Understanding the Properties of the Quadratic Function f(x) = x²: A Detailed Analysis

f(x)=x²

The function f(x) = x² is a quadratic function, which means it is a polynomial of degree 2

The function f(x) = x² is a quadratic function, which means it is a polynomial of degree 2. The graph of this quadratic function is a parabola that opens upwards.

To understand the properties of the function f(x) = x², let’s look at a few aspects:

1. Domain and Range:
The domain of the function is all real numbers, since any value of x can be squared. The range of the function is all non-negative real numbers, including zero, because squaring any real number results in a positive or zero value.

2. Symmetry:
The function f(x) = x² is symmetric with respect to the y-axis. This means that if you replace x with -x in the function, the function remains the same. In other words, f(-x) = f(x).

3. Vertex:
The vertex of the graph of f(x) = x² is at the point (0, 0). The vertex represents the minimum point of the parabola. Since the parabola opens upwards, the vertex is the lowest point on the graph.

4. Axis of Symmetry:
The axis of symmetry of the graph of f(x) = x² is the vertical line x = 0. The axis of symmetry is the line that divides the parabola into two symmetrical halves.

5. Increasing and Decreasing Intervals:
The function f(x) = x² is strictly increasing for x > 0, meaning that as x increases, the corresponding values of f(x) also increase. Similarly, the function is strictly decreasing for x < 0. 6. Zeros: The only zero of the function f(x) = x² is x = 0. This means that the parabola intersects the x-axis only at the origin (0, 0). These are some of the key properties and characteristics of the function f(x) = x². Understanding these properties can help you analyze the behavior of the function and interpret its graph.

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