The Graph Of A Cube Root Function Has A Horizontal Translation That Is Three Times The Vertical Translation. The Vertical Translation Is Negative.a. Write A Function \[$ G \$\] That Has These Attributes.b. Graph Your Function And The Parent

Introduction

In this article, we will explore the graph of a cube root function with a horizontal translation that is three times the vertical translation. The vertical translation is negative. We will write a function that has these attributes and graph it along with its parent function.

Understanding Cube Root Functions

A cube root function is a type of function that has the form $f(x) = \sqrt[3]{x}$. The cube root function is the inverse of the cube function, which is defined as $f(x) = x^3$. The cube root function has a few key properties that make it useful in mathematics and other fields.

Properties of Cube Root Functions

One of the key properties of cube root functions is that they are one-to-one functions. This means that each value of the function corresponds to exactly one value of the input. This property makes it easy to find the inverse of a cube root function.

Another key property of cube root functions is that they are continuous functions. This means that the function can be drawn without lifting the pencil from the paper. This property makes it easy to graph cube root functions.

Writing a Function with Horizontal and Vertical Translation

To write a function with a horizontal translation that is three times the vertical translation, we need to use the following formula:

$$g(x) = a\sqrt[3]{x - h} + k$$

where $a$ is the vertical translation, $h$ is the horizontal translation, and $k$ is the vertical translation.

In this case, we are given that the vertical translation is negative, so we can set $a = -1$. We are also given that the horizontal translation is three times the vertical translation, so we can set $h = 3k$.

Substituting these values into the formula, we get:

$$g(x) = -\sqrt[3]{x - 3k} - k$$

Graphing the Function and the Parent Function

To graph the function and the parent function, we need to use a graphing calculator or a computer program. We can set $k = -1$ to get:

$$g(x) = -\sqrt[3]{x + 3} - 1$$

The parent function is the cube root function, which is defined as $f(x) = \sqrt[3]{x}$. We can graph both functions using a graphing calculator or a computer program.

Graphing the Function

To graph the function, we can use a graphing calculator or a computer program. We can set the window to $[-10, 10]$ and $[-10, 10]$ to get a good view of the graph.

The graph of the function is a cube root function with a horizontal translation of $3$ units to the left and a vertical translation of $1$ unit down.

Graphing the Parent Function

To graph the parent function, we can use a graphing calculator or a computer program. We can set the window to $[-10, 10]$ and $[-10, 10]$ to get a good view of the graph.

The graph of the parent function is a cube root function with no horizontal or vertical translation.

Comparison of the Graphs

The graph of the function and the graph of the parent function are similar, but they are not identical. The graph of the function has a horizontal translation of $3$ units to the left and a vertical translation of $1$ unit down.

Conclusion

In this article, we have explored the graph of a cube root function with a horizontal translation that is three times the vertical translation. The vertical translation is negative. We have written a function that has these attributes and graphed it along with its parent function.

References

• [1] "Cube Root Functions" by Math Open Reference
• [2] "Graphing Functions" by Math Is Fun

Code

import numpy as np
import matplotlib.pyplot as plt

# Define the function
def g(x):
    return -np.cbrt(x + 3) - 1

# Define the parent function
def f(x):
    return np.cbrt(x)

# Create an array of x values
x = np.linspace(-10, 10, 400)

# Create an array of y values for the function
y = g(x)

# Create an array of y values for the parent function
y_parent = f(x)

# Create a plot
plt.plot(x, y, label='g(x)')
plt.plot(x, y_parent, label='f(x)')

# Add a title and labels
plt.title('Graph of a Cube Root Function with Horizontal and Vertical Translation')
plt.xlabel('x')
plt.ylabel('y')

# Add a legend
plt.legend()

# Show the plot
plt.show()

$$$$

Q: What is a cube root function?

A: A cube root function is a type of function that has the form $f(x) = \sqrt[3]{x}$. The cube root function is the inverse of the cube function, which is defined as $f(x) = x^3$.

Q: What are the key properties of cube root functions?

A: One of the key properties of cube root functions is that they are one-to-one functions. This means that each value of the function corresponds to exactly one value of the input. Another key property of cube root functions is that they are continuous functions. This means that the function can be drawn without lifting the pencil from the paper.

Q: How do you write a function with a horizontal translation that is three times the vertical translation?

A: To write a function with a horizontal translation that is three times the vertical translation, you need to use the following formula:

$$g(x) = a\sqrt[3]{x - h} + k$$

where $a$ is the vertical translation, $h$ is the horizontal translation, and $k$ is the vertical translation.

Q: What is the formula for a cube root function with a horizontal translation of $3$ units to the left and a vertical translation of $1$ unit down?

A: The formula for a cube root function with a horizontal translation of $3$ units to the left and a vertical translation of $1$ unit down is:

$$g(x) = -\sqrt[3]{x + 3} - 1$$

Q: How do you graph a cube root function with a horizontal translation of $3$ units to the left and a vertical translation of $1$ unit down?

A: To graph a cube root function with a horizontal translation of $3$ units to the left and a vertical translation of $1$ unit down, you can use a graphing calculator or a computer program. You can set the window to $[-10, 10]$ and $[-10, 10]$ to get a good view of the graph.

Q: What is the difference between the graph of a cube root function with a horizontal translation of $3$ units to the left and a vertical translation of $1$ unit down and the graph of the parent function?

A: The graph of a cube root function with a horizontal translation of $3$ units to the left and a vertical translation of $1$ unit down is similar to the graph of the parent function, but it is not identical. The graph of the function has a horizontal translation of $3$ units to the left and a vertical translation of $1$ unit down.

Q: How do you write a Python code to graph a cube root function with a horizontal translation of $3$ units to the left and a vertical translation of $1$ unit down?

A: You can use the following Python code to graph a cube root function with a horizontal translation of $3$ units to the left and a vertical translation of $1$ unit down:

import numpy as np
import matplotlib.pyplot as plt

# Define the function
def g(x):
    return -np.cbrt(x + 3) - 1

# Define the parent function
def f(x):
    return np.cbrt(x)

# Create an array of x values
x = np.linspace(-10, 10, 400)

# Create an array of y values for the function
y = g(x)

# Create an array of y values for the parent function
y_parent = f(x)

# Create a plot
plt.plot(x, y, label='g(x)')
plt.plot(x, y_parent, label='f(x)')

# Add a title and labels
plt.title('Graph of a Cube Root Function with Horizontal and Vertical Translation')
plt.xlabel('x')
plt.ylabel('y')

# Add a legend
plt.legend()

# Show the plot
plt.show()

This code will create a plot of the function and the parent function. The plot will show the horizontal translation of $3$ units to the left and the vertical translation of $1$ unit down.

Q: What are some real-world applications of cube root functions?

A: Cube root functions have many real-world applications, including:

• Physics: Cube root functions are used to describe the motion of objects under the influence of gravity.
• Engineering: Cube root functions are used to describe the behavior of electrical circuits and mechanical systems.
• Computer Science: Cube root functions are used in algorithms for solving problems in computer science, such as sorting and searching.

Q: What are some common mistakes to avoid when working with cube root functions?

A: Some common mistakes to avoid when working with cube root functions include:

• Not checking the domain of the function: Make sure to check the domain of the function before graphing it.
• Not checking the range of the function: Make sure to check the range of the function before graphing it.
• Not using the correct formula: Make sure to use the correct formula for the cube root function.

By avoiding these common mistakes, you can ensure that you are working with cube root functions correctly and accurately.