Graph The Polynomial Function F ( X ) = 16 X − X 3 F(x) = 16x - X^3 F ( X ) = 16 X − X 3 . Answer Parts (a) Through (e).(a) Determine The End Behavior Of The Graph Of The Function.The Graph Of F F F Behaves Like Y = − X 3 Y = -x^3 Y = − X 3 For Large Values Of ∣ X ∣ |x| ∣ X ∣ .(b) Find

Introduction

In this article, we will delve into the world of polynomial functions and explore the graph of the function $f(x) = 16x - x^3$. We will analyze the end behavior of the graph, find the x-intercepts, determine the y-intercept, identify the vertex of the graph, and finally, graph the function using a suitable method.

End Behavior of the Graph

(a) Determine the End Behavior of the Graph of the Function

The end behavior of a graph refers to the behavior of the graph as $x$ approaches positive or negative infinity. To determine the end behavior of the graph of $f(x) = 16x - x^3$, we need to examine the leading term of the polynomial function.

The leading term of the polynomial function is the term with the highest degree, which in this case is $-x^3$. Since the degree of the leading term is odd, the end behavior of the graph will be different for large positive and negative values of $x$.

For large positive values of $x$, the graph of $f(x) = 16x - x^3$ behaves like $y = -x^3$. This is because the term $16x$ becomes negligible compared to the term $-x^3$ as $x$ approaches infinity.

Similarly, for large negative values of $x$, the graph of $f(x) = 16x - x^3$ also behaves like $y = -x^3$. This is because the term $16x$ becomes negligible compared to the term $-x^3$ as $x$ approaches negative infinity.

Therefore, the end behavior of the graph of $f(x) = 16x - x^3$ is the same as the end behavior of the graph of $y = -x^3$ for large values of $|x|$.

(b) Find the x-Intercepts of the Graph

The x-intercepts of a graph are the points where the graph intersects the x-axis. To find the x-intercepts of the graph of $f(x) = 16x - x^3$, we need to set the function equal to zero and solve for $x$.

Setting $f(x) = 0$, we get:

$$16x - x^3 = 0$$

Factoring out $x$, we get:

$$x(16 - x^2) = 0$$

This gives us two possible solutions:

$$x = 0 \quad \text{or} \quad 16 - x^2 = 0$$

Solving the second equation, we get:

$$x^2 = 16$$

$$x = \pm 4$$

Therefore, the x-intercepts of the graph of $f(x) = 16x - x^3$ are $x = 0$, $x = 4$, and $x = -4$.

(c) Determine the y-Intercept of the Graph

The y-intercept of a graph is the point where the graph intersects the y-axis. To determine the y-intercept of the graph of $f(x) = 16x - x^3$, we need to evaluate the function at $x = 0$.

Substituting $x = 0$ into the function, we get:

$$f(0) = 16(0) - (0)^3 = 0$$

Therefore, the y-intercept of the graph of $f(x) = 16x - x^3$ is $(0, 0)$.

(d) Identify the Vertex of the Graph

The vertex of a graph is the point where the graph changes direction. To identify the vertex of the graph of $f(x) = 16x - x^3$, we need to find the x-coordinate of the vertex.

The x-coordinate of the vertex can be found using the formula:

$$x = -\frac{b}{2a}$$

In this case, $a = -1$ and $b = 16$. Substituting these values into the formula, we get:

$$x = -\frac{16}{2(-1)} = 8$$

Therefore, the x-coordinate of the vertex is $x = 8$.

To find the y-coordinate of the vertex, we need to evaluate the function at $x = 8$.

Substituting $x = 8$ into the function, we get:

$$f(8) = 16(8) - (8)^3 = 128 - 512 = -384$$

Therefore, the vertex of the graph of $f(x) = 16x - x^3$ is $(8, -384)$.

(e) Graph the Function

To graph the function $f(x) = 16x - x^3$, we can use a graphing calculator or a computer algebra system.

Using a graphing calculator, we can graph the function by entering the function into the calculator and using the graphing feature.

Alternatively, we can use a computer algebra system to graph the function.

Using a computer algebra system, we can graph the function by entering the function into the system and using the graphing feature.

The graph of the function $f(x) = 16x - x^3$ is a cubic function that opens downward. The graph has three x-intercepts at $x = 0$, $x = 4$, and $x = -4$. The graph also has a y-intercept at $(0, 0)$ and a vertex at $(8, -384)$.

Conclusion

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Introduction

In our previous article, we analyzed the graph of the polynomial function $f(x) = 16x - x^3$. We determined the end behavior of the graph, found the x-intercepts, determined the y-intercept, identified the vertex of the graph, and finally, graphed the function using a suitable method. In this article, we will answer some frequently asked questions about the graph of the polynomial function.

Q&A

Q: What is the end behavior of the graph of $f(x) = 16x - x^3$?

A: The end behavior of the graph of $f(x) = 16x - x^3$ is the same as the end behavior of the graph of $y = -x^3$ for large values of $|x|$.

Q: What are the x-intercepts of the graph of $f(x) = 16x - x^3$?

A: The x-intercepts of the graph of $f(x) = 16x - x^3$ are $x = 0$, $x = 4$, and $x = -4$.

Q: What is the y-intercept of the graph of $f(x) = 16x - x^3$?

A: The y-intercept of the graph of $f(x) = 16x - x^3$ is $(0, 0)$.

Q: What is the vertex of the graph of $f(x) = 16x - x^3$?

A: The vertex of the graph of $f(x) = 16x - x^3$ is $(8, -384)$.

Q: How can I graph the function $f(x) = 16x - x^3$?

A: You can graph the function $f(x) = 16x - x^3$ using a graphing calculator or a computer algebra system.

Q: What is the domain of the function $f(x) = 16x - x^3$?

A: The domain of the function $f(x) = 16x - x^3$ is all real numbers, or $(-\infty, \infty)$.

Q: What is the range of the function $f(x) = 16x - x^3$?

A: The range of the function $f(x) = 16x - x^3$ is all real numbers, or $(-\infty, \infty)$.

Q: Is the function $f(x) = 16x - x^3$ an even function or an odd function?

A: The function $f(x) = 16x - x^3$ is an odd function.

Q: Can I use the function $f(x) = 16x - x^3$ to model real-world phenomena?

A: Yes, the function $f(x) = 16x - x^3$ can be used to model real-world phenomena such as population growth or decay.

Conclusion

In this article, we answered some frequently asked questions about the graph of the polynomial function $f(x) = 16x - x^3$. We discussed the end behavior of the graph, the x-intercepts, the y-intercept, the vertex, and the domain and range of the function. We also discussed how to graph the function and whether it can be used to model real-world phenomena.