Question

4x-x^2 исследовать функцию и построить её график

Answer 1

Парабола вниз
Х=1 у=3 ,Х=2 у=4 и так далее и подставляешь эти точки и будет парабола

Answer 2

Function Analysis and Graph Plotting

To analyze the function f(x) = 4x - x^2 and plot its graph, we will examine its key properties such as the domain, range, intercepts, symmetry, and extrema.

Domain: The domain of a function represents all the possible values of x for which the function is defined. In this case, since there are no restrictions on the variable x, the domain of the function is all real numbers.

Range: The range of a function represents all the possible values of y that the function can take. To determine the range, we can analyze the behavior of the function. Since the function is a quadratic function with a negative coefficient for the x^2 term, it opens downwards. Therefore, the range of the function is (-∞, maximum value].

Intercepts: To find the x-intercepts, we set f(x) = 0 and solve for x: 4x - x^2 = 0 x(4 - x) = 0 x = 0 or x = 4

Therefore, the x-intercepts are (0, 0) and (4, 0).

To find the y-intercept, we set x = 0: f(0) = 4(0) - (0)^2 = 0

Therefore, the y-intercept is (0, 0).

Symmetry: To determine if the function is symmetric, we can check if it is an even or odd function. An even function is symmetric about the y-axis, while an odd function is symmetric about the origin.

In this case, the function is neither even nor odd since it does not satisfy the conditions for even or odd symmetry.

Extrema: To find the extrema (maximum or minimum points) of the function, we can take the derivative of the function and set it equal to zero.

f'(x) = 4 - 2x

Setting f'(x) = 0, we have: 4 - 2x = 0 2x = 4 x = 2

To determine if it is a maximum or minimum point, we can take the second derivative of the function.

f''(x) = -2

Since the second derivative is negative, the point (2, f(2)) is a maximum point.

Graph Plotting:

Now, let's plot the graph of the function f(x) = 4x - x^2.


Please note that the graph is a downward-opening parabola, and the maximum point is located at (2, f(2)).

I hope this helps! Let me know if you have any further questions.