At Which X Root Does the Graph of F(X) = (X – 5)3(X + 2)2 Touch the X-Axis? Understanding the Critical Points: -5, -2, 2, or 5

At Which Root Does The Graph Of F(X) = (X – 5)3(X + 2)2 Touch The X-Axis? –5 –2 2 5

Find the roots of f(x)=(x-5)^3(x+2)^2 to determine where its graph touches the x-axis. The options are -5, -2, 2, and 5.

Are you struggling to solve the equation F(x) = (x – 5)3(x + 2)2? Do you want to know at which root the graph of the function touches the x-axis? If so, then this article is for you. In this article, we will guide you through the steps to find out the answer to this question. So, sit back and relax while we take you on a journey to discover the roots of the function.

Before diving into finding the roots, let's first understand what it means when a graph touches the x-axis. When a graph touches the x-axis, it means that the y-coordinate of the point where the graph intersects the x-axis is zero. In other words, the value of the function at that point is zero. Now, let's move on to finding the roots of the given function.

To find the roots of the function F(x) = (x – 5)3(x + 2)2, we need to set the function equal to zero and solve for x. This is because the roots of the function are the values of x for which the function equals zero. So, let's write the equation:

(x – 5)3(x + 2)2 = 0

Now, we can use the zero product property to solve for x. According to this property, if the product of two factors is zero, then at least one of the factors must be zero. So, we can set each factor equal to zero and solve for x:

(x – 5)3 = 0 or (x + 2)2 = 0

Solving for x in the first equation, we get:

x – 5 = 0

x = 5

So, we have found one root of the function, which is x = 5. But, what about the second equation? Let's solve for x in the second equation:

x + 2 = 0

x = –2

So, we have found another root of the function, which is x = –2. Now, we need to check whether these roots make the graph touch the x-axis or not. To do this, we need to plot the graph of the function and see where it intersects the x-axis.

To plot the graph of the function, we can use a graphing calculator or software. The graph looks like this:

Insert Graph Here

From the graph, we can see that the function touches the x-axis at x = 5 and x = –2. So, the answer to the question at which root does the graph of F(x) = (x – 5)3(x + 2)2 touch the x-axis? is both –2 and 5.

In conclusion, finding the roots of a function is an important task in mathematics. By setting the function equal to zero and solving for x, we can find the values of x for which the function equals zero. These values are called the roots of the function. In this article, we solved the equation F(x) = (x – 5)3(x + 2)2 and found that the graph of the function touches the x-axis at x = –2 and x = 5. We hope this article has helped you understand how to find the roots of a function and how to use them to analyze the graph of the function.

The Equation of F(x)

F(x) is a polynomial function with a degree of 5, which means that it has 5 roots. To find the x-intercepts (or where the graph touches the x-axis), we need to solve for the roots of the equation F(x) = 0. The equation for F(x) can be expressed as: F(x) = (x – 5)^3(x + 2)^2. This equation represents a curve that intersects the x-axis at various points.

Understanding Roots

The roots of a polynomial function are the values of x that make the function equal to zero. The roots can be found by setting the function equal to zero and solving for x. If the function crosses the x-axis at a specific point, then that point is a root of the function. If the function touches the x-axis without crossing it, then that point is a double root.

Finding the Roots of F(x)

To find the roots of F(x), we need to set the equation equal to zero and solve for x. Therefore, we have:

(x – 5)^3(x + 2)^2 = 0

We can simplify this equation by factoring out the common factors:

(x – 5)^3(x + 2)^2 = (x – 5)(x – 5)(x – 5)(x + 2)(x + 2) = 0

From this factorization, we can see that the roots of F(x) are x = 5 and x = -2. Both of these roots have a multiplicity of 3 and 2, respectively. This means that the graph of F(x) touches the x-axis at x = 5 and x = -2, but does not cross it.

Visualizing the Graph of F(x)

To visualize the graph of F(x), we can use a graphing calculator or software. The graph of F(x) is a polynomial curve that has two turning points at x = -2 and x = 5. The curve also touches the x-axis at these two points, as we have found earlier.

The Point of Contact

The question asks us to find the point where the graph of F(x) touches the x-axis. This means that we need to find the coordinates of the two points of contact, which are (5,0) and (-2,0). These points represent the double roots of the equation, as we have discussed earlier.

The Behavior of the Graph Near the Roots

When a function has a root with an odd multiplicity (such as 1 or 3), the graph crosses the x-axis at that point. When a function has a root with an even multiplicity (such as 2 or 4), the graph touches the x-axis at that point without crossing it.

In the case of F(x), both roots have an even multiplicity, which means that the graph touches the x-axis at these points without crossing it. Therefore, the behavior of the graph near the roots is a horizontal tangent line.

The Sign of F(x) Near the Roots

The sign of F(x) near the roots can tell us whether the graph is above or below the x-axis. When F(x) is positive near a root, the graph is above the x-axis. When F(x) is negative near a root, the graph is below the x-axis.

In the case of F(x), when x is less than -2, F(x) is negative. When x is between -2 and 5, F(x) is positive. When x is greater than 5, F(x) is negative. Therefore, the graph of F(x) is below the x-axis to the left of x = -2, above the x-axis between x = -2 and x = 5, and below the x-axis to the right of x = 5.

The Behavior of the Graph at Infinity

The behavior of the graph of F(x) at infinity can tell us whether the graph is going up or down as it approaches infinity in either direction. This can be determined by looking at the leading coefficient of the polynomial function. The leading coefficient is the coefficient of the term with the highest degree.

In the case of F(x), the leading coefficient is positive, which means that the graph of F(x) is going up as it approaches infinity in either direction.

The Shape of the Graph Near the Turning Points

The turning points of a polynomial function are the points where the graph changes direction from increasing to decreasing or vice versa. These points can be found by taking the derivative of the function and setting it equal to zero.

In the case of F(x), the turning points are at x = -2 and x = 5. The shape of the graph near these points is a horizontal inflection point.

Conclusion

In conclusion, the graph of F(x) = (x – 5)^3(x + 2)^2 touches the x-axis at x = -2 and x = 5. Both of these points are double roots with an even multiplicity. The behavior of the graph near these points is a horizontal tangent line. The sign of F(x) near the roots tells us whether the graph is above or below the x-axis. The behavior of the graph at infinity tells us that the graph is going up as it approaches infinity in either direction. The shape of the graph near the turning points is a horizontal inflection point.

Understanding the Nature of F(x)

To determine at which root the graph of f(x) = (x – 5)3(x + 2)2 touches the x-axis, we first need to understand the nature of the function. F(x) is a polynomial function with two factors: (x – 5) and (x + 2). The exponent of each factor represents the number of times it is multiplied by itself. In this case, (x – 5) is raised to the power of 3, and (x + 2) is raised to the power of 2.

Analyzing the Roots of F(x)

The roots of f(x) are the values of x that make the function equal to zero. To find the roots of f(x), we set the function equal to zero and solve for x. So, we have:

f(x) = 0

(x – 5)3(x + 2)2 = 0

Using the zero product property, we know that if a product of factors equals zero, then at least one of the factors must be zero. Therefore, we can set each factor equal to zero and solve for x:

(x – 5) = 0 or (x + 2) = 0

x = 5 or x = –2

Exploring the Different Roots of F(x)

As we can see, f(x) has two roots: x = 5 and x = –2. These roots represent the points where the graph of f(x) intersects the x-axis. When x = 5, one of the factors of f(x) is equal to zero, which means the graph of f(x) touches the x-axis at x = 5. Similarly, when x = –2, the other factor of f(x) is equal to zero, which means the graph of f(x) touches the x-axis at x = –2.

Investigating the Point Where F(x) Touches the X-Axis

The point where f(x) touches the x-axis is called the root or the zero of the function. In this case, we have two roots: x = 5 and x = –2. When the graph of f(x) touches the x-axis at a particular root, it means that the function has a real and equal solution for that value of x. This is because when the graph touches the x-axis, it means that the y-value of the function is zero.

Understanding What it Means for F(x) to Touch the X-Axis

When the graph of f(x) touches the x-axis, it means that the function has a real and equal solution for the root where it intersects the x-axis. This is because the x-axis represents the values of x where the y-value of the function is equal to zero. Therefore, when the graph touches the x-axis, it means that the function has a root at that point.

Identifying the Key Points on F(x)

To identify the key points on f(x), we need to analyze the behavior of the graph of the function. The graph of f(x) is a polynomial function, which means that it is continuous and smooth. It also has a degree of 5, which means that it has up to 4 turning points or inflection points. The key points on f(x) are the x-intercepts, the y-intercept, and any turning points or inflection points.

Analyzing the Graph of F(x)

The graph of f(x) is a polynomial function with two roots: x = 5 and x = –2. The graph of the function intersects the x-axis at these two points. At x = 5, the graph touches the x-axis with a minimum turning point. This means that the function changes from decreasing to increasing at this point. At x = –2, the graph touches the x-axis with a maximum turning point. This means that the function changes from increasing to decreasing at this point.

Using Algebra to Solve for the Root of F(x)

To solve for the root of f(x) at x = 5, we can substitute x = 5 into the function:

f(5) = (5 – 5)3(5 + 2)2 = 0

Therefore, the root of f(x) at x = 5 is zero.

To solve for the root of f(x) at x = –2, we can substitute x = –2 into the function:

f(–2) = (–2 – 5)3(–2 + 2)2 = 0

Therefore, the root of f(x) at x = –2 is also zero.

Interpreting the Solution for F(x)

The solution for f(x) tells us that the graph of the function touches the x-axis at x = 5 and x = –2. This means that the function has real and equal solutions for these values of x. The roots of f(x) are also the x-intercepts of the graph of the function. The fact that the roots of f(x) are both zero means that the graph of the function touches the x-axis but does not cross it.

Applying the Solution to Real World Scenarios

The solution for f(x) can be applied to real-world scenarios where polynomial functions are used to model data. For example, a polynomial function may be used to model the growth of a population over time. The roots of the function would represent the points where the population stops growing or starts declining. In this case, if the roots of the function are both zero, it means that the population reaches a steady state where it is neither growing nor declining.

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

A Quest for the Point of Intersection

Ah, the joys of mathematics! The thrill of solving complex equations, the satisfaction of discovering the perfect solution, and the frustration of getting stuck on a seemingly impossible problem.

But for those who love numbers and patterns, there is nothing quite like the feeling of uncovering the root of an equation. And that is exactly what I set out to do when faced with the task of finding where the graph of F(X) = (X – 5)3(X + 2)2 touches the x-axis.

The Journey Begins

Armed with my trusty calculator and a determination to succeed, I began my quest. First, I had to understand the basics of the equation itself. F(X) is simply a polynomial function with two factors: (X – 5)3 and (X + 2)2. This means that the graph of the function will have two roots, or points where it intersects the x-axis.

But which roots are they? To find out, I needed to set F(X) equal to zero and solve for X:

F(X) = (X – 5)3(X + 2)2 = 0

Expanding the equation, I got:

X⁵ – 3X⁴ - 34X³ + 94X² + 120X – 200 = 0

The Final Destination

Now came the tricky part. I needed to factor the equation and solve for X. After several attempts and a lot of scribbling, I finally discovered the two roots:

X = -2
X = 5

These were the points where the graph of F(X) would intersect the x-axis. But which one was the point where it touched, rather than crossed, the axis?

To find out, I had to look at the behavior of the function around each root. At X = -2, the graph of F(X) would cross the x-axis from negative to positive values, indicating that it did not touch the axis at that point. But at X = 5, the graph of F(X) would pass through the x-axis with a zero slope, indicating that it did indeed touch the axis at that point.

The Joy of Discovery

And there it was, the answer to my quest: the root at X = 5 was the point where the graph of F(X) = (X – 5)3(X + 2)2 touched the x-axis. It was a small victory, but one that filled me with pride and satisfaction.

For those who love numbers and patterns, there is nothing quite like the feeling of uncovering the root of an equation. And for me, the joy of discovery is what makes the journey worth it.

Keywords	Definition
Root	A value of X that makes the equation equal to zero
Polynomial Function	An expression that consists of variables and coefficients, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents
Factor	A number or expression that divides another number or expression evenly, leaving no remainder
X-Axis	The horizontal line on a graph that represents the values of X
Intersection	The point where two lines or curves meet or cross each other

Closing Message: Discovering the Roots of a Graph

Thank you for taking the time to read through our article on the roots of the graph of f(x)= (x-5)3(x+2)2. We hope that we have provided valuable insights and explanations that helped you understand how to solve problems like these. Remember, understanding the roots of a graph is crucial in many fields, especially in science, engineering, and mathematics.

It is essential to note that finding the roots of a graph requires patience, practice, and a positive mindset. The process may seem daunting at first, but with a little bit of effort and perseverance, you can master it. We encourage you to keep practicing and applying what you learned today.

Furthermore, it is worth mentioning that the use of technology can help you find the roots of a graph more efficiently. There are many software programs and online tools available that can help you visualize and analyze graphs. However, we advise caution when using these tools, as they may not always provide accurate results, especially if you do not know how to use them correctly.

Lastly, we would like to invite you to explore more topics related to mathematics and science. Learning never stops, and there is always something new to discover. We believe that by continuously expanding your knowledge, you can become a well-rounded individual who can contribute positively to society.

Once again, thank you for visiting our blog. We hope that you have gained valuable insights and that you will share this information with others. Remember, understanding the roots of a graph is an essential skill that can help you succeed in many fields of study and work.

Keep exploring, keep learning, and keep growing!

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