The given quadratic function can be used to model real-world situations, such as the motion of an object under the influence of gravity.

Q: How can I use the given quadratic function in real-world applications?

A: You can use the given quadratic function to model real-world situations, such as the motion of an object under the influence of gravity. For example, you can use the given quadratic function to model the trajectory of a projectile, such as a ball thrown upwards.

Q: What are some common applications of quadratic functions?

A: Quadratic functions have many common applications in mathematics and science. Some examples include:

Modeling the motion of an object under the influence of gravity
Modeling the trajectory of a projectile
Modeling the growth or decay of a population
Modeling the stress on a beam or a bridge
Modeling the motion of a pendulum

Q: How can I learn more about quadratic functions and their applications?

A: You can learn more about quadratic functions and their applications by reading books, articles, and online resources. You can also take online courses or attend workshops to learn more about quadratic functions and their applications. Additionally, you can practice solving problems and working with quadratic functions to develop your skills and understanding.