Question

y=x^2-6x+4 ось симметрии,вершины пар-лы,нули функции,промежутки знакопостоянства,промежутки возростния и убывания,вершина пар-лы,направление вершинвсе это мне нужно узнать

Answer 1

сначала преобразовываем.....у=х в квадрате-2*3+3-3+4  , у=(х-3)в кв +1  , вершина параболы(3;1)  , ось симметрии х=3 нули ф: нет   , пром. знакопостоянства: незнаю что это, пром убывания: х принадлежит( -бесконечность;3] возрастания х принадлежит [3; +бесконечности)

 

Answer 2

Equation Analysis: y = x^2 - 6x + 4

To analyze the equation y = x^2 - 6x + 4, we will determine the axis of symmetry, vertex, zeros of the function, intervals of sign, intervals of increase and decrease, and the direction of the vertex.

Axis of Symmetry:

The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves. For a quadratic equation in the form y = ax^2 + bx + c, the equation for the axis of symmetry is given by x = -b/2a. In our case, the equation is y = x^2 - 6x + 4, so the axis of symmetry is x = -(-6)/(2*1) = 3.

Vertex:

The vertex is the point on the parabola where it reaches its minimum or maximum value. The x-coordinate of the vertex is the same as the x-coordinate of the axis of symmetry. To find the y-coordinate of the vertex, substitute the x-coordinate into the equation. In our case, the x-coordinate is 3. Substituting x = 3 into the equation y = x^2 - 6x + 4, we get y = 3^2 - 6(3) + 4 = 9 - 18 + 4 = -5. Therefore, the vertex is (3, -5).

Zeros of the Function:

The zeros of the function are the values of x for which y = 0. To find the zeros, we set the equation y = x^2 - 6x + 4 equal to zero and solve for x. In our case, we have x^2 - 6x + 4 = 0. To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. Factoring is not possible in this case, so we will use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a). Plugging in the values from our equation, we get x = (6 ± √((-6)^2 - 4(1)(4))) / (2(1)) = (6 ± √(36 - 16)) / 2 = (6 ± √20) / 2 = 3 ± √5. Therefore, the zeros of the function are x = 3 + √5 and x = 3 - √5.

Intervals of Sign:

To determine the intervals of sign, we need to find the values of x for which the function is positive or negative. We can do this by analyzing the sign of the quadratic expression x^2 - 6x + 4. We know that the parabola opens upwards because the coefficient of x^2 is positive. The sign of the quadratic expression depends on the value of x. We can use the zeros of the function to divide the x-axis into intervals. Testing a value from each interval will determine the sign of the expression in that interval.

- For x < 3 - √5, the expression is positive. - For 3 - √5 < x < 3 + √5, the expression is negative. - For x > 3 + √5, the expression is positive.

Intervals of Increase and Decrease:

To determine the intervals of increase and decrease, we need to analyze the slope of the function. The slope of a quadratic function is given by the derivative of the function. Taking the derivative of y = x^2 - 6x + 4, we get y' = 2x - 6. The derivative is a linear function, and its sign determines the slope of the quadratic function.

- For x < 3, the derivative is negative, indicating a decreasing slope. - For x > 3, the derivative is positive, indicating an increasing slope.

Therefore, the function y = x^2 - 6x + 4 is decreasing on the interval (-∞, 3) and increasing on the interval (3, +∞).

Direction of the Vertex:

The direction of the vertex can be determined by the coefficient of x^2. If the coefficient is positive, the parabola opens upwards, and if it is negative, the parabola opens downwards. In our case, the coefficient of x^2 is positive, so the parabola opens upwards.

To summarize: - Axis of Symmetry: x = 3 - Vertex: (3, -5) - Zeros of the Function: x = 3 + √5 and x = 3 - √5 - Intervals of Sign: (-∞, 3 - √5), (3 - √5, 3 + √5), (3 + √5, +∞) - Intervals of Increase and Decrease: Decreasing on (-∞, 3), Increasing on (3, +∞) - Direction of the Vertex: Upwards

Please let me know if there's anything else I can help you with!