Which Statement Describes The Graph Of F ( X ) = − 4 X 3 − 28 X 2 − 32 X + 64 F(x)=-4x^3-28x^2-32x+64 F ( X ) = − 4 X 3 − 28 X 2 − 32 X + 64 ?A. The Graph Crosses The X X X -axis At X = 4 X=4 X = 4 And Touches The X X X -axis At X = − 1 X=-1 X = − 1 .B. The Graph Touches The X X X -axis At X = 4 X=4 X = 4 And

Which Statement Describes the Graph of $f(x)=-4x^3-28x^2-32x+64$?

Understanding the Problem

The given problem involves analyzing the graph of a cubic function, $f(x)=-4x^3-28x^2-32x+64$. To determine which statement describes the graph, we need to examine the function's behavior, particularly its intersections with the x-axis.

Analyzing the Function

The given function is a cubic function, which means it can have at most three x-intercepts. To find the x-intercepts, we need to set the function equal to zero and solve for x.

f(x) = -4x^3 - 28x^2 - 32x + 64 = 0

Factoring the Function

To simplify the function and make it easier to analyze, we can try to factor it. Factoring the function, we get:

f(x) = -4(x^3 + 7x^2 + 8x - 16)

Finding the Roots

To find the roots of the function, we need to set the factored expression equal to zero and solve for x.

x^3 + 7x^2 + 8x - 16 = 0

Using the Rational Root Theorem

The rational root theorem states that if a rational number p/q is a root of the polynomial, then p must be a factor of the constant term, and q must be a factor of the leading coefficient. In this case, the constant term is -16, and the leading coefficient is 1. Therefore, the possible rational roots are ±1, ±2, ±4, ±8, and ±16.

Finding the First Root

Using the rational root theorem, we can try to find the first root by substituting the possible rational roots into the polynomial.

x = -4

Dividing the Polynomial

After finding the first root, we can divide the polynomial by (x + 4) to get a quadratic expression.

x^2 + 3x - 4 = 0

Finding the Second and Third Roots

To find the second and third roots, we can use the quadratic formula to solve the quadratic expression.

x = (-b ± √(b^2 - 4ac)) / 2a

Solving for x

Substituting the values of a, b, and c into the quadratic formula, we get:

x = (-3 ± √(9 + 16)) / 2
x = (-3 ± √25) / 2
x = (-3 ± 5) / 2

Finding the Second and Third Roots

Simplifying the expression, we get two possible values for x:

x = (-3 + 5) / 2 = 1
x = (-3 - 5) / 2 = -4

Analyzing the Graph

Now that we have found the roots of the function, we can analyze the graph. The graph touches the x-axis at x = -4 and x = 1, but it does not cross the x-axis at x = 4.

Conclusion

Based on the analysis, the correct statement that describes the graph of $f(x)=-4x^3-28x^2-32x+64$ is:

A. The graph touches the x-axis at x = -4 and x = 1.

Discussion

The given problem involves analyzing the graph of a cubic function. To determine which statement describes the graph, we need to examine the function's behavior, particularly its intersections with the x-axis. By factoring the function and finding its roots, we can analyze the graph and determine the correct statement.

Key Takeaways

• The given function is a cubic function, which means it can have at most three x-intercepts.
• To find the x-intercepts, we need to set the function equal to zero and solve for x.
• Factoring the function can simplify it and make it easier to analyze.
• The rational root theorem can be used to find the possible rational roots of the polynomial.
• Dividing the polynomial by (x + 4) can give us a quadratic expression that can be solved using the quadratic formula.
• The graph touches the x-axis at x = -4 and x = 1, but it does not cross the x-axis at x = 4.

Final Answer

The final answer is A. The graph touches the x-axis at x = -4 and x = 1.
Q&A: Understanding the Graph of $f(x)=-4x^3-28x^2-32x+64$

Q: What is the main goal of analyzing the graph of $f(x)=-4x^3-28x^2-32x+64$?

A: The main goal of analyzing the graph of $f(x)=-4x^3-28x^2-32x+64$ is to determine its behavior, particularly its intersections with the x-axis.

Q: What is the significance of the x-intercepts in the graph of $f(x)=-4x^3-28x^2-32x+64$?

A: The x-intercepts in the graph of $f(x)=-4x^3-28x^2-32x+64$ represent the points where the graph touches or crosses the x-axis. These points are crucial in understanding the behavior of the function.

Q: How can we find the x-intercepts of the graph of $f(x)=-4x^3-28x^2-32x+64$?

A: To find the x-intercepts of the graph of $f(x)=-4x^3-28x^2-32x+64$, we need to set the function equal to zero and solve for x.

Q: What is the rational root theorem, and how can it be used to find the roots of the polynomial?

A: The rational root theorem states that if a rational number p/q is a root of the polynomial, then p must be a factor of the constant term, and q must be a factor of the leading coefficient. This theorem can be used to find the possible rational roots of the polynomial.

Q: How can we divide the polynomial by (x + 4) to get a quadratic expression?

A: We can divide the polynomial by (x + 4) using polynomial long division or synthetic division to get a quadratic expression.

Q: What is the quadratic formula, and how can it be used to solve the quadratic expression?

A: The quadratic formula is a mathematical formula that can be used to solve quadratic equations. It is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

This formula can be used to solve the quadratic expression obtained by dividing the polynomial by (x + 4).

Q: What is the significance of the graph touching the x-axis at x = -4 and x = 1?

A: The graph touching the x-axis at x = -4 and x = 1 indicates that these points are the x-intercepts of the graph. This means that the graph crosses the x-axis at these points.

Q: What is the final answer to the problem?

A: The final answer to the problem is A. The graph touches the x-axis at x = -4 and x = 1.

Q: What are the key takeaways from this problem?

A: The key takeaways from this problem are:

• The given function is a cubic function, which means it can have at most three x-intercepts.
• To find the x-intercepts, we need to set the function equal to zero and solve for x.
• Factoring the function can simplify it and make it easier to analyze.
• The rational root theorem can be used to find the possible rational roots of the polynomial.
• Dividing the polynomial by (x + 4) can give us a quadratic expression that can be solved using the quadratic formula.
• The graph touches the x-axis at x = -4 and x = 1, but it does not cross the x-axis at x = 4.

Q: What is the importance of understanding the graph of $f(x)=-4x^3-28x^2-32x+64$?

A: Understanding the graph of $f(x)=-4x^3-28x^2-32x+64$ is important because it helps us analyze the behavior of the function, particularly its intersections with the x-axis. This knowledge can be applied to various fields, such as mathematics, science, and engineering.