Which Statement Describes The Graph Of F ( X ) = − X 4 + 3 X 3 + 10 X 2 F(x)=-x^4+3x^3+10x^2 F ( X ) = − X 4 + 3 X 3 + 10 X 2 ?A. The Graph Crosses The X X X -axis At X = 0 X=0 X = 0 And Touches The X X X -axis At X = 5 X=5 X = 5 And X = − 2 X=-2 X = − 2 .B. The Graph Touches The X X X -axis At

Introduction

When analyzing the graph of a polynomial function, it's essential to understand the behavior of the function, including its roots, turning points, and asymptotes. In this article, we will delve into the graph of the function $f(x)=-x^4+3x^3+10x^2$ and determine which statement accurately describes its behavior.

Analyzing the Function

The given function is a quartic polynomial, which means it has a degree of 4. The general form of a quartic polynomial is $f(x)=ax^4+bx^3+cx^2+dx+e$. In this case, the function is $f(x)=-x^4+3x^3+10x^2$. To analyze the graph of this function, we need to find its roots, which are the values of $x$ where the function intersects the $x$-axis.

Finding the Roots

To find the roots of the function, we need to set $f(x)=0$ and solve for $x$. This means we need to solve the equation $-x^4+3x^3+10x^2=0$. We can start by factoring out the greatest common factor, which is $x^2$. This gives us $x^2(-x^2+3x+10)=0$. We can then set each factor equal to zero and solve for $x$.

Solving for $x$

Setting $x^2=0$ gives us $x=0$. This is one of the roots of the function. To find the other roots, we need to solve the quadratic equation $-x^2+3x+10=0$. We can use the quadratic formula to solve this equation: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. In this case, $a=-1$, $b=3$, and $c=10$. Plugging these values into the formula gives us $x=\frac{-3\pm\sqrt{3^2-4(-1)(10)}}{2(-1)}$. Simplifying this expression gives us $x=\frac{-3\pm\sqrt{49}}{-2}$. This gives us two possible values for $x$: $x=\frac{-3+\sqrt{49}}{-2}$ and $x=\frac{-3-\sqrt{49}}{-2}$.

Simplifying the Roots

Simplifying the roots gives us $x=\frac{-3+7}{-2}$ and $x=\frac{-3-7}{-2}$. This gives us $x=\frac{4}{-2}$ and $x=\frac{-10}{-2}$. Simplifying further gives us $x=-2$ and $x=5$.

Analyzing the Graph

Now that we have found the roots of the function, we can analyze the graph of the function. The graph of a polynomial function is a continuous curve that can have various shapes, including a single peak or a series of peaks and valleys. In this case, the graph of the function $f(x)=-x^4+3x^3+10x^2$ has a single peak at $x=5$ and a single valley at $x=-2$.

Determining the Correct Statement

Based on our analysis of the graph of the function, we can determine which statement accurately describes its behavior. Statement A says that the graph crosses the $x$-axis at $x=0$ and touches the $x$-axis at $x=5$ and $x=-2$. This statement is accurate, as the graph of the function does cross the $x$-axis at $x=0$ and touches the $x$-axis at $x=5$ and $x=-2$.

Conclusion

In conclusion, the graph of the function $f(x)=-x^4+3x^3+10x^2$ crosses the $x$-axis at $x=0$ and touches the $x$-axis at $x=5$ and $x=-2$. This is the correct statement that describes the graph of the function.

References

• [1] "Polynomial Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/polynomial.html
• [2] "Quadratic Formula" by Math Is Fun. Retrieved from https://www.mathisfun.com/algebra/quadratic-formula.html

Additional Resources

• [1] "Graphing Polynomial Functions" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra2/x2f5f7d7/x2f5f7d7:graphing-polynomial-functions
• [2] "Polynomial Functions" by Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/PolynomialFunction.html
  

Introduction

In our previous article, we analyzed the graph of the function $f(x)=-x^4+3x^3+10x^2$ and determined that it crosses the $x$-axis at $x=0$ and touches the $x$-axis at $x=5$ and $x=-2$. In this article, we will answer some frequently asked questions about the graph of a polynomial function.

Q: What is the difference between a root and a turning point?

A: A root is a value of $x$ where the function intersects the $x$-axis, while a turning point is a point on the graph where the function changes direction.

Q: How do I find the roots of a polynomial function?

A: To find the roots of a polynomial function, you need to set the function equal to zero and solve for $x$. This can be done using various methods, including factoring, the quadratic formula, and numerical methods.

Q: What is the significance of the degree of a polynomial function?

A: The degree of a polynomial function is the highest power of $x$ in the function. It determines the number of turning points and the shape of the graph.

Q: Can a polynomial function have a negative degree?

A: No, a polynomial function cannot have a negative degree. The degree of a polynomial function is always a non-negative integer.

Q: How do I determine the number of turning points of a polynomial function?

A: The number of turning points of a polynomial function is equal to the degree of the function minus one.

Q: Can a polynomial function have a horizontal asymptote?

A: Yes, a polynomial function can have a horizontal asymptote. This occurs when the degree of the function is greater than the degree of the numerator.

Q: How do I find the horizontal asymptote of a polynomial function?

A: To find the horizontal asymptote of a polynomial function, you need to compare the degrees of the numerator and the denominator. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Q: Can a polynomial function have a vertical asymptote?

A: Yes, a polynomial function can have a vertical asymptote. This occurs when the denominator of the function is equal to zero.

Q: How do I find the vertical asymptote of a polynomial function?

A: To find the vertical asymptote of a polynomial function, you need to set the denominator equal to zero and solve for $x$.

Q: Can a polynomial function have a hole in its graph?

A: Yes, a polynomial function can have a hole in its graph. This occurs when there is a factor in the numerator and denominator that cancels out.

Q: How do I find the hole in the graph of a polynomial function?

A: To find the hole in the graph of a polynomial function, you need to factor the numerator and denominator and cancel out any common factors.

Conclusion

In conclusion, the graph of a polynomial function can have various features, including roots, turning points, horizontal and vertical asymptotes, and holes. Understanding these features is essential for analyzing and graphing polynomial functions.

References

• [1] "Polynomial Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/polynomial.html
• [2] "Graphing Polynomial Functions" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra2/x2f5f7d7/x2f5f7d7:graphing-polynomial-functions
• [3] "Polynomial Functions" by Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/PolynomialFunction.html

Additional Resources

• [1] "Graphing Polynomial Functions" by Math Is Fun. Retrieved from https://www.mathisfun.com/algebra/graphing-polynomial-functions.html
• [2] "Polynomial Functions" by Purplemath. Retrieved from https://www.purplemath.com/modules/polygraph.htm