Determining the Behavior of the Graph

Oct 13, 2025 by talynt.org 140 views

Introduction

In mathematics, graphing polynomial functions is a crucial aspect of understanding their behavior and characteristics. The given function, $f(x)=4x^7+40x^6+100x^5$ , is a polynomial of degree 7, which means it can have up to 7 real roots or x-intercepts. In this article, we will analyze the graph of this function and determine which statement accurately describes its behavior.

Analyzing the Function

To begin, let's examine the function $f(x)=4x^7+40x^6+100x^5$ . We can see that it is a polynomial of degree 7, which means it can have up to 7 real roots or x-intercepts. The leading coefficient of the function is 4, which is positive. This means that the function will have a positive leading term, and its graph will be increasing for large values of x.

Finding the x-Intercepts

To find the x-intercepts of the function, we need to set $f(x)=0$ and solve for x. This means that we need to find the values of x that make the function equal to zero. We can start by factoring out the greatest common factor (GCF) of the function, which is $4x^5$ . Factoring out $4x^5$ gives us:

f(x)=4x^5(10x^2+40x+25)

Now, we can see that the function has a factor of $4x^5$ , which means that it will have a root at $x=0$ . This is because when $x=0$ , the factor $4x^5$ becomes zero, and the entire function becomes zero.

Determining the Behavior of the Graph

Now that we have found the x-intercept at $x=0$ , we need to determine the behavior of the graph near this point. To do this, we can use the fact that the function has a factor of $4x^5$ . This means that the graph will have a vertical tangent line at $x=0$ , which is a line that is tangent to the graph at the point where $x=0$ .

Conclusion

Based on our analysis, we can conclude that the graph of the function $f(x)=4x^7+40x^6+100x^5$ touches the x-axis at $x=0$ and has a vertical tangent line at this point. This means that the graph will have a sharp turn at $x=0$ , and it will not cross the x-axis at this point.

Answer

The correct answer is B. The graph touches the x-axis at $x=0$ and has a vertical tangent line at this point.

Additional Analysis

To further analyze the graph, we can use the fact that the function has a factor of $4x^5$ . This means that the graph will have a root at $x=0$ , and it will have a vertical tangent line at this point. We can also use the fact that the function has a positive leading coefficient, which means that the graph will be increasing for large values of x.

Graphical Representation

The graph of the function $f(x)=4x^7+40x^6+100x^5$ can be represented graphically as follows:

The graph will have a root at $x=0$ , which means that it will touch the x-axis at this point.
The graph will have a vertical tangent line at $x=0$ , which means that it will have a sharp turn at this point.
The graph will be increasing for large values of x, which means that it will have a positive leading term.

Conclusion

Q: What is the degree of the polynomial function $f(x)=4x^7+40x^6+100x^5$ ?

A: The degree of the polynomial function $f(x)=4x^7+40x^6+100x^5$ is 7, which means it can have up to 7 real roots or x-intercepts.

Q: What is the leading coefficient of the polynomial function $f(x)=4x^7+40x^6+100x^5$ ?

A: The leading coefficient of the polynomial function $f(x)=4x^7+40x^6+100x^5$ is 4, which is a positive number.

Q: What is the x-intercept of the polynomial function $f(x)=4x^7+40x^6+100x^5$ ?

A: The x-intercept of the polynomial function $f(x)=4x^7+40x^6+100x^5$ is $x=0$ , which means that the graph touches the x-axis at this point.

Q: What is the behavior of the graph of the polynomial function $f(x)=4x^7+40x^6+100x^5$ near the x-intercept?

A: The graph of the polynomial function $f(x)=4x^7+40x^6+100x^5$ has a vertical tangent line at the x-intercept $x=0$ , which means that it has a sharp turn at this point.

Q: Is the graph of the polynomial function $f(x)=4x^7+40x^6+100x^5$ increasing or decreasing for large values of x?

A: The graph of the polynomial function $f(x)=4x^7+40x^6+100x^5$ is increasing for large values of x, which means that it has a positive leading term.

Q: Can the graph of the polynomial function $f(x)=4x^7+40x^6+100x^5$ cross the x-axis at multiple points?

A: No, the graph of the polynomial function $f(x)=4x^7+40x^6+100x^5$ cannot cross the x-axis at multiple points. It can only touch the x-axis at one point, which is the x-intercept $x=0$ .

Q: How can I determine the behavior of the graph of a polynomial function?

A: To determine the behavior of the graph of a polynomial function, you can use the following steps:

Find the x-intercepts of the function by setting it equal to zero and solving for x.
Determine the behavior of the graph near the x-intercepts by using the fact that the function has a factor of $x-a$ if it has a root at $x=a$ .
Use the leading coefficient of the function to determine the behavior of the graph for large values of x.

Q: What is the significance of the leading coefficient of a polynomial function?

A: The leading coefficient of a polynomial function determines the behavior of the graph for large values of x. If the leading coefficient is positive, the graph will be increasing for large values of x. If the leading coefficient is negative, the graph will be decreasing for large values of x.

Q: Can the graph of a polynomial function have multiple vertical tangent lines?

A: No, the graph of a polynomial function cannot have multiple vertical tangent lines. It can only have one vertical tangent line at the x-intercept.

Q: How can I graph a polynomial function?

A: To graph a polynomial function, you can use the following steps:

Find the x-intercepts of the function by setting it equal to zero and solving for x.
Determine the behavior of the graph near the x-intercepts by using the fact that the function has a factor of $x-a$ if it has a root at $x=a$ .
Use the leading coefficient of the function to determine the behavior of the graph for large values of x.
Plot the graph using a graphing calculator or a computer algebra system.

Conclusion

In conclusion, the graph of the polynomial function $f(x)=4x^7+40x^6+100x^5$ touches the x-axis at $x=0$ and has a vertical tangent line at this point. This means that the graph will have a sharp turn at $x=0$ , and it will not cross the x-axis at this point.