Determine The Real Number 𝑎 For Point A(1; 0) On Function 𝑓(x) = X² + 𝑎x + 1

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In mathematics, particularly in the study of functions and graphs, a fundamental problem involves determining the parameters of a function given certain conditions. One such condition is that a specific point lies on the graph of the function. This article delves into the process of finding the real number a for which the point A(1; 0) belongs to the graph of the function f: ℝ → ℝ, defined by f(x) = x² + ax + 1. This exploration is crucial for understanding how parameters influence the behavior and graphical representation of functions.

Understanding the Problem

The core of the problem lies in understanding the relationship between a function, its graph, and the points that lie on that graph. A point (x, y) belongs to the graph of a function f(x) if and only if f(x) = y. In simpler terms, when we substitute the x-coordinate of the point into the function, the output must be equal to the y-coordinate of the point. In our case, we are given the function f(x) = x² + ax + 1 and the point A(1; 0). This means that for point A to lie on the graph of f, the equation f(1) = 0 must hold true. Our task is to find the value of a that satisfies this condition. This involves a straightforward application of algebraic principles, specifically the substitution of values and the solving of a linear equation. This is a common type of problem encountered in algebra and pre-calculus courses, serving as a building block for more complex concepts in mathematical analysis. The ability to solve such problems is essential for students to develop a strong foundation in mathematics, enabling them to tackle more advanced topics such as calculus and differential equations.

Step-by-Step Solution

To determine the value of a, we will follow a step-by-step approach:

  1. Substitute the x-coordinate of point A into the function: Since point A has coordinates (1; 0), we substitute x = 1 into the function f(x) = x² + ax + 1. This gives us f(1) = (1)² + a(1) + 1.

  2. Set the function equal to the y-coordinate of point A: We know that for point A to belong to the graph of f, f(1) must equal 0. Therefore, we set up the equation: (1)² + a(1) + 1 = 0.

  3. Simplify the equation: Simplifying the equation, we get 1 + a + 1 = 0, which further simplifies to a + 2 = 0.

  4. Solve for a: To isolate a, we subtract 2 from both sides of the equation: a = -2.

Therefore, the real number a for which the point A(1; 0) belongs to the graph of the function f(x) = x² + ax + 1 is a = -2. This result is crucial as it uniquely defines the quadratic function that passes through the given point. The process involves a clear application of algebraic principles and emphasizes the importance of understanding the relationship between a function and its graph. This type of problem not only tests the student's ability to manipulate algebraic expressions but also reinforces the fundamental concept of how points are related to functions in a coordinate plane. Mastering this concept is essential for further studies in mathematics, particularly in calculus and analytical geometry.

Verifying the Solution

To ensure the accuracy of our solution, it's crucial to verify that the value of a we found indeed satisfies the given condition. We can do this by substituting a = -2 back into the function and checking if f(1) equals 0. This verification step is a fundamental practice in mathematics, ensuring that the solution obtained is not only mathematically correct but also consistent with the initial problem statement. By performing this check, we can confidently confirm that our solution is valid and that no errors were made during the solution process. This practice also reinforces the understanding of the relationship between a function's parameters and its behavior, providing a deeper insight into the function's properties and its graphical representation.

  1. Substitute a = -2 into the function: We have f(x) = x² + ax + 1. Substituting a = -2, we get f(x) = x² - 2x + 1.

  2. Evaluate f(1): Now, we substitute x = 1 into the modified function: f(1) = (1)² - 2(1) + 1.

  3. Simplify: Simplifying the expression, we get f(1) = 1 - 2 + 1 = 0.

Since f(1) = 0, our solution a = -2 is verified. This confirms that the point A(1; 0) indeed lies on the graph of the function f(x) = x² - 2x + 1. The verification process highlights the importance of double-checking solutions in mathematical problems, especially in algebraic manipulations and function evaluations. This practice not only ensures the correctness of the answer but also helps in identifying any potential errors in the reasoning or calculations. Moreover, it strengthens the student's understanding of the underlying mathematical principles and concepts.

Graphical Interpretation

Visualizing the problem graphically provides a deeper understanding of the solution. The function f(x) = x² + ax + 1 represents a parabola. The value of a affects the shape and position of the parabola in the coordinate plane. When a = -2, the function becomes f(x) = x² - 2x + 1, which can be further simplified as f(x) = (x - 1)². This is a parabola that opens upwards and has its vertex at the point (1, 0). The fact that point A(1; 0) lies on the graph means that this point is the vertex of the parabola.

The graphical interpretation helps to connect the algebraic solution with the geometric representation of the function. It illustrates how changing the parameter a can shift and deform the parabola. In this specific case, setting a to -2 results in a parabola that touches the x-axis at the point (1, 0), indicating a unique solution where the function has a double root. This visual understanding is crucial for developing a strong intuition for functions and their properties. By visualizing the graph, students can better comprehend the relationship between the algebraic form of a function and its geometric behavior, which is a fundamental skill in mathematics.

Implications and Applications

Determining the parameters of a function based on given conditions has significant implications and applications in various fields. In mathematics, this skill is essential for curve fitting, where we find a function that best represents a set of data points. In physics, it can be used to model trajectories and other physical phenomena. In engineering, it is crucial for designing systems that meet specific performance requirements. The ability to find the value of a in this problem is a basic example of a broader class of problems where parameters of functions are determined based on specific conditions.

This type of problem also serves as a foundation for more advanced topics in mathematics. For example, in calculus, similar techniques are used to find the equation of a tangent line to a curve or to determine the maximum or minimum values of a function. In linear algebra, the concept of parameter determination is extended to systems of equations and matrices. Therefore, mastering this basic skill is crucial for students who plan to pursue further studies in mathematics, science, or engineering. The problem-solving approach used here, which involves substituting values, simplifying equations, and verifying solutions, is a fundamental strategy that can be applied to a wide range of mathematical problems. This highlights the importance of developing a strong foundation in basic algebraic principles and problem-solving techniques.

Conclusion

In conclusion, we have successfully determined that the real number a for which the point A(1; 0) belongs to the graph of the function f(x) = x² + ax + 1 is a = -2. This was achieved by substituting the coordinates of point A into the function, setting the result equal to 0, and solving the resulting equation for a. We then verified our solution by substituting a = -2 back into the function and confirming that f(1) = 0. Furthermore, we explored the graphical interpretation of the problem, which provided a visual understanding of the solution and its implications.

This problem illustrates the fundamental relationship between a function, its graph, and the points that lie on the graph. It also demonstrates the importance of algebraic manipulation, equation solving, and solution verification in mathematics. The problem-solving techniques used here are applicable to a wide range of mathematical problems, making this a valuable exercise for students studying algebra and pre-calculus. The ability to determine the parameters of a function based on given conditions is a crucial skill in various fields, including mathematics, physics, engineering, and computer science. Mastering this skill provides a strong foundation for more advanced studies and practical applications.