Question

Y=2x^2+5x+2 Исследовать функцию Решить задачу

Answer 1

Точек перегиба нет

 

6) Асимптот нет

7) Непериодическая

8) Построим график (во вложении)

Answer 2

Function Analysis

The given function is Y = 2x^2 + 5x + 2. Let's analyze this function step by step.

1. Degree of the Function: The degree of the function is the highest power of x in the function. In this case, the highest power of x is 2, so the degree of the function is 2.

2. Vertex: The vertex of a quadratic function in the form of y = ax^2 + bx + c can be found using the formula x = -b/2a. In this case, a = 2, b = 5, and c = 2. Plugging these values into the formula, we get x = -5/(2*2) = -5/4. To find the y-coordinate of the vertex, we substitute this value of x back into the function: y = 2*(-5/4)^2 + 5*(-5/4) + 2. Evaluating this expression, we get y = 2.25. Therefore, the vertex of the function is (-5/4, 2.25).

3. Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex of the function. In this case, the axis of symmetry is the line x = -5/4.

4. Discriminant: The discriminant of a quadratic function in the form of y = ax^2 + bx + c is given by the formula D = b^2 - 4ac. In this case, a = 2, b = 5, and c = 2. Plugging these values into the formula, we get D = 5^2 - 4*2*2 = 25 - 16 = 9. The discriminant is positive, which means the function has two distinct real roots.

5. Roots: The roots of a quadratic function can be found using the quadratic formula x = (-b ± √D) / (2a). In this case, a = 2, b = 5, and c = 2. Plugging these values into the formula, we get x = (-5 ± √9) / (2*2) = (-5 ± 3) / 4. Simplifying further, we get x1 = -2/4 = -1/2 and x2 = -8/4 = -2. Therefore, the roots of the function are x = -1/2 and x = -2.

6. Y-Intercept: The y-intercept is the point where the function intersects the y-axis. To find the y-intercept, we substitute x = 0 into the function: y = 2*0^2 + 5*0 + 2 = 2. Therefore, the y-intercept of the function is (0, 2).

7. Increasing and Decreasing Intervals: To determine the intervals where the function is increasing or decreasing, we need to examine the sign of the coefficient of x^2, which is positive in this case (a = 2). Since the coefficient is positive, the function is increasing for all values of x.

8. Concavity: The concavity of a quadratic function can be determined by examining the sign of the coefficient of x^2. In this case, the coefficient is positive (a = 2), which means the function is concave up.

Solution to the Problem

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