Understanding Quadratic Functions | Exploring the Equation y = x² and its Properties

y=x²

The equation y = x² represents a quadratic function

The equation y = x² represents a quadratic function. In this equation, x represents the input variable and y represents the output variable. The function describes a parabola, a U-shaped curve, which opens upward.

To understand the behavior of the quadratic function, we can analyze its properties. First, let’s discuss the shape of the graph. Since the coefficient of x² is positive (1), the parabola opens upward. The vertex of the parabola is at the origin (0,0) since there is no constant term.

The axis of symmetry is a vertical line passing through the vertex. In this case, the axis of symmetry is the y-axis, represented by the equation x = 0.

We can find the x-intercepts (also called the zeros or roots) of the quadratic function by setting y to zero and solving for x. So, setting y = x² to zero, we get x² = 0, which means x = 0. Thus, the parabola intersects the x-axis only at the origin (0,0).

The y-intercept is the point where the parabola intersects the y-axis. To find it, we substitute x = 0 into the equation y = x². This gives us y = 0², which means y = 0. Hence, the y-intercept is also at the origin (0,0).

Additionally, the quadratic function y = x² is symmetric about the y-axis because if you substitute -x for x in the equation, you get the same equation. This symmetry implies that for every point (x, y) on the graph, the point (-x, y) is also on the graph.

As you move further away from the vertex, the values of y increase as x becomes more positive or negative. The rate of increase of y depends on the value of x. For example, if x increases by 1, y increases by 1^2 = 1. If x increases by 2, y increases by 2^2 = 4.

Overall, the equation y = x² represents a simple and fundamental quadratic function that helps understand the behavior of parabolas and quadratic relationships.

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