Exploring the Characteristics and Properties of the Quadratic Function f(x) = x²

f(x)=x² (quadratic)

The given equation represents a quadratic function

The given equation represents a quadratic function. A quadratic function is a polynomial function of degree 2, which means it has the highest power of x as 2. In this case, f(x) = x², the function is defined as the square of x.

Graphically, the graph of a quadratic function is a curve called a parabola. For the function f(x) = x², the parabola opens upwards. The vertex of the parabola is at the point (0, 0), which corresponds to the minimum value of the function.

To analyze the properties of the quadratic function f(x) = x², we can look at its key characteristics:

1. Vertex: The vertex of a quadratic function is the point where the parabola reaches its minimum (if it opens upwards) or maximum (if it opens downwards). For f(x) = x², the vertex is at (0, 0).

2. Axis of symmetry: The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two symmetric halves. The axis of symmetry for f(x) = x² is the line x = 0.

3. Y-intercept: The point where the graph of a function intersects the y-axis is called the y-intercept. For f(x) = x², the y-intercept is (0, 0).

4. X-intercepts: The points where the graph of a function intersects the x-axis are called the x-intercepts or roots of the equation. To find the x-intercepts of f(x) = x², we set f(x) equal to zero and solve for x: x² = 0. The only solution is x = 0, so the parabola touches the x-axis at (0, 0).

5. Symmetry: A quadratic function is symmetric with respect to its axis of symmetry. For f(x) = x², the graph is symmetric with respect to the y-axis.

6. Increasing and decreasing intervals: The quadratic function f(x) = x² is always increasing for x > 0 and always decreasing for x < 0. There is no maximum or minimum. 7. Range: The range of f(x) = x² is all real numbers greater than or equal to 0. In other words, the function takes on all non-negative values. These are some of the important characteristics and properties of the quadratic function f(x) = x².

More Answers: