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

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Introduction

In mathematics, graph analysis is a crucial aspect of understanding the behavior of functions. When it comes to cubic functions, their graphs can exhibit various characteristics, such as crossing or touching the x-axis. In this article, we will delve into the graph analysis of the given cubic function f(x)=−4x3−28x2−32x+64f(x) = -4x^3 - 28x^2 - 32x + 64 and determine which statement accurately describes its graph.

Understanding the Function

Before we proceed with the graph analysis, let's take a closer look at the given function. The function is a cubic function, which means it has a degree of 3. The general form of a cubic function is f(x)=ax3+bx2+cx+df(x) = ax^3 + bx^2 + cx + d, where aa, bb, cc, and dd are constants. In this case, the function is f(x)=−4x3−28x2−32x+64f(x) = -4x^3 - 28x^2 - 32x + 64.

Graph Analysis

To analyze the graph of the function, we need to find the x-intercepts, which are the points where the graph crosses the x-axis. We can find the x-intercepts by setting the function equal to zero and solving for x.

Finding the X-Intercepts

To find the x-intercepts, we set the function equal to zero:

−4x3−28x2−32x+64=0-4x^3 - 28x^2 - 32x + 64 = 0

We can start by factoring out the greatest common factor (GCF), which is -4:

−4(x3+7x2+8x−16)=0-4(x^3 + 7x^2 + 8x - 16) = 0

Now, we can divide both sides by -4:

x3+7x2+8x−16=0x^3 + 7x^2 + 8x - 16 = 0

Factoring the Cubic Expression

To factor the cubic expression, we can try to find a rational root using the Rational Root Theorem. The 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.

Applying the Rational Root Theorem

The constant term is -16, and the leading coefficient is 1. The factors of -16 are ±1, ±2, ±4, ±8, and ±16. The factors of 1 are ±1.

Finding the Rational Root

We can try each of the possible rational roots by substituting them into the cubic expression:

x3+7x2+8x−16=0x^3 + 7x^2 + 8x - 16 = 0

Let's try x = 2:

(2)3+7(2)2+8(2)−16=8+28+16−16=36(2)^3 + 7(2)^2 + 8(2) - 16 = 8 + 28 + 16 - 16 = 36

Since 36 is not equal to 0, x = 2 is not a root.

Finding the Rational Root (Continued)

Let's try x = -2:

(−2)3+7(−2)2+8(−2)−16=−8+28−16−16=−12( -2)^3 + 7( -2)^2 + 8( -2) - 16 = -8 + 28 - 16 - 16 = -12

Since -12 is not equal to 0, x = -2 is not a root.

Finding the Rational Root (Continued)

Let's try x = 4:

(4)3+7(4)2+8(4)−16=64+112+32−16=192(4)^3 + 7(4)^2 + 8(4) - 16 = 64 + 112 + 32 - 16 = 192

Since 192 is not equal to 0, x = 4 is not a root.

Finding the Rational Root (Continued)

Let's try x = -4:

(−4)3+7(−4)2+8(−4)−16=−64+112−32−16=0( -4)^3 + 7( -4)^2 + 8( -4) - 16 = -64 + 112 - 32 - 16 = 0

Since -64 + 112 - 32 - 16 = 0, x = -4 is a root.

Finding the Other Roots

Now that we have found one root, x = -4, we can factor the cubic expression:

x3+7x2+8x−16=(x+4)(x2+3x−4)x^3 + 7x^2 + 8x - 16 = (x + 4)(x^2 + 3x - 4)

We can further factor the quadratic expression:

(x+4)(x2+3x−4)=(x+4)(x+4)(x−1)(x + 4)(x^2 + 3x - 4) = (x + 4)(x + 4)(x - 1)

Finding the X-Intercepts

Now that we have factored the cubic expression, we can find the x-intercepts by setting each factor equal to zero:

(x+4)=0⇒x=−4(x + 4) = 0 \Rightarrow x = -4

(x+4)=0⇒x=−4(x + 4) = 0 \Rightarrow x = -4

(x−1)=0⇒x=1(x - 1) = 0 \Rightarrow x = 1

Conclusion

In conclusion, the graph of the function f(x)=−4x3−28x2−32x+64f(x) = -4x^3 - 28x^2 - 32x + 64 crosses the x-axis at x = -4 and x = 1.

Answer

The correct answer is A. The graph crosses the x-axis at x = 4 and touches the x-axis at x = -1 is incorrect.

Introduction

In our previous article, we analyzed the graph of the cubic function f(x)=−4x3−28x2−32x+64f(x) = -4x^3 - 28x^2 - 32x + 64 and determined that the graph crosses the x-axis at x = -4 and x = 1. In this article, we will answer some frequently asked questions (FAQs) related to the graph analysis of the cubic function.

Q: What is the degree of the cubic function?

A: The degree of the cubic function is 3, which means it has a cubic term.

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

A: The general form of a cubic function is f(x)=ax3+bx2+cx+df(x) = ax^3 + bx^2 + cx + d, where aa, bb, cc, and dd are constants.

Q: How do you 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.

Q: What is the Rational Root Theorem?

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.

Q: How do you apply the Rational Root Theorem?

A: To apply the Rational Root Theorem, you need to find the factors of the constant term and the leading coefficient, and then try each possible rational root by substituting it into the polynomial.

Q: What is the significance of the x-intercepts in graph analysis?

A: The x-intercepts are the points where the graph crosses the x-axis, and they are significant in graph analysis because they help determine the behavior of the function.

Q: Can you provide an example of a cubic function with a different degree?

A: Yes, an example of a cubic function with a different degree is f(x)=2x2−5x+1f(x) = 2x^2 - 5x + 1. This function has a degree of 2, which is less than the degree of the original function.

Q: How do you determine the behavior of a cubic function?

A: To determine the behavior of a cubic function, you need to analyze the graph of the function, including the x-intercepts, the y-intercept, and the end behavior.

Q: What is the y-intercept of the cubic function?

A: The y-intercept of the cubic function is the point where the graph crosses the y-axis. To find the y-intercept, you need to substitute x = 0 into the function.

Q: Can you provide an example of a cubic function with a different leading coefficient?

A: Yes, an example of a cubic function with a different leading coefficient is f(x)=3x3−2x2+x−1f(x) = 3x^3 - 2x^2 + x - 1. This function has a leading coefficient of 3, which is different from the leading coefficient of the original function.

Q: How do you factor a cubic expression?

A: To factor a cubic expression, you need to find the greatest common factor (GCF) and then factor the remaining expression.

Q: What is the significance of the greatest common factor (GCF) in factoring a cubic expression?

A: The GCF is significant in factoring a cubic expression because it helps simplify the expression and make it easier to factor.

Conclusion

In conclusion, the graph analysis of the cubic function f(x)=−4x3−28x2−32x+64f(x) = -4x^3 - 28x^2 - 32x + 64 is a complex process that involves finding the x-intercepts, factoring the cubic expression, and analyzing the graph of the function. We hope this Q&A article has provided you with a better understanding of the graph analysis of cubic functions.

Additional Resources

For more information on graph analysis of cubic functions, we recommend the following resources:

We hope this article has been helpful in your understanding of graph analysis of cubic functions. If you have any further questions or need additional clarification, please don't hesitate to ask.