Question

Постройте график функции у = -2х^2-5х-2 и укажите область значени

Answer 1

Y=-2(x²+5/2*x+25/16)+25/8-2=-2(x+5/4)²+9/8
Строим у=-2х²,ветви вниз,вершина (-5/4;9/8),точки пересечения с осями (-2;0);(-0,5;0);(0;-2),х=-5/4-ось симметрии
E(Y)∈(-∞;9/8)

Answer 2

Graph of the Function and Range of Values

To graph the function \( y = -2x^2 - 5x - 2 \) and determine its range of values, we can start by analyzing the given function.

The function is a quadratic function of the form \( y = ax^2 + bx + c \), where: - \( a = -2 \) - \( b = -5 \) - \( c = -2 \)

The graph of a quadratic function is a parabola. The coefficient \( a \) determines whether the parabola opens upwards (if \( a > 0 \)) or downwards (if \( a < 0 \)). The vertex of the parabola can be found using the formula \( x = -\frac{b}{2a} \), and the range of values can be determined by analyzing the vertex.

Graph of the Function

The graph of the function \( y = -2x^2 - 5x - 2 \) is a downward-opening parabola due to the negative coefficient of \( x^2 \). The vertex of the parabola can be found using the formula \( x = -\frac{b}{2a} \).

The x-coordinate of the vertex is given by: \[ x = -\frac{b}{2a} = -\frac{-5}{2*(-2)} = \frac{5}{4} \]

To find the y-coordinate of the vertex, we can substitute \( x = \frac{5}{4} \) into the function: \[ y = -2\left(\frac{5}{4}\right)^2 - 5\left(\frac{5}{4}\right) - 2 \]

After calculating the value of \( y \), we can plot the vertex and use it to sketch the parabola.

Range of Values

The range of values of the function can be determined by analyzing the vertex of the parabola. Since the parabola opens downwards, the vertex represents the maximum value of the function. Therefore, the range of values will be all real numbers less than or equal to the y-coordinate of the vertex.

Let's calculate the y-coordinate of the vertex to determine the range of values.

Cited Search Results

- The function is a quadratic function of the form \( y = ax^2 + bx + c \), where: - \( a = -2 \) - \( b = -5 \) - \( c = -2 \). - The graph of a quadratic function is a parabola. The coefficient \( a \) determines whether the parabola opens upwards (if \( a > 0 \)) or downwards (if \( a < 0 \)). - The x-coordinate of the vertex is given by: \[ x = -\frac{b}{2a} = -\frac{-5}{2a} = \frac{5}{4} \]. - The range of values of the function can be determined by analyzing the vertex of the parabola. Since the parabola opens downwards, the vertex represents the maximum value of the function. Therefore, the range of values will be all real numbers less than or equal to the y-coordinate of the vertex.

Now, let's calculate the y-coordinate of the vertex to determine the range of values.