Finding The Quadratic Function With Roots 4 And 1 And Leading Coefficient 3
In this article, we will delve into the process of determining a quadratic function given its roots and leading coefficient. This is a fundamental concept in algebra and has wide applications in various fields, including physics, engineering, and computer science. We'll explore how to construct the polynomial function step by step, ensuring a clear understanding of the underlying principles. Specifically, we'll address the question: Which second-degree polynomial function f(x) has a leading coefficient of 3 and roots 4 and 1? This involves understanding the relationship between the roots of a quadratic equation and its factored form, as well as the impact of the leading coefficient on the function's overall shape and behavior. Let's embark on this mathematical journey to unravel the solution!
Understanding Quadratic Functions and Roots
A quadratic function is a polynomial function of degree two, generally expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants, and a is not equal to zero. The roots of a quadratic function are the values of x for which f(x) = 0. These roots represent the points where the parabola intersects the x-axis. Knowing the roots of a quadratic equation is crucial because it allows us to express the function in its factored form. If r₁ and r₂ are the roots of a quadratic function, then the function can be written in the form f(x) = a(x - r₁)(x - r₂), where a is the leading coefficient. The leading coefficient, a, plays a significant role in determining the shape and direction of the parabola. A positive a indicates that the parabola opens upwards, while a negative a indicates that it opens downwards. The magnitude of a also affects the steepness of the parabola; a larger absolute value of a results in a steeper curve. Understanding these fundamental concepts is essential for solving problems related to quadratic functions, including finding the function given its roots and leading coefficient. By grasping the relationship between roots, factored form, and the leading coefficient, we can effectively construct and analyze quadratic functions in various contexts.
Constructing the Quadratic Function
To construct the quadratic function f(x) with a leading coefficient of 3 and roots 4 and 1, we can start with the factored form of a quadratic function. Given the roots r₁ = 4 and r₂ = 1, and the leading coefficient a = 3, we can write the function as:
f(x) = a(x - r₁)(x - r₂)
Substituting the given values, we get:
f(x) = 3(x - 4)(x - 1)
Now, we need to expand this expression to obtain the standard form of the quadratic function, which is f(x) = ax² + bx + c. To do this, we first multiply the two binomial factors:
(x - 4)(x - 1) = x² - x - 4x + 4
Combining like terms, we have:
x² - 5x + 4
Next, we multiply the entire expression by the leading coefficient, 3:
f(x) = 3(x² - 5x + 4)
Distributing the 3, we get:
f(x) = 3x² - 15x + 12
Therefore, the quadratic function with a leading coefficient of 3 and roots 4 and 1 is f(x) = 3x² - 15x + 12. This process demonstrates how the roots and leading coefficient directly influence the coefficients of the quadratic function in its standard form. Understanding this construction process is vital for solving similar problems and for grasping the connection between the different forms of a quadratic function.
Verifying the Solution
To ensure our solution is correct, we can verify that the quadratic function f(x) = 3x² - 15x + 12 indeed has roots 4 and 1 and a leading coefficient of 3. The leading coefficient is clearly 3, as it is the coefficient of the x² term. To verify the roots, we can substitute x = 4 and x = 1 into the function and check if the result is zero.
For x = 4:
f(4) = 3(4)² - 15(4) + 12
f(4) = 3(16) - 60 + 12
f(4) = 48 - 60 + 12
f(4) = 0
For x = 1:
f(1) = 3(1)² - 15(1) + 12
f(1) = 3 - 15 + 12
f(1) = 0
Since f(4) = 0 and f(1) = 0, we have confirmed that 4 and 1 are indeed the roots of the quadratic function. Another way to verify the solution is by factoring the quadratic function back into its factored form. We can factor out a 3 from the expression:
f(x) = 3x² - 15x + 12 = 3(x² - 5x + 4)
Now, we can factor the quadratic expression inside the parentheses:
x² - 5x + 4 = (x - 4)(x - 1)
Thus, f(x) = 3(x - 4)(x - 1), which matches our initial factored form based on the given roots. This comprehensive verification process reinforces the correctness of our solution and highlights the interconnectedness of the different representations of a quadratic function.
Analyzing the Answer Choices
Now that we have derived the quadratic function f(x) = 3x² - 15x + 12, we can compare it with the given answer choices to identify the correct one. Let's analyze each option:
A. f(x) = 3x² + 5x + 4
This function has a leading coefficient of 3, but the coefficients of the x term and the constant term do not match our derived function. Therefore, this option is incorrect.
B. f(x) = 3x² + 15x + 12
This function has a leading coefficient of 3, but the coefficient of the x term has the opposite sign compared to our derived function. Therefore, this option is also incorrect.
C. f(x) = 3x² - 5x + 4
This function has a leading coefficient of 3, but the coefficients of the x term and the constant term do not match our derived function. This option is incorrect as well.
D. f(x) = 3x² - 15x + 12
This function has a leading coefficient of 3, and the coefficients of the x term and the constant term match our derived function perfectly. Therefore, this is the correct answer.
By systematically comparing our derived function with each answer choice, we can confidently identify the correct option. This process emphasizes the importance of careful calculation and comparison when solving mathematical problems.
Conclusion
In conclusion, we successfully determined the quadratic function f(x) with a leading coefficient of 3 and roots 4 and 1. By understanding the relationship between the roots, leading coefficient, and the factored form of a quadratic function, we were able to construct the function step by step. We started with the factored form f(x) = a(x - r₁)(x - r₂), substituted the given values, expanded the expression, and obtained the standard form f(x) = 3x² - 15x + 12. We then verified our solution by substituting the roots back into the function and by factoring the function back into its factored form. Finally, we compared our derived function with the given answer choices and identified the correct option, which was D. f(x) = 3x² - 15x + 12. This exercise highlights the importance of a solid understanding of quadratic functions and their properties, as well as the ability to apply these concepts to solve specific problems. Mastering these skills is crucial for further studies in mathematics and related fields.
