Question

Построить график функции и описать основные свойства. y=x^2-3x-4

Answer 1

Решение во вложения, удачи.

Answer 2

Graph of the Function y = x^2 - 3x - 4

To graph the function y = x^2 - 3x - 4, we can plot several points and connect them to form a curve. Let's find some key points on the graph:

1. When x = 0: - y = (0)^2 - 3(0) - 4 = -4 - So, we have the point (0, -4).

2. When x = 2: - y = (2)^2 - 3(2) - 4 = -2 - So, we have the point (2, -2).

3. When x = 4: - y = (4)^2 - 3(4) - 4 = 0 - So, we have the point (4, 0).

Now, let's plot these points on a graph:

| x | y | |---|---| | 0 | -4| | 2 | -2| | 4 | 0|

We can connect these points to form a curve. The graph of the function y = x^2 - 3x - 4 looks like this:


Properties of the Function y = x^2 - 3x - 4

1. Vertex: The vertex of a quadratic function in the form y = ax^2 + bx + c is given by the formula x = -b/2a. In this case, a = 1, b = -3, and c = -4. Plugging these values into the formula, we get x = -(-3)/(2*1) = 3/2. Therefore, the vertex of the function y = x^2 - 3x - 4 is (3/2, -25/4).

2. Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex of the parabola. For the function y = x^2 - 3x - 4, the axis of symmetry is x = 3/2.

3. Intercepts: To find the x-intercepts, we set y = 0 and solve for x. For the function y = x^2 - 3x - 4, we can factor the quadratic equation or use the quadratic formula to find the x-intercepts. The x-intercepts are (-1, 0) and (4, 0).

4. Minimum/Maximum: Since the coefficient of x^2 is positive (a = 1), the parabola opens upward, and the vertex represents the minimum point on the graph. Therefore, the function y = x^2 - 3x - 4 has a minimum value at the vertex (3/2, -25/4).

5. Symmetry: The function y = x^2 - 3x - 4 is symmetric with respect to the axis of symmetry x = 3/2. This means that if we reflect any point on the graph across the axis of symmetry, we will get another point on the graph.

6. Increasing/Decreasing Intervals: The function y = x^2 - 3x - 4 is increasing to the left of the vertex (3/2, -25/4) and decreasing to the right of the vertex. The vertex represents the minimum point on the graph.

These are the main properties of the function y = x^2 - 3x - 4.