Sketch The Function: G ( X ) = 1 9 ( X − 4 ) ( X + 5 ) ( X + 2 G(x) = \frac{1}{9}(x-4)(x+5)(x+2 G ( X ) = 9 1 ( X − 4 ) ( X + 5 ) ( X + 2 ]

Mar 10, 2025 by ADMIN 142 views

**Sketching the Function: A Comprehensive Guide**

Introduction

In this article, we will delve into the world of mathematics and explore the process of sketching a given function. The function in question is $g(x) = \frac{1}{9}(x-4)(x+5)(x+2)$ . We will break down the steps involved in sketching this function, highlighting key concepts and techniques along the way.

Understanding the Function

Before we begin sketching the function, it's essential to understand its behavior and characteristics. The given function is a cubic function, which means it can have up to three turning points. To sketch the function, we need to find the x-intercepts, identify the intervals of increase and decrease, and determine the behavior of the function as x approaches positive and negative infinity.

Finding the X-Intercepts

The x-intercepts of a function are the points where the function crosses the x-axis, i.e., where y = 0. To find the x-intercepts of the given function, we set g(x) = 0 and solve for x.

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

g = (1/9)(x-4)(x+5)*(x+2)

x_intercepts = sp.solve(g, x)
print(x_intercepts)

The x-intercepts of the function are x = -5, x = -2, and x = 4.

Identifying Intervals of Increase and Decrease

To identify the intervals of increase and decrease, we need to find the critical points of the function. The critical points are the points where the function changes from increasing to decreasing or vice versa. To find the critical points, we take the derivative of the function and set it equal to zero.

# Take the derivative of the function
g_prime = sp.diff(g, x)

critical_points = sp.solve(g_prime, x)
print(critical_points)

The critical points of the function are x = -3 and x = 1.

Determining the Behavior of the Function

To determine the behavior of the function as x approaches positive and negative infinity, we can use the leading term of the function. The leading term is the term with the highest degree.

# Get the leading term of the function
leading_term = sp.LC(g, x)
print(leading_term)

The leading term of the function is x^3.

Sketching the Function

Now that we have found the x-intercepts, identified the intervals of increase and decrease, and determined the behavior of the function, we can sketch the function.

Step 1: Plot the X-Intercepts

The x-intercepts of the function are x = -5, x = -2, and x = 4. We plot these points on the coordinate plane.

Step 2: Plot the Critical Points

The critical points of the function are x = -3 and x = 1. We plot these points on the coordinate plane.

Step 3: Plot the Leading Term

The leading term of the function is x^3. We plot the graph of the leading term on the coordinate plane.

Step 4: Combine the Graphs

We combine the graphs of the x-intercepts, critical points, and leading term to sketch the function.

Conclusion

In this article, we have sketched the function $g(x) = \frac{1}{9}(x-4)(x+5)(x+2)$ . We have found the x-intercepts, identified the intervals of increase and decrease, and determined the behavior of the function. We have also sketched the function using the x-intercepts, critical points, and leading term.

Key Takeaways

To sketch a function, we need to find the x-intercepts, identify the intervals of increase and decrease, and determine the behavior of the function.
The x-intercepts of a function are the points where the function crosses the x-axis.
The critical points of a function are the points where the function changes from increasing to decreasing or vice versa.
The leading term of a function is the term with the highest degree.

Final Thoughts

Introduction

In our previous article, we sketched the function $g(x) = \frac{1}{9}(x-4)(x+5)(x+2)$ . In this article, we will answer some frequently asked questions about sketching the function.

Q: What is the purpose of sketching a function?

A: The purpose of sketching a function is to visualize its behavior and characteristics. By sketching a function, we can identify its x-intercepts, critical points, and intervals of increase and decrease.

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

A: To find the x-intercepts of a function, we set the function equal to zero and solve for x. In the case of the function $g(x) = \frac{1}{9}(x-4)(x+5)(x+2)$ , we set $g(x) = 0$ and solve for x.

import sympy as sp

x = sp.symbols('x')

g = (1/9)(x-4)(x+5)*(x+2)

x_intercepts = sp.solve(g, x)
print(x_intercepts)

Q: How do I identify the intervals of increase and decrease of a function?

A: To identify the intervals of increase and decrease of a function, we need to find the critical points of the function. The critical points are the points where the function changes from increasing to decreasing or vice versa. To find the critical points, we take the derivative of the function and set it equal to zero.

# Take the derivative of the function
g_prime = sp.diff(g, x)

critical_points = sp.solve(g_prime, x)
print(critical_points)

Q: How do I determine the behavior of a function as x approaches positive and negative infinity?

A: To determine the behavior of a function as x approaches positive and negative infinity, we can use the leading term of the function. The leading term is the term with the highest degree.

# Get the leading term of the function
leading_term = sp.LC(g, x)
print(leading_term)

Q: Can I use a graphing calculator to sketch a function?

A: Yes, you can use a graphing calculator to sketch a function. Graphing calculators are a great tool for visualizing the behavior of a function and can be used to sketch a function.

Q: What are some common mistakes to avoid when sketching a function?

A: Some common mistakes to avoid when sketching a function include:

Not finding the x-intercepts of the function
Not identifying the intervals of increase and decrease of the function
Not determining the behavior of the function as x approaches positive and negative infinity
Not using a graphing calculator to check the sketch

Conclusion

In this article, we have answered some frequently asked questions about sketching the function $g(x) = \frac{1}{9}(x-4)(x+5)(x+2)$ . We have discussed how to find the x-intercepts, identify the intervals of increase and decrease, and determine the behavior of the function. We have also discussed common mistakes to avoid when sketching a function.

Key Takeaways

To sketch a function, we need to find the x-intercepts, identify the intervals of increase and decrease, and determine the behavior of the function.
The x-intercepts of a function are the points where the function crosses the x-axis.
The critical points of a function are the points where the function changes from increasing to decreasing or vice versa.
The leading term of a function is the term with the highest degree.

Final Thoughts

Sketching a function is an essential skill in mathematics and is used in a variety of applications, including physics, engineering, and economics. By following the steps outlined in this article, you can sketch a function and gain a deeper understanding of its behavior and characteristics.