У=х^2 -6х+3

To prepare for university exams, it is important to familiarize yourself with various mathematical concepts and techniques. In this case, the equation given is a quadratic equation in the form of y = x^2 - 6x + 3.

To further analyze this equation, you can determine its vertex, axis of symmetry, and whether it opens upward or downward.

1. Vertex: The vertex of a quadratic equation in the form y = ax^2 + bx + c can be determined using the formula x = -b / (2a). In this case, a = 1, b = -6, and c = 3. Plugging these values into the formula, we get x = -(-6) / (2*1), which simplifies to x = 3. To find the y-coordinate of the vertex, you can substitute x = 3 into the equation. y = (3)^2 - 6(3) + 3, which simplifies to y = 3 - 18 + 3, resulting in y = -12.

So, the vertex of the given equation is (3, -12).

2. Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex of a parabola. For a quadratic equation in the form y = ax^2 + bx + c, the equation of the axis of symmetry is x = -b / (2a). In this case, substituting the values a = 1 and b = -6 into the formula, we get x = -(-6) / (2*1), which simplifies to x = 3. Therefore, the axis of symmetry for the given equation is x = 3.

3. Direction of Opening: The coefficient a in the quadratic equation determines the direction of the parabola's opening. If a is positive, the parabola opens upward, and if a is negative, the parabola opens downward. In this case, the coefficient a is 1, which is positive. Therefore, the parabola opens upward.

In summary, the given equation y = x^2 - 6x + 3 has a vertex at (3, -12), an axis of symmetry at x = 3, and the parabola opens upwards. These insights can help you solve problems related to this equation and related concepts in your university exams.