EVALUARE NAȚIONALĂ MATEMATICĂ 2025 Solution Function F(x) = 2x - 4

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EVALUARE NAȚIONALĂ MATEMATICĂ 2025 Sub. III. 3. Function f(x) = 2x - 4. a) Show f(2) - f(0) = 4. b) Find coordinates A and B where the function's graph intersects Ox and Oy. c) Calculate the area of the triangle formed by the function's graph and Ox, Oy axes.

Problem Statement

Consider the function f:R→R, defined by f(x) = 2x - 4.

a) Show that f(2) - f(0) = 4. b) The geometric representation of the graph of the function f intersects the Ox and Oy axes of the orthogonal coordinate system xOy at points A and B, respectively. Determine the coordinates of points A and B. c) Calculate the area of the triangle formed by the graph of the function f and the axes Ox and Oy of the xOy coordinate system.

Solution

a) Show that f(2) - f(0) = 4.

To demonstrate that f(2) - f(0) = 4, we first need to evaluate f(2) and f(0) using the given function f(x) = 2x - 4. Let's start by calculating f(2):

f(2) = 2 * (2) - 4 f(2) = 4 - 4 f(2) = 0

Now, let's calculate f(0):

f(0) = 2 * (0) - 4 f(0) = 0 - 4 f(0) = -4

Now that we have the values of f(2) and f(0), we can compute f(2) - f(0):

f(2) - f(0) = 0 - (-4) f(2) - f(0) = 0 + 4 f(2) - f(0) = 4

Therefore, we have shown that f(2) - f(0) = 4, as required. This confirms the initial statement by direct calculation and substitution into the function's formula. This first part of the problem emphasizes the importance of direct substitution and basic arithmetic operations in function evaluation. By correctly applying the function's definition to the given inputs, we arrive at the desired conclusion. The ability to accurately perform these calculations is fundamental to solving more complex problems involving functions. Furthermore, this exercise reinforces the understanding of how the function's output changes with different inputs, setting the stage for further analysis of the function's properties and behavior. Understanding this foundational aspect is vital for mastering more advanced concepts in mathematics and for applying these concepts to real-world scenarios. The simplicity of this part of the problem belies its significance in building a strong mathematical foundation. The result also gives a first glimpse into the nature of this particular function, a linear function, where consistent changes in input result in consistent changes in output.

b) Determine the coordinates of points A and B.

To determine the coordinates of points A and B, where the graph of the function f(x) = 2x - 4 intersects the Ox and Oy axes, we need to understand what these intersections represent. The point where the graph intersects the Ox axis (point A) has a y-coordinate of 0, and the point where the graph intersects the Oy axis (point B) has an x-coordinate of 0.

Let's find the coordinates of point A, which lies on the Ox axis. To do this, we set f(x) = 0 and solve for x:

2x - 4 = 0 2x = 4 x = 2

So, the coordinates of point A are (2, 0). This calculation involves a simple algebraic manipulation to isolate x, demonstrating a core skill in solving equations. Finding the x-intercept is crucial in understanding the function's behavior and its relationship to the coordinate axes. This point represents the root of the equation f(x) = 0, and its graphical representation is where the line crosses the horizontal axis. The ability to find such intercepts is fundamental in various mathematical applications, including graphing, optimization, and root-finding algorithms. The precise determination of point A allows us to visualize one of the key features of the function's graph, its point of intersection with the x-axis, which is a vital step in understanding the function's overall characteristics.

Now, let's find the coordinates of point B, which lies on the Oy axis. To do this, we set x = 0 and solve for f(x):

f(0) = 2 * (0) - 4 f(0) = -4

So, the coordinates of point B are (0, -4). Determining the y-intercept is equally important as finding the x-intercept. The y-intercept provides insight into the function's value when x is zero, which is often a significant point in many practical applications. For instance, in a linear cost function, the y-intercept might represent the fixed costs, while the slope represents the variable costs. Geometrically, the y-intercept marks the point where the function's graph intersects the vertical axis, giving us another anchor point to understand its behavior. The calculation involves direct substitution and reveals the function's value at a critical point, making it a fundamental concept in function analysis. The coordinates of point B, (0, -4), define this key intersection and are essential for sketching the graph and understanding the function's vertical position.

In conclusion, the coordinates of point A are (2, 0), and the coordinates of point B are (0, -4). These two points define the intersections of the function's graph with the coordinate axes and are crucial for understanding its graphical representation and behavior. Finding these coordinates is a fundamental skill in algebra and calculus and provides a basis for further analysis of functions and their properties. This step connects the algebraic representation of the function with its geometric interpretation, allowing for a comprehensive understanding of the function's characteristics and behavior. The accuracy in determining these points is essential for the subsequent calculation of the area of the triangle formed by the function's graph and the axes.

c) Calculate the area of the triangle formed by the graph of the function f and the axes Ox and Oy of the xOy coordinate system.

To calculate the area of the triangle formed by the graph of the function f(x) = 2x - 4 and the axes Ox and Oy, we first need to visualize this triangle. The vertices of this triangle are the origin (0, 0), point A (2, 0), and point B (0, -4). The triangle is a right-angled triangle, with the right angle at the origin.

The base of the triangle is the distance from the origin to point A, which is the absolute value of the x-coordinate of A, i.e., |2| = 2 units. The height of the triangle is the distance from the origin to point B, which is the absolute value of the y-coordinate of B, i.e., |-4| = 4 units.

The area of a triangle is given by the formula:

Area = (1/2) * base * height

In this case, the base is 2 units, and the height is 4 units. Plugging these values into the formula, we get:

Area = (1/2) * 2 * 4 Area = (1/2) * 8 Area = 4 square units

Therefore, the area of the triangle formed by the graph of the function f(x) = 2x - 4 and the axes Ox and Oy is 4 square units. Calculating the area of a triangle formed by a linear function and the coordinate axes is a common problem in coordinate geometry. The key to solving this is to first identify the vertices of the triangle, which are the intersection points of the function's graph with the axes and the origin. The lengths of the base and height can then be determined from the coordinates of these points. The use of absolute values ensures that the lengths are positive, regardless of the signs of the coordinates. Applying the area formula is a straightforward process once the base and height are known. This calculation demonstrates the practical application of coordinate geometry principles and the understanding of how geometric shapes can be defined and measured within a coordinate system. The final answer, 4 square units, represents the space enclosed by the function's graph and the axes, providing a concrete measure of a geometric property derived from the function.

In conclusion, by finding the points of intersection with the axes and applying the formula for the area of a triangle, we successfully calculated the area formed by the graph of the function and the coordinate axes. This problem illustrates the interconnectedness of algebraic functions and geometric shapes, highlighting the importance of visualizing mathematical concepts and applying appropriate formulas to solve problems.