Question

1. Дана функция у = -x² - 4x - 4, найдите а) координаты вершин параболы b) ось симметрии параболы с) точки пересечения параболы с осью Ох d) точки пересечения параболы с осью Оу e) постройте график функции

Answer 1

Ответ:а)(-2;0)

б)-2

c)(-2;0)

Объяснение:a) Хв=4/-2=-2

Ув=-4+8-4=0

б)Х=4/-2=-2

c)y=0

d)(0;-4)

-x²-4x-4=0

x²+4x+4=0

(x+2)²=0

x=-2

d)x=0

y=0-0-4=-4

Answer 2

Given Function:

The given function is u = -x² - 4x - 4.

a) Coordinates of the Vertex:

To find the coordinates of the vertex of the parabola, we can use the formula x = -b/2a. In this case, a = -1 and b = -4.

Substituting the values into the formula, we get: x = -(-4) / (2 * -1) = 4 / -2 = -2.

To find the y-coordinate of the vertex, we substitute the value of x into the given function: u = -(-2)² - 4(-2) - 4 = -4 + 8 - 4 = 0.

Therefore, the coordinates of the vertex are (-2, 0).

b) Axis of Symmetry:

The axis of symmetry of a parabola is a vertical line that passes through the vertex. The equation of the axis of symmetry is x = -b/2a.

In this case, the equation of the axis of symmetry is x = -(-4) / (2 * -1) = 4 / -2 = -2.

Therefore, the axis of symmetry is the vertical line x = -2.

c) Points of Intersection with the x-axis:

To find the points of intersection of the parabola with the x-axis, we set the value of u to zero and solve for x.

Setting u = 0, we have: 0 = -x² - 4x - 4.

To solve this quadratic equation, we can factor it or use the quadratic formula. Factoring is not possible in this case, so we'll use the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a).

Substituting the values into the formula, we get: x = (-(-4) ± √((-4)² - 4 * (-1) * (-4))) / (2 * (-1)).

Simplifying further, we have: x = (4 ± √(16 - 16)) / (-2), x = (4 ± √0) / (-2), x = (4 ± 0) / (-2).

Since the discriminant (√0) is zero, there is only one real root: x = 4 / -2 = -2.

Therefore, the parabola intersects the x-axis at the point (-2, 0).

d) Point of Intersection with the y-axis:

To find the point of intersection of the parabola with the y-axis, we set the value of x to zero and solve for u.

Setting x = 0, we have: u = -(0)² - 4(0) - 4, u = -4.

Therefore, the parabola intersects the y-axis at the point (0, -4).

e) Graph of the Function:

To plot the graph of the function, we can use the coordinates of the vertex, the points of intersection with the x-axis and y-axis, and the axis of symmetry.

The vertex is (-2, 0), the point of intersection with the x-axis is (-2, 0), and the point of intersection with the y-axis is (0, -4). The axis of symmetry is x = -2.

Here is the graph of the function:

   ^
   |
   |
   |
   |
   |
   |
   |
   |       *
   |      *
   |     *
   |    *
   |   *
   |  *
   | *
   |*
   +--------------------------------->
         -4  -3  -2  -1   0   1   2   3   4

The graph is a downward-opening parabola with the vertex at (-2, 0).

Note: The graph is a rough representation and may not be to scale.

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