Question

Иследовать график функции y=0,5x^2+4x+10

Answer 1

Пишем уравнение функции.
Y = 0.5*X² + 4X+10
1) Область определения - R - все по Х.
2) Находим ДИСКРИМИНАНТ квадратного уравнения и видим, что D= - 4
Вывод действительных корней - точек пересечения с осью Х - НЕТ
3) Для поиска экстремума находим производную
Y' = X+4.  Минимум при Х= - 4 равен Y=2
4) Строим график  для проверки
И - всё готово.

Answer 2

Graph of the Function y = 0.5x^2 + 4x + 10

To analyze the graph of the function y = 0.5x^2 + 4x + 10, we can examine its key features such as the vertex, axis of symmetry, and the behavior of the graph.

The general form of a quadratic function is y = ax^2 + bx + c, where a, b, and c are constants. In this case, a = 0.5, b = 4, and c = 10.

To find the vertex of the parabola, we can use the formula x = -b/2a. Plugging in the values, we get:

x = -(4)/(2*0.5) = -4/1 = -4

To find the corresponding y-coordinate, we substitute x = -4 into the equation:

y = 0.5(-4)^2 + 4(-4) + 10 = 0.5(16) - 16 + 10 = 8 - 16 + 10 = 2

Therefore, the vertex of the parabola is (-4, 2).

The axis of symmetry is a vertical line that passes through the vertex. In this case, the equation of the axis of symmetry is x = -4.

To determine the behavior of the graph, we can look at the coefficient of the x^2 term. Since the coefficient a = 0.5 is positive, the parabola opens upwards. This means that as x approaches positive or negative infinity, the y-values of the function also increase without bound.

Now let's plot some points to get a better understanding of the graph:

- When x = -3: y = 0.5(-3)^2 + 4(-3) + 10 = 0.5(9) - 12 + 10 = 4.5 - 12 + 10 = 2.5 - 2 = 0.5 So, the point (-3, 0.5) lies on the graph.

- When x = 0: y = 0.5(0)^2 + 4(0) + 10 = 0 + 0 + 10 = 10 So, the point (0, 10) lies on the graph.

- When x = 1: y = 0.5(1)^2 + 4(1) + 10 = 0.5 + 4 + 10 = 14.5 So, the point (1, 14.5) lies on the graph.

- When x = 2: y = 0.5(2)^2 + 4(2) + 10 = 2 + 8 + 10 = 20 So, the point (2, 20) lies on the graph.

- When x = 3: y = 0.5(3)^2 + 4(3) + 10 = 4.5 + 12 + 10 = 26.5 So, the point (3, 26.5) lies on the graph.

- When x = 4: y = 0.5(4)^2 + 4(4) + 10 = 8 + 16 + 10 = 34 So, the point (4, 34) lies on the graph.

Based on these points, we can plot the graph of the function y = 0.5x^2 + 4x + 10. The graph is a parabola that opens upwards, with the vertex at (-4, 2) and the axis of symmetry at x = -4. The graph passes through the points (-3, 0.5), (0, 10), (1, 14.5), (2, 20), (3, 26.5), and (4, 34).


Please note that the graph is a visual representation and may not be perfectly accurate. It is always recommended to use graphing software or a graphing calculator for precise graphs.