At Which Point Does the Graph of F(X) = (X – 5)3(X + 2)2 Touch the X-Axis? Solving for Roots -5, -2, 2, and 5

Have you ever come across a math problem that seemed impossible to solve? Well, fret not because with a little bit of practice and guidance, any problem can be solved. Today, we will be discussing a problem that has puzzled many students and mathematicians alike. The problem is: At which root does the graph of f(x) = (x – 5)³(x + 2)² touch the x-axis? Is it at -5, -2, 2, or 5?

Before we delve into the solution, let us first understand what the question is asking. The graph of f(x) is a cubic function, which means it has three roots. These roots are the points where the graph intersects the x-axis. In this question, we are asked to find the root at which the graph touches the x-axis. This is different from finding the roots of the equation, which would give us all three points where the graph intersects the x-axis.

Now, let us begin solving the problem. The first step is to factorize the equation f(x). We can do this by using the distributive property:

f(x) = (x – 5)³(x + 2)²

= (x – 5)(x – 5)(x – 5)(x + 2)(x + 2)

Next, we can plot the graph of f(x) using the roots we already know, which are x = -5, -2, 2, and 5. We can do this by using a graphing calculator or by hand. When we plot the graph, we will notice that it does not touch the x-axis at x = -5 or x = -2. This is because these roots are repeated, which means the graph only touches the x-axis but does not cross it.

However, when we plot the graph at x = 2 and x = 5, we will notice that the graph touches the x-axis at these points. This means that these are the roots we were looking for. But which one is the correct answer?

To find out, we need to analyze the behavior of the graph around these points. When we approach x = 2 from the left, we will notice that the graph is negative. As we get closer to x = 2, the graph starts to increase until it touches the x-axis. After this point, the graph becomes positive. This means that x = 2 is a local minimum of the function.

On the other hand, when we approach x = 5 from the left, we will notice that the graph is positive. As we get closer to x = 5, the graph starts to decrease until it touches the x-axis. After this point, the graph becomes negative. This means that x = 5 is a local maximum of the function.

Therefore, the answer to our question is that the graph of f(x) = (x – 5)³(x + 2)² touches the x-axis at x = 2 and x = 5. However, x = 2 is the root where the graph touches the x-axis for the first time and is therefore the answer we were looking for.

In conclusion, this problem may have seemed daunting at first, but with a little bit of practice and understanding of the behavior of functions, we can solve any problem. So, don't be afraid to tackle challenging math problems because they can be solved with the right approach.

Introduction

Mathematics is an essential subject that deals with numbers, quantities, and shapes. It plays a significant role in our daily lives, from calculating our expenses to building skyscrapers. Graphs are one of the essential tools used in mathematics to analyze and represent data. They can help us visualize the relationship between two variables and predict future outcomes. In this article, we will discuss the graph of the function F(x) = (x-5)3(x+2)2 and determine at which root it touches the X-axis.

Overview of the Function F(x) = (x-5)3(x+2)2

The function F(x) = (x-5)3(x+2)2 is a polynomial function that has two factors, (x-5) and (x+2). The exponents 3 and 2 indicate that these factors are multiplied by themselves three and two times, respectively. The function is a third-degree polynomial because the highest exponent is 3. When plotted on a graph, the function F(x) creates a curve that passes through various points.

What is the X-Axis?

The X-axis is the horizontal line on the coordinate plane that represents the values of the independent variable, x. It is the line on which the y-value or dependent variable is equal to zero. In other words, if a point lies on the X-axis, its y-coordinate is zero.

Determining the X-Intercepts of the Graph

To determine the roots of the function F(x), we need to find the values of x that make the function equal to zero. These values are called the x-intercepts or roots of the graph. To find the x-intercepts of the graph, we need to set the function equal to zero and solve for x.

F(x) = (x-5)3(x+2)2 = 0

Since the product of two factors is equal to zero if and only if one or both of the factors are zero, we can set each factor equal to zero and solve for x.

(x-5)3 = 0 or (x+2)2 = 0

Solving each equation, we get:

x = 5 or x = -2

What Does it Mean for a Graph to Touch the X-Axis?

For a graph to touch the X-axis, it means that the corresponding y-value or dependent variable is equal to zero at that point. In other words, the graph intersects the X-axis at that point.

Determining at which Root the Graph Touches the X-Axis

To determine at which root the graph of F(x) = (x-5)3(x+2)2 touches the X-axis, we need to substitute each root into the equation and check whether the output is equal to zero.

When x = 5, F(5) = (5-5)3(5+2)2 = 0

When x = -2, F(-2) = (-2-5)3(-2+2)2 = 0

Therefore, the graph of F(x) touches the X-axis at x = 5 and x = -2.

What is the Significance of the Roots in a Graph?

The roots of a graph are significant because they represent the points at which the graph crosses or touches the X-axis. They also indicate the solutions to equations that involve the graph. For example, if the graph represents a real-world problem, the roots might represent the times or values of a variable that satisfy the conditions of the problem.

Conclusion

In conclusion, the graph of the function F(x) = (x-5)3(x+2)2 touches the X-axis at x = 5 and x = -2. The roots of a graph are essential because they represent the points at which the graph crosses or touches the X-axis and indicate the solutions to equations that involve the graph. Understanding the basics of graphs and their properties can help us analyze and interpret data more effectively.

Introduction: Understanding the Graph of F(X) = (X – 5)3(X + 2)2

When given a function in the form of an equation, it is possible to graph it and analyze its behavior. One such function is F(X) = (X – 5)3(X + 2)2. This is a polynomial function that has five roots, which are the values of X that make the function equal to zero. By understanding the behavior of the roots, it is possible to identify where the graph touches the X-axis.

The Importance of Roots in Graphing

The roots of a function are essential in graphing because they indicate where the graph intersects the X-axis. These points are also known as zeros, and they help us determine the behavior of the function. When a function crosses the X-axis at a specific point, it changes signs. Therefore, it is crucial to identify the roots of a function to understand how it behaves.

Identifying the X Axis Touch Point

To identify where the graph of F(X) = (X – 5)3(X + 2)2 touches the X-axis, we need to find the roots of the function. The roots are -5, -2, 2, and 5. To determine where the graph touches the X-axis, we need to test each root and see if the function changes signs.

Solving for X with the Quadratic Formula

To test the roots, we can use the quadratic formula to solve for X. If the value of X is a real number, then the root exists, and the graph touches the X-axis at that point. The quadratic formula is X = [-b ± √(b^2 - 4ac)]/2a, where a, b, and c are the coefficients of the equation.

Testing the Negative Root of -5

Let's start by testing the negative root of -5. When we plug in -5 for X, we get F(-5) = (-10)3(-3)2 = -2700. Since the value is negative, the function changes signs and crosses the X-axis at that point. Therefore, the graph of F(X) touches the X-axis at -5.

Analyzing the Root of -2

Next, let's test the root of -2. When we plug in -2 for X, we get F(-2) = (3)32 = 27. Since the value is positive, the function does not change signs and does not cross the X-axis at that point. Therefore, the graph of F(X) does not touch the X-axis at -2.

Testing the Positive Root of 2

Now, let's test the positive root of 2. When we plug in 2 for X, we get F(2) = (-3)32 = -27. Since the value is negative, the function changes signs and crosses the X-axis at that point. Therefore, the graph of F(X) touches the X-axis at 2.

Investigating the Root of 5

Lastly, let's test the root of 5. When we plug in 5 for X, we get F(5) = (0)32 = 0. Since the value is zero, the function does not change signs and does not cross the X-axis at that point. Therefore, the graph of F(X) does not touch the X-axis at 5.

Confirming the Touch Point with Calculus

To confirm our findings, we can use calculus to find the critical points of the function. The critical points are where the derivative of the function is zero or undefined. When the derivative is zero, the function has a maximum or minimum value. When the derivative is undefined, the function has a vertical tangent.When we take the derivative of F(X), we get F'(X) = 5(X – 5)2(2X + 3). To find the critical points, we need to set the derivative equal to zero and solve for X. We get X = 5/2 and X = -3/2. These are the locations of the maximum and minimum values of the function, respectively.Since there are no critical points at -5, -2, 2, and 5, we can confirm that the graph of F(X) touches the X-axis at -5 and 2 and does not touch the X-axis at -2 and 5.

Conclusion: Understanding and Applying Knowledge of Roots and Graphing

In conclusion, understanding the roots of a function is essential in graphing and analyzing its behavior. By testing the roots and identifying where the function changes signs, we can determine where the graph touches the X-axis. Additionally, using calculus to find the critical points can confirm our findings and help us understand the behavior of the function better. By applying this knowledge, we can create accurate graphs and make informed decisions in various fields, such as finance, engineering, and science.

Discovering the Roots of F(X) = (X – 5)3(X + 2)2

The Question

At which root does the graph of F(X) = (X – 5)3(X + 2)2 touch the X axis? Is it at –5, –2, 2, or 5?

The Search for Answers

To find the answer to this question, we need to look at the properties of the function F(X). One of the most important properties is its roots, which are the values of X that make F(X) equal to zero. At these points, the graph of F(X) intersects with the X axis.

To begin our search for the roots of F(X), let's start by factoring the function:

F(X) = (X – 5)3(X + 2)2

By factoring, we can see that the function has two distinct roots: X = 5 and X = –2. However, these roots do not necessarily correspond to the points where the graph of F(X) touches the X axis. To find those points, we need to determine the multiplicity of each root.

The multiplicity of a root is the number of times it appears as a factor in the function. If a root has an odd multiplicity, the graph of the function crosses the X axis at that point. If a root has an even multiplicity, the graph touches the X axis but does not cross it.

Using our factored form of F(X), we can see that X = 5 has a multiplicity of 3, while X = –2 has a multiplicity of 2. This means that the graph of F(X) touches the X axis at X = –2, but does not cross it. The graph also crosses the X axis at X = 5, but we do not know whether it touches the X axis at that point or not.

To determine whether the graph of F(X) touches the X axis at X = 5, we need to look at the derivative of the function. The derivative tells us about the rate of change of the function at each point, and can help us identify points of inflection where the graph changes from concave up to concave down or vice versa.

Using the product rule and chain rule, we can find the derivative of F(X):

F'(X) = 3(X – 5)2(X + 2)2 + 2(X – 5)3(X + 2)

We can see that F'(X) is equal to zero at X = 5, which means that the graph of F(X) has a horizontal tangent line at that point. However, we cannot determine from this information alone whether the graph touches the X axis or not.

Table of Information

Function: F(X) = (X – 5)3(X + 2)2
Roots: X = 5, X = –2
Multiplicity: X = 5 (multiplicity 3), X = –2 (multiplicity 2)
Derivative: F'(X) = 3(X – 5)2(X + 2)2 + 2(X – 5)3(X + 2)
X-intercepts: X = –2 (touches), X = 5 (unknown)

The Empathic Voice and Tone

We can imagine that someone who is struggling with this question might feel frustrated or confused. They may be unsure how to approach the problem, and may feel overwhelmed by the complexity of the function.

As we explore the properties of F(X) and search for its roots, we can use an empathic voice and tone to help guide the reader through the process. We might acknowledge the difficulty of the question and offer encouragement, reminding the reader that they are capable of solving it.

We might also use a clear and concise writing style, breaking down complex concepts into simpler terms and using examples to illustrate our points. By taking the time to explain each step in the process, we can help the reader gain a deeper understanding of the problem and build their confidence in tackling similar questions in the future.

With patience, persistence, and a willingness to learn, anyone can discover the roots of F(X) and solve even the most challenging mathematical problems.

Closing Message:

Thank you for taking the time to read our article on At Which Root Does The Graph Of F(X) = (X – 5)3(X + 2)2 Touch The X Axis?. We hope that you have found it informative and useful in your studies of mathematics.

It is important to remember that when working with functions and graphs, it is crucial to understand the concepts and equations involved. By knowing the key terms and formulas, we can better analyze and interpret the data presented to us.

Throughout this article, we have discussed various methods of finding the roots of a function and how to determine where the graph touches the x-axis. We have explored the different techniques used to solve equations, including factoring, using the quadratic formula, and graphing on a coordinate plane.

We have also highlighted the importance of using clear and concise language when describing mathematical concepts. By breaking down complex equations into simpler terms and explaining each step in detail, we can help readers better understand the material.

Furthermore, we have emphasized the need for practice and perseverance when studying math. It is essential to work through problems systematically and to seek help when needed. By practicing regularly and seeking out additional resources, such as tutors or online forums, we can develop a deeper understanding of the subject matter.

Finally, we want to remind our readers that math is an essential subject that plays a vital role in many areas of our lives. From calculating taxes to designing buildings and bridges, math is used in countless applications. By mastering the concepts and skills presented in this article, we can become more confident and effective problem-solvers in our daily lives.

Once again, thank you for visiting our blog and reading our article on At Which Root Does The Graph Of F(X) = (X – 5)3(X + 2)2 Touch The X Axis?. We hope that you have found it informative and engaging. If you have any questions or comments, please feel free to leave them below.