Discovering the Point where the Graph of F(X) = (X - 5)3(X + 2)2 Intersects the X-Axis - An Insightful Guide
Learn how to find the root of F(x) = (x-5)^3(x+2)^2 and pinpoint where its graph touches the x-axis. Mathematical problem-solving made easy!
If you're a math enthusiast, you know how fascinating it is to solve complex equations and problems. One such problem is to find the point where the graph of f(x) = (x – 5)3(x + 2)2 touches the x-axis. This problem requires a deep understanding of calculus and the properties of graphs. So, if you're ready to dive into the world of math, let's explore this intriguing problem together.
Firstly, we need to understand what it means for a graph to touch the x-axis. When a graph touches the x-axis, it means that the y-value of that point is zero. In other words, we need to find the value of x when f(x) equals zero. This value is also known as the x-intercept of the graph.
To solve this problem, we need to factorize the given equation. By using the distributive property, we can expand the equation to get:
f(x) = (x – 5)(x – 5)(x – 5)(x + 2)(x + 2)
Now, we can see that the roots of f(x) are x = 5 and x = -2. But, we need to find at which root does the graph touch the x-axis. To do this, we need to look at the multiplicity of each root.
The multiplicity of a root is the number of times it appears in the factorization of the equation. In this case, the root x = 5 has a multiplicity of 3, and the root x = -2 has a multiplicity of 2. The multiplicity of a root determines the behavior of the graph near that point.
When a root has an odd multiplicity, the graph crosses the x-axis at that point. When a root has an even multiplicity, the graph touches the x-axis but does not cross it. In this case, since the root x = 5 has a multiplicity of 3 (which is odd), the graph crosses the x-axis at x = 5.
Now, let's take a closer look at the behavior of the graph near x = -2. Since the root x = -2 has a multiplicity of 2 (which is even), the graph touches the x-axis at x = -2 but does not cross it. To determine whether the graph touches the x-axis from above or below at x = -2, we need to look at the sign of f(x) on either side of x = -2.
If f(x) is positive on both sides of x = -2, then the graph touches the x-axis from above. If f(x) is negative on both sides of x = -2, then the graph touches the x-axis from below. If f(x) changes sign at x = -2, then the graph crosses the x-axis at that point.
To find the sign of f(x) on either side of x = -2, we can use a test point. Let's choose x = -3 as our test point. Plugging x = -3 into the equation, we get:
f(-3) = (-8)(1) = -8
Since f(-3) is negative, we know that the graph touches the x-axis from below at x = -2. Therefore, the answer to our problem is x = -2.
In conclusion, we have solved the problem of finding the point where the graph of f(x) = (x – 5)3(x + 2)2 touches the x-axis. By factorizing the equation and analyzing the multiplicity of each root, we determined that the graph crosses the x-axis at x = 5 and touches it from below at x = -2. This problem required a deep understanding of calculus and the properties of graphs, but with patience and perseverance, we were able to solve it.
Introduction
Mathematics is a subject that requires logical thinking and problem-solving skills. One of the most common problems that students face in mathematics is finding the roots of a function. In this article, we will discuss the graph of f(x) = (x – 5)3(x + 2)2 and at which root it touches the x-axis.
Finding the Roots of a Function
The roots of a function are the values of x that make the function equal to zero. To find the roots of a function, we need to set the function equal to zero and solve for x. In this case, we have the function f(x) = (x – 5)3(x + 2)2, so we need to solve for x when f(x) = 0.
The Zero Product Property
The zero product property states that if ab = 0, then either a or b must be equal to zero. Using this property, we can solve for the roots of the function by setting each factor equal to zero.
x – 5 = 0
To find the first root, we set x – 5 equal to zero and solve for x.
x – 5 = 0
x = 5
x + 2 = 0
To find the second root, we set x + 2 equal to zero and solve for x.
x + 2 = 0
x = -2
The Behavior of the Function
Now that we have found the roots of the function, we can analyze the behavior of the function near these points. The behavior of the function can be determined by looking at the sign of each factor.
When x < -2
When x is less than -2, both factors are negative. Since two negative numbers multiplied together give a positive result, f(x) is positive when x < -2.
When -2 < x < 5
When x is between -2 and 5, one factor is negative and one factor is positive. Since one negative number and one positive number multiplied together give a negative result, f(x) is negative when -2 < x < 5.
When x > 5
When x is greater than 5, both factors are positive. Since two positive numbers multiplied together give a positive result, f(x) is positive when x > 5.
The Graph of the Function
Based on the behavior of the function, we can sketch the graph of f(x) = (x – 5)3(x + 2)2. The graph will touch the x-axis at the root x = 5 since the function changes sign from positive to negative at this point. The graph will also cross the x-axis at the root x = -2 since the function changes sign from negative to positive at this point.
Conclusion
In conclusion, the function f(x) = (x – 5)3(x + 2)2 touches the x-axis at the root x = 5. To find this root, we used the zero product property to solve for x when f(x) = 0. We also analyzed the behavior of the function near the roots to sketch the graph of the function. With this knowledge, we can better understand how functions behave and solve problems related to finding the roots of a function.
Understanding the Problem at Hand
Before we can solve the problem of finding where the graph of F(x) = (x – 5)3(x + 2)2 touches the x-axis, we must first understand what we are looking for and what information we have been given.The Function F(x)
The function F(x) is a polynomial of degree 5, represented by the expression F(x) = (x – 5)3(x + 2)2.The X-Axis
The x-axis is the horizontal line that the graph of F(x) intersects at points where y = 0.Finding the X-Intercepts
To find where the graph of F(x) intersects the x-axis, we need to find the values of x where F(x) = 0.Factoring the Expression
To solve for x, we will need to factor the expression F(x) = (x – 5)3(x + 2)2.Exploring the Factors
When exploring the factors of the expression, we can see that there are two factors: (x – 5) and (x + 2).Necessary Conditions for Intersection
For the graph of F(x) to touch the x-axis, the factors of the expression must have a common root or zero.Evaluating the Factors
To find where the factors intersect, we set each factor equal to zero and solve for x.Analyzing the Results
After solving the factors, we obtain x = 5 and x = -2. These represent the values where the factors intersect and where the graph of F(x) touches the x-axis.Conclusion
In conclusion, we found that the graph of F(x) = (x – 5)3(x + 2)2 touches the x-axis at two points, x = 5 and x = -2. By understanding the problem at hand, exploring the factors of the expression, and evaluating the factors to find their intersection, we were able to determine where the graph of F(x) intersects the x-axis.Storytelling: At Which Root Does The Graph Of F(X) = (X – 5)3(X + 2)2 Touch The X Axis?
The Search for the Root
Once upon a time, there was a curious mathematician named Alex. Alex loved solving complex equations and finding the roots of graphs. One day, Alex stumbled upon a graph with the equation f(x) = (x – 5)3(x + 2)2. As Alex studied the graph, they noticed that it didn't touch the x-axis at all points. Alex was determined to find the root at which the graph touched the x-axis.The Process of Deduction
Alex started by analyzing the equation and breaking it down into its factors. They found that the equation had two factors: (x – 5) and (x + 2). Next, Alex plotted the graph of each factor separately. They noticed that the graph of (x – 5) touched the x-axis at x = 5, while the graph of (x + 2) touched the x-axis at x = -2.Alex then analyzed the graph of the entire equation and realized that the graph touched the x-axis when the two factors intersected. Alex deduced that the graph would touch the x-axis at the root where the two factors intersected.The Discovery
Alex used their knowledge of algebraic equations to solve for the root where the two factors intersected. They set the two factors equal to each other and solved for x. (x – 5) = -(x + 2)2x = 3x = 3/2Alex's deduction was correct! The graph of f(x) = (x – 5)3(x + 2)2 touches the x-axis at the root x = 3/2.Point of View: At Which Root Does The Graph Of F(X) = (X – 5)3(X + 2)2 Touch The X Axis?
The Importance of Finding the Root
As a student of mathematics, I have always been fascinated by the idea of finding roots of graphs. It is an essential concept in mathematics and has several real-world applications. When I came across the graph of f(x) = (x – 5)3(x + 2)2, I was intrigued by its shape and structure. However, I noticed that it didn't touch the x-axis at all points. I knew that there had to be a root where the graph touched the x-axis, and I was determined to find it.The Process of Deduction
I started by analyzing the equation and breaking it down into its factors. I plotted the graph of each factor separately and noticed that they intersected at two points - x=5 and x=-2. Next, I analyzed the graph of the entire equation and realized that the graph touched the x-axis when the two factors intersected. I deduced that the graph would touch the x-axis at the root where the two factors intersected.Using algebraic equations, I solved for the root where the two factors intersected.The Discovery
After solving for the root, I discovered that the graph of f(x) = (x – 5)3(x + 2)2 touches the x-axis at x=3/2. It was a moment of triumph for me as I had successfully found the root of the graph.Table Information: {{keywords}}
Here is a table with information about the keywords related to the story:
| Keyword | Definition |
|---|---|
| Root | The point where a graph intersects with the x-axis |
| Factor | An expression that can be multiplied with another expression to produce a product |
| Deduction | The process of using logic and reasoning to arrive at a conclusion |
| Equation | A mathematical statement that shows the equality of two expressions |
| Graph | A visual representation of an equation or function |
Understanding these keywords is essential to solving mathematical problems and finding roots of complex equations.
Closing Message: Thank you for exploring the root of F(x) with us!
As we conclude our journey to discover the root of F(x) = (x-5)^3(x+2)^2, we want to thank you for joining us on this exciting learning adventure. We hope that you have gained valuable insights into how to identify roots and interpret graphs.
Our exploration began by breaking down the equation into its basic components, understanding the concept of roots and how they are represented on a graph. We then delved into the different types of roots, including real, complex, and imaginary roots, and how to determine their coordinates.
The next step was to obtain the derivative of F(x), which gave us the slope of the tangent line at any given point on the graph. By analyzing the behavior of the derivative at various points, we were able to determine the nature of the roots and their relationship with each other.
With this knowledge, we were able to identify the critical points and inflection points of the graph, which helped us understand the behavior of the function as it approaches the x-axis.
We also explored the significance of symmetry in graphs and how it can be used to predict the behavior of the function. By observing the symmetry of the graph of F(x), we were able to determine that it touches the x-axis at a single point.
Finally, we used algebraic techniques to determine the exact location of the root and confirm our findings. By setting F(x) equal to zero and solving the resulting equation, we found that the graph touches the x-axis at x = 5 and x = -2.
As we wrap up our discussion, we want to reiterate the importance of understanding roots and their relationship with graphs. This knowledge can be applied to a wide range of fields, including engineering, physics, and computer science, making it a valuable tool for anyone looking to pursue a career in STEM.
Once again, we thank you for joining us on this journey, and we hope that you have found our discussion informative and engaging. We look forward to exploring more exciting topics with you in the future!
People Also Ask About At Which Root Does The Graph Of F(X) = (X – 5)3(X + 2)2 Touch The X Axis?
What is the meaning of touching the x-axis in a graph?
When a graph of a function touches the x-axis, it means that the corresponding y-value of that point is equal to zero. This point is called the x-intercept or the root of the function.
How do you find the roots of a function?
To find the roots of a function, we need to set the value of y (or f(x)) equal to zero and solve for x. In other words, we need to find the values of x that make the function equal to zero. These values are called the roots or the zeros of the function.
What is the degree of the function f(x) = (x – 5)3(x + 2)2?
The degree of a polynomial function is the highest power of the variable in the function. In this case, the highest power of x is 3, so the degree of the function f(x) = (x – 5)3(x + 2)2 is 5.
How do you determine if a function touches the x-axis at a certain point?
If a function touches the x-axis at a certain point, it means that the value of the function at that point is equal to zero. To determine if a function touches the x-axis at a certain point, we need to substitute that point's x-value into the function and see if the result is zero.
What is the root of the function f(x) = (x – 5)3(x + 2)2?
Since the function f(x) = (x – 5)3(x + 2)2 is a polynomial function, its roots correspond to the x-intercepts of its graph. To find the roots of the function, we need to set the function equal to zero and solve for x:
(x – 5)3(x + 2)2 = 0
Setting each factor equal to zero gives us:
- x – 5 = 0, which gives us x = 5
- x + 2 = 0, which gives us x = -2
Therefore, the graph of the function f(x) = (x – 5)3(x + 2)2 touches the x-axis at the points (5,0) and (-2,0).
Conclusion
The roots or x-intercepts of a function correspond to the points where the graph of the function touches the x-axis. To find the roots of a polynomial function, we need to set the function equal to zero and solve for x. In the case of the function f(x) = (x – 5)3(x + 2)2, the graph touches the x-axis at the points (5,0) and (-2,0).