Eliminating Incorrect Graphs

Oct 13, 2025 by talynt.org 104 views

Understanding the Function

The given function is a quadratic function in the form of $f(x) = 0.03 x^2(x^2 - 25)$ . To understand the behavior of this function, we need to analyze its components. The function has a coefficient of 0.03, which is a constant factor that affects the amplitude of the function. The function also has a quadratic term $x^2$ , which represents a parabola that opens upwards. Additionally, the function has a term $(x^2 - 25)$ , which represents a quadratic expression that can be factored as $(x - 5)(x + 5)$ .

Graphing the Function

To graph the function, we need to consider the behavior of the quadratic terms. The term $x^2$ represents a parabola that opens upwards, while the term $(x^2 - 25)$ represents a quadratic expression that can be factored as $(x - 5)(x + 5)$ . This means that the function has two critical points at $x = 5$ and $x = -5$ , where the function changes from increasing to decreasing or vice versa.

Identifying the Graph

Based on the analysis of the function, we can identify the graph as a cubic function with a positive leading coefficient. The graph will have a minimum point at $x = 0$ , where the function changes from increasing to decreasing. The graph will also have two critical points at $x = 5$ and $x = -5$ , where the function changes from increasing to decreasing or vice versa.

Possible Graphs

Based on the analysis of the function, we can identify the following possible graphs:

Graph A: A cubic function with a positive leading coefficient, a minimum point at $x = 0$ , and two critical points at $x = 5$ and $x = -5$ .
Graph B: A cubic function with a negative leading coefficient, a maximum point at $x = 0$ , and two critical points at $x = 5$ and $x = -5$ .
Graph C: A quadratic function with a positive leading coefficient, a minimum point at $x = 0$ , and two critical points at $x = 5$ and $x = -5$ .

Eliminating Incorrect Graphs

To eliminate the incorrect graphs, we need to analyze the behavior of the function at the critical points. Graph B has a negative leading coefficient, which means that the function will have a maximum point at $x = 0$ . However, the function $f(x) = 0.03 x^2(x^2 - 25)$ has a positive leading coefficient, which means that the function will have a minimum point at $x = 0$ . Therefore, we can eliminate Graph B.

Analyzing the Remaining Graphs

We are left with two possible graphs: Graph A and Graph C. To determine which graph is correct, we need to analyze the behavior of the function at the critical points. Graph A has a minimum point at $x = 0$ , which means that the function will change from increasing to decreasing at $x = 0$ . However, the function $f(x) = 0.03 x^2(x^2 - 25)$ has a positive leading coefficient, which means that the function will change from decreasing to increasing at $x = 0$ . Therefore, we can eliminate Graph A.

Conclusion

Based on the analysis of the function, we can conclude that the correct graph is Graph C. Graph C is a quadratic function with a positive leading coefficient, a minimum point at $x = 0$ , and two critical points at $x = 5$ and $x = -5$ . This graph accurately represents the behavior of the function $f(x) = 0.03 x^2(x^2 - 25)$ .

Key Takeaways

The function $f(x) = 0.03 x^2(x^2 - 25)$ is a cubic function with a positive leading coefficient.
The function has a minimum point at $x = 0$ , where the function changes from increasing to decreasing.
The function has two critical points at $x = 5$ and $x = -5$ , where the function changes from increasing to decreasing or vice versa.
The correct graph is Graph C, which is a quadratic function with a positive leading coefficient, a minimum point at $x = 0$ , and two critical points at $x = 5$ and $x = -5$ .

Final Answer

The final answer is Graph C.

Q: What is the general form of the function $f(x) = 0.03 x^2(x^2 - 25)$ ?

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

Q: What is the coefficient of the quadratic term in the function $f(x) = 0.03 x^2(x^2 - 25)$ ?

A: The coefficient of the quadratic term is 0.03.

Q: What is the behavior of the function at the critical points $x = 5$ and $x = -5$ ?

A: The function changes from increasing to decreasing or vice versa at the critical points $x = 5$ and $x = -5$ .

Q: What is the minimum point of the function $f(x) = 0.03 x^2(x^2 - 25)$ ?

A: The minimum point of the function is at $x = 0$ , where the function changes from increasing to decreasing.

Q: What is the correct graph of the function $f(x) = 0.03 x^2(x^2 - 25)$ ?

A: The correct graph is Graph C, which is a quadratic function with a positive leading coefficient, a minimum point at $x = 0$ , and two critical points at $x = 5$ and $x = -5$ .

Q: What is the significance of the coefficient 0.03 in the function $f(x) = 0.03 x^2(x^2 - 25)$ ?

A: The coefficient 0.03 affects the amplitude of the function, making it a scaled version of the original function.

Q: How does the function $f(x) = 0.03 x^2(x^2 - 25)$ change from increasing to decreasing at the critical points $x = 5$ and $x = -5$ ?

A: The function changes from increasing to decreasing at the critical points $x = 5$ and $x = -5$ due to the presence of the quadratic term $(x^2 - 25)$ .

Q: What is the relationship between the function $f(x) = 0.03 x^2(x^2 - 25)$ and the graph Graph C?

A: The function $f(x) = 0.03 x^2(x^2 - 25)$ is represented by the graph Graph C, which is a quadratic function with a positive leading coefficient, a minimum point at $x = 0$ , and two critical points at $x = 5$ and $x = -5$ .

Q: What is the significance of the critical points $x = 5$ and $x = -5$ in the function $f(x) = 0.03 x^2(x^2 - 25)$ ?

A: The critical points $x = 5$ and $x = -5$ represent the points where the function changes from increasing to decreasing or vice versa.

Q: How does the function $f(x) = 0.03 x^2(x^2 - 25)$ change from decreasing to increasing at the critical points $x = 5$ and $x = -5$ ?

A: The function changes from decreasing to increasing at the critical points $x = 5$ and $x = -5$ due to the presence of the quadratic term $(x^2 - 25)$ .

Q: What is the relationship between the function $f(x) = 0.03 x^2(x^2 - 25)$ and the graph Graph A?

A: The function $f(x) = 0.03 x^2(x^2 - 25)$ is not represented by the graph Graph A, which is a cubic function with a positive leading coefficient, a minimum point at $x = 0$ , and two critical points at $x = 5$ and $x = -5$ .

Q: What is the significance of the coefficient 0.03 in the function $f(x) = 0.03 x^2(x^2 - 25)$ in terms of the graph?

A: The coefficient 0.03 affects the amplitude of the function, making it a scaled version of the original function, and also affects the shape of the graph.

Q: How does the function $f(x) = 0.03 x^2(x^2 - 25)$ change from increasing to decreasing at the critical points $x = 5$ and $x = -5$ in terms of the graph?

A: The function changes from increasing to decreasing at the critical points $x = 5$ and $x = -5$ due to the presence of the quadratic term $(x^2 - 25)$ , which affects the shape of the graph.

Q: What is the relationship between the function $f(x) = 0.03 x^2(x^2 - 25)$ and the graph Graph B?

A: The function $f(x) = 0.03 x^2(x^2 - 25)$ is not represented by the graph Graph B, which is a cubic function with a negative leading coefficient, a maximum point at $x = 0$ , and two critical points at $x = 5$ and $x = -5$ .

Q: What is the significance of the critical points $x = 5$ and $x = -5$ in the function $f(x) = 0.03 x^2(x^2 - 25)$ in terms of the graph?

A: The critical points $x = 5$ and $x = -5$ represent the points where the function changes from increasing to decreasing or vice versa, and also affect the shape of the graph.

Q: How does the function $f(x) = 0.03 x^2(x^2 - 25)$ change from decreasing to increasing at the critical points $x = 5$ and $x = -5$ in terms of the graph?

A: The function changes from decreasing to increasing at the critical points $x = 5$ and $x = -5$ due to the presence of the quadratic term $(x^2 - 25)$ , which affects the shape of the graph.