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 = □ Y = \square Y = □ For Large Values Of ∣ X ∣ |x| ∣ X ∣ .

Graphing the Polynomial Function $f(x)=16x-x^3$

Understanding the End Behavior of the Graph

The end behavior of a polynomial function is determined by the degree and the leading coefficient of the polynomial. In this case, we have a cubic polynomial, which means it has a degree of 3. The leading coefficient is -1, which is negative.

Determining the End Behavior

To determine the end behavior of the graph of the function, we need to consider the behavior of the function as x approaches positive infinity and negative infinity.

• As x approaches positive infinity: Since the degree of the polynomial is odd (3) and the leading coefficient is negative (-1), the graph of the function will behave like a negative linear function for large values of x. This means that as x approaches positive infinity, the graph of the function will approach negative infinity.
• As x approaches negative infinity: Since the degree of the polynomial is odd (3) and the leading coefficient is negative (-1), the graph of the function will behave like a positive linear function for large values of x. This means that as x approaches negative infinity, the graph of the function will approach positive infinity.

Graphing the Function

To graph the function, we can start by plotting the x-intercepts, which are the points where the graph of the function crosses the x-axis. In this case, the x-intercepts are the solutions to the equation f(x) = 0.

• Finding the x-intercepts: To find the x-intercepts, we need to solve the equation 16x - x^3 = 0. We can factor out x to get x(16 - x^2) = 0. This gives us two possible solutions: x = 0 and 16 - x^2 = 0. Solving the second equation, we get x^2 = 16, which gives us x = ±4.
• Plotting the x-intercepts: We can plot the x-intercepts by marking the points (0, 0) and (-4, 0) and (4, 0) on the graph.

Determining the Y-Intercept

To determine the y-intercept, we need to find the value of the function at x = 0.

• Finding the y-intercept: We can find the y-intercept by substituting x = 0 into the function f(x) = 16x - x^3. This gives us f(0) = 16(0) - 0^3 = 0.

Graphing the Function

Now that we have the x-intercepts and the y-intercept, we can graph the function by plotting the points and drawing a smooth curve through them.

• Graphing the function: We can graph the function by plotting the points (0, 0), (-4, 0), and (4, 0) and drawing a smooth curve through them. The graph of the function will be a cubic curve that opens downward.

Answering Part (b)

Part (b) asks us to determine the x-intercepts of the graph of the function.

• Determining the x-intercepts: We have already determined the x-intercepts in part (a). The x-intercepts are the points where the graph of the function crosses the x-axis. In this case, the x-intercepts are the solutions to the equation f(x) = 0. We found that the x-intercepts are x = 0 and x = ±4.

Answering Part (c)

Part (c) asks us to determine the y-intercept of the graph of the function.

• Determining the y-intercept: We have already determined the y-intercept in part (a). The y-intercept is the value of the function at x = 0. We found that the y-intercept is f(0) = 0.

Answering Part (d)

Part (d) asks us to determine the axis of symmetry of the graph of the function.

• Determining the axis of symmetry: Since the graph of the function is a cubic curve that opens downward, the axis of symmetry is the vertical line x = 0.

Answering Part (e)

Part (e) asks us to determine the vertex of the graph of the function.

• Determining the vertex: Since the graph of the function is a cubic curve that opens downward, the vertex is the point where the graph changes direction. We can find the vertex by finding the x-coordinate of the vertex, which is the x-coordinate of the point where the graph changes direction. The x-coordinate of the vertex is x = -b / 2a, where a and b are the coefficients of the quadratic term and the linear term, respectively. In this case, a = -1 and b = 16, so the x-coordinate of the vertex is x = -16 / (2(-1)) = 8. The y-coordinate of the vertex is f(8) = 16(8) - 8^3 = 128 - 512 = -384. Therefore, the vertex is the point (8, -384).

Conclusion

In conclusion, we have graphed the polynomial function f(x) = 16x - x^3 and determined the end behavior of the graph, the x-intercepts, the y-intercept, the axis of symmetry, and the vertex. We have also answered parts (a) through (e) of the problem.
Graphing the Polynomial Function $f(x)=16x-x^3$: Q&A

Understanding the Graph of the Function

In our previous article, we graphed the polynomial function $f(x)=16x-x^3$ and determined the end behavior of the graph, the x-intercepts, the y-intercept, the axis of symmetry, and the vertex. In this article, we will answer some common questions about the graph of the function.

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

A: The degree of the polynomial function $f(x)=16x-x^3$ is 3, which means it is a cubic polynomial.

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

A: The leading coefficient of the polynomial function $f(x)=16x-x^3$ is -1, which is negative.

Q: What is the end behavior of the graph of the function?

A: The end behavior of the graph of the function is that it behaves like a negative linear function for large values of x. As x approaches positive infinity, the graph of the function approaches negative infinity. As x approaches negative infinity, the graph of the function approaches positive infinity.

Q: What are the x-intercepts of the graph of the function?

A: The x-intercepts of the graph of the function are the points where the graph of the function crosses the x-axis. In this case, the x-intercepts are x = 0 and x = ±4.

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

A: The y-intercept of the graph of the function is the value of the function at x = 0. In this case, the y-intercept is f(0) = 0.

Q: What is the axis of symmetry of the graph of the function?

A: The axis of symmetry of the graph of the function is the vertical line x = 0.

Q: What is the vertex of the graph of the function?

A: The vertex of the graph of the function is the point where the graph changes direction. In this case, the vertex is the point (8, -384).

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

A: To graph the polynomial function $f(x)=16x-x^3$, you can start by plotting the x-intercepts, which are the points where the graph of the function crosses the x-axis. Then, you can plot the y-intercept, which is the value of the function at x = 0. Finally, you can draw a smooth curve through the points to graph the function.

Q: What are some common mistakes to avoid when graphing the polynomial function $f(x)=16x-x^3$?

A: Some common mistakes to avoid when graphing the polynomial function $f(x)=16x-x^3$ include:

• Not plotting the x-intercepts correctly
• Not plotting the y-intercept correctly
• Not drawing a smooth curve through the points
• Not considering the end behavior of the graph

Conclusion

In conclusion, we have answered some common questions about the graph of the polynomial function $f(x)=16x-x^3$. We have also provided some tips for graphing the function and avoiding common mistakes.