Determine Quadratic Function G(x) With G(2)=0, G(6)=0, And G(4)=12
In the realm of mathematics, quadratic functions hold a significant position due to their widespread applications in various fields, including physics, engineering, and economics. 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 ≠ 0. Understanding how to determine a specific quadratic function that meets certain conditions is a fundamental skill in algebra and calculus.
This article delves into the process of determining the mathematical sentence (or equation) of a quadratic function, often denoted as g(x), that satisfies specific conditions. We will explore the steps involved in finding the coefficients a, b, and c given certain values of the function at particular points. Specifically, we will address the problem of finding the quadratic function g(x) that satisfies the conditions g(2) = 0, g(6) = 0, and g(4) = 12. These conditions provide us with valuable information about the function's behavior, allowing us to uniquely identify its equation.
Understanding these concepts is crucial for students and professionals alike. Whether you're solving physics problems, modeling economic trends, or simply expanding your mathematical knowledge, mastering the determination of quadratic functions is a valuable asset. This article aims to provide a clear and comprehensive guide to this process, equipping you with the skills and knowledge to tackle similar problems with confidence. By the end of this discussion, you will be able to methodically approach the task of finding a quadratic function that satisfies given conditions, and you will appreciate the power and elegance of quadratic functions in mathematical modeling.
To effectively determine a mathematical sentence for a quadratic function, it's essential to first understand the fundamental properties and forms of these functions. A quadratic function is defined as a polynomial function of degree two, meaning the highest power of the variable x is 2. The general form of a quadratic function is expressed as:
g(x) = ax² + bx + c
where a, b, and c are constants, and a ≠ 0. The coefficient a determines the direction of the parabola's opening: if a > 0, the parabola opens upwards, and if a < 0, it opens downwards. The vertex of the parabola represents the minimum or maximum point of the function, depending on the sign of a. The b coefficient influences the position of the parabola's axis of symmetry, which is the vertical line that passes through the vertex. The c coefficient represents the y-intercept of the parabola, which is the point where the parabola intersects the y-axis.
Quadratic functions can also be expressed in other forms, each highlighting different characteristics of the function. One important form is the vertex form:
g(x) = a(x - h)² + k
where (h, k) represents the coordinates of the vertex of the parabola. This form is particularly useful when the vertex of the parabola is known or needs to be easily identified. Another form is the factored form, also known as the intercept form:
g(x) = a(x - r₁)(x - r₂)
where r₁ and r₂ are the roots (or zeros) of the quadratic function, which are the x-values where the function equals zero. This form is especially helpful when the roots of the quadratic function are known or need to be determined. Understanding the relationship between these different forms of quadratic functions is crucial for solving various problems, including the one we will address in this article.
In summary, quadratic functions possess unique characteristics and can be represented in different forms, each providing valuable insights into the function's behavior. The general form highlights the coefficients that determine the parabola's shape and position, the vertex form emphasizes the vertex coordinates, and the factored form reveals the roots of the function. By grasping these concepts, we lay a strong foundation for determining the specific equation of a quadratic function that satisfies given conditions.
Now that we have a solid understanding of quadratic functions, let's formally state the problem we aim to solve. We are tasked with determining the mathematical sentence (equation) of a quadratic function, denoted as g(x), that satisfies the following conditions:
- g(2) = 0
- g(6) = 0
- g(4) = 12
These conditions provide us with three key pieces of information about the function g(x). The first two conditions, g(2) = 0 and g(6) = 0, tell us that the function has roots (or zeros) at x = 2 and x = 6. In other words, the parabola intersects the x-axis at these two points. This is a crucial piece of information because it allows us to express the quadratic function in its factored form.
The third condition, g(4) = 12, provides us with an additional point on the parabola. This point does not lie on the x-axis, but it gives us a specific (x, y) coordinate (4, 12) that the function must pass through. This condition is essential for determining the leading coefficient a of the quadratic function, which controls the parabola's vertical stretch or compression.
To summarize, we have two roots and one additional point on the parabola. This information is sufficient to uniquely determine the equation of the quadratic function. The challenge lies in strategically using these conditions to find the coefficients of the function. We will explore the methods and techniques involved in solving this problem in the subsequent sections. By carefully applying the properties of quadratic functions and algebraic manipulation, we can arrive at the solution and express the function in its mathematical sentence form. This problem serves as a practical example of how mathematical concepts can be applied to solve real-world problems, highlighting the importance of understanding and mastering quadratic functions.
To determine the mathematical sentence of the quadratic function g(x) satisfying the given conditions, we will employ a systematic approach that leverages the factored form of quadratic functions. Given that we know the roots of the function, g(2) = 0 and g(6) = 0, we can express the function in the following form:
g(x) = a(x - 2)(x - 6)
where a is a constant coefficient that we need to determine. This form directly incorporates the information about the roots, making it a convenient starting point.
Our next step is to use the third condition, g(4) = 12, to solve for the unknown coefficient a. By substituting x = 4 into the equation, we get:
g(4) = a(4 - 2)(4 - 6)
We know that g(4) = 12, so we can substitute this value into the equation:
12 = a(2)(-2)
Simplifying the equation, we have:
12 = -4a
Now, we can solve for a by dividing both sides of the equation by -4:
a = -3
With the value of a determined, we can now write the complete factored form of the quadratic function:
g(x) = -3(x - 2)(x - 6)
While this form is a valid representation of the quadratic function, it is often desirable to express the function in its general form, g(x) = ax² + bx + c. To do this, we need to expand the factored form:
g(x) = -3(x² - 6x - 2x + 12)
g(x) = -3(x² - 8x + 12)
Distributing the -3, we obtain the general form:
g(x) = -3x² + 24x - 36
Therefore, the quadratic function that satisfies the given conditions is g(x) = -3x² + 24x - 36. This methodical approach demonstrates how we can effectively utilize the properties of quadratic functions and algebraic techniques to solve problems involving specific conditions. In the next section, we will verify our solution to ensure its accuracy and consistency.
After determining the mathematical sentence of the quadratic function, it is crucial to verify the solution to ensure its accuracy. This step involves substituting the given values of x into the derived function and checking if the resulting values of g(x) match the conditions provided in the problem statement. Our derived function is:
g(x) = -3x² + 24x - 36
Let's start by verifying the first condition, g(2) = 0. Substituting x = 2 into the function, we get:
g(2) = -3(2)² + 24(2) - 36
g(2) = -3(4) + 48 - 36
g(2) = -12 + 48 - 36
g(2) = 0
The result matches the given condition, so the function satisfies g(2) = 0.
Next, let's verify the second condition, g(6) = 0. Substituting x = 6 into the function, we get:
g(6) = -3(6)² + 24(6) - 36
g(6) = -3(36) + 144 - 36
g(6) = -108 + 144 - 36
g(6) = 0
This result also matches the given condition, confirming that the function satisfies g(6) = 0.
Finally, let's verify the third condition, g(4) = 12. Substituting x = 4 into the function, we get:
g(4) = -3(4)² + 24(4) - 36
g(4) = -3(16) + 96 - 36
g(4) = -48 + 96 - 36
g(4) = 12
This result also matches the given condition, confirming that the function satisfies g(4) = 12.
Since the derived function g(x) = -3x² + 24x - 36 satisfies all three conditions provided in the problem statement, we can confidently conclude that our solution is correct. This verification process is essential in mathematics to ensure the accuracy and reliability of the results. By systematically checking the solution against the given conditions, we gain confidence in the correctness of our approach and the validity of the derived function. In the next section, we will summarize our findings and discuss the implications of the solution in a broader context.
In this article, we successfully determined the mathematical sentence of a quadratic function, g(x), that satisfies the specified conditions: g(2) = 0, g(6) = 0, and g(4) = 12. We started by understanding the fundamental properties of quadratic functions, including their general form, vertex form, and factored form. We then formulated the problem and outlined a systematic approach to solve it.
Our methodology involved leveraging the factored form of quadratic functions, which allowed us to directly incorporate the information about the roots (x = 2 and x = 6). By substituting the given values and solving for the unknown coefficient a, we obtained the factored form of the function: g(x) = -3(x - 2)(x - 6). To express the function in its general form, we expanded the factored form and arrived at the equation g(x) = -3x² + 24x - 36.
To ensure the accuracy of our solution, we performed a thorough verification process. We substituted the given values of x into the derived function and confirmed that the resulting values of g(x) matched the conditions provided in the problem statement. This verification step is crucial in mathematics to build confidence in the correctness of the solution.
The quadratic function g(x) = -3x² + 24x - 36 is a unique solution that satisfies the given conditions. This solution demonstrates the power of quadratic functions in modeling various phenomena. Quadratic functions are widely used in physics to describe projectile motion, in engineering to design parabolic structures, and in economics to model cost and revenue functions. The ability to determine the equation of a quadratic function that satisfies specific conditions is a valuable skill in many fields.
In summary, this article provided a comprehensive guide to determining the equation of a quadratic function given specific conditions. We explored the properties of quadratic functions, developed a systematic approach to solve the problem, and verified the accuracy of our solution. This process highlights the importance of understanding mathematical concepts, applying them strategically, and verifying the results to ensure their validity. The skills and knowledge gained from this discussion can be applied to a wide range of problems involving quadratic functions and mathematical modeling.