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

Introduction

In mathematics, a cubic function is a polynomial function of degree three, which means the highest power of the variable is three. The general form of a cubic function is $f(x) = ax^3 + bx^2 + cx + d$, where $a$, $b$, $c$, and $d$ are constants. In this article, we will analyze the graph of the cubic function $f(x) = -4x^3 - 28x^2 - 32x + 64$ and determine which statement describes its graph.

Understanding the Function

The given function is $f(x) = -4x^3 - 28x^2 - 32x + 64$. To analyze its graph, we need to find the x-intercepts, which are the points where the graph crosses the x-axis. We can do this by setting $f(x) = 0$ and solving for $x$.

Finding the x-Intercepts

To find the x-intercepts, we set $f(x) = 0$ and solve for $x$:

$$-4x^3 - 28x^2 - 32x + 64 = 0$$

We can factor out $-4$ from the equation:

$$-4(x^3 + 7x^2 + 8x - 16) = 0$$

Now, we can factor the cubic expression inside the parentheses:

$$-4(x - 2)(x^2 + 9x + 8) = 0$$

We can further factor the quadratic expression inside the parentheses:

$$-4(x - 2)(x + 1)(x + 8) = 0$$

Now, we can set each factor equal to zero and solve for $x$:

$$x - 2 = 0 \Rightarrow x = 2$$

$$x + 1 = 0 \Rightarrow x = -1$$

$$x + 8 = 0 \Rightarrow x = -8$$

Therefore, the x-intercepts of the graph are $x = 2$, $x = -1$, and $x = -8$.

Analyzing the Graph

Now that we have found the x-intercepts, we can analyze the graph. The graph of a cubic function can have at most two turning points, which are the points where the graph changes direction. We can find the turning points by taking the derivative of the function and setting it equal to zero.

Finding the Turning Points

To find the turning points, we take the derivative of the function:

$$f'(x) = -12x^2 - 56x - 32$$

Now, we set the derivative equal to zero and solve for $x$:

$$-12x^2 - 56x - 32 = 0$$

We can factor out $-4$ from the equation:

$$-4(3x^2 + 14x + 8) = 0$$

Now, we can factor the quadratic expression inside the parentheses:

$$-4(3x + 2)(x + 4) = 0$$

We can set each factor equal to zero and solve for $x$:

$$3x + 2 = 0 \Rightarrow x = -\frac{2}{3}$$

$$x + 4 = 0 \Rightarrow x = -4$$

Therefore, the turning points of the graph are $x = -\frac{2}{3}$ and $x = -4$.

Conclusion

In conclusion, the graph of the cubic function $f(x) = -4x^3 - 28x^2 - 32x + 64$ has x-intercepts at $x = 2$, $x = -1$, and $x = -8$. The graph also has turning points at $x = -\frac{2}{3}$ and $x = -4$. Therefore, the statement that describes the graph is:

The graph crosses the x-axis at x = 2 and touches the x-axis at x = -1 and x = -8.

$$$$$$

Introduction

In our previous article, we analyzed the graph of the cubic function $f(x) = -4x^3 - 28x^2 - 32x + 64$. We found the x-intercepts and turning points of the graph, and determined that the statement "The graph crosses the x-axis at x = 2 and touches the x-axis at x = -1 and x = -8" accurately describes the graph. In this article, we will answer some frequently asked questions about the graph of a cubic function.

Q: What is the general form of a cubic function?

A: The general form of a cubic function is $f(x) = ax^3 + bx^2 + cx + d$, where $a$, $b$, $c$, and $d$ are constants.

Q: How do I find the x-intercepts of a cubic function?

A: To find the x-intercepts of a cubic function, you need to set the function equal to zero and solve for $x$. You can use factoring or the quadratic formula to solve for $x$.

Q: What are the turning points of a cubic function?

A: The turning points of a cubic function are the points where the graph changes direction. You can find the turning points by taking the derivative of the function and setting it equal to zero.

Q: How do I determine the number of x-intercepts of a cubic function?

A: The number of x-intercepts of a cubic function can be determined by the degree of the function. A cubic function can have at most three x-intercepts.

Q: Can a cubic function have more than two turning points?

A: No, a cubic function can have at most two turning points.

Q: How do I determine the direction of the graph of a cubic function?

A: You can determine the direction of the graph of a cubic function by looking at the leading coefficient of the function. If the leading coefficient is positive, the graph will open upward. If the leading coefficient is negative, the graph will open downward.

Q: Can a cubic function have a horizontal asymptote?

A: Yes, a cubic function can have a horizontal asymptote. The horizontal asymptote is the horizontal line that the graph approaches as $x$ approaches infinity.

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

A: To find the horizontal asymptote of a cubic function, you need to look at the leading term of the function. The horizontal asymptote is the value of the leading term.

Q: Can a cubic function have a vertical asymptote?

A: No, a cubic function cannot have a vertical asymptote.

Q: How do I determine the domain of a cubic function?

A: The domain of a cubic function is all real numbers, unless there are any restrictions on the function.

Q: Can a cubic function be a rational function?

A: No, a cubic function cannot be a rational function.

Conclusion

In conclusion, the graph of a cubic function can be analyzed by finding the x-intercepts and turning points of the function. The number of x-intercepts and turning points can be determined by the degree of the function. The direction of the graph can be determined by the leading coefficient of the function. The horizontal asymptote can be found by looking at the leading term of the function.