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Quadratic functions are a fundamental concept in algebra that take the form \( y = ax^2 + bx + c \). In simpler terms, they are polynomials of degree two, meaning the highest power of \( x \) is two. A characteristic feature of quadratic functions is the presence of a parabola, which is the curve produced when these functions are graphed.
Quadratic functions can open upwards or downwards depending on the coefficient \( a \):

• If \( a > 0 \), the parabola opens upwards, resembling a U shape.
• If \( a < 0 \), it opens downwards, resembling an upside-down U.

When \( a = 1 \) and \( b = 0 \), as in the case of the function \( y = x^2 \), the vertex, or the lowest point when opening upwards, is at the origin, \( (0,0) \). This function is symmetric about the y-axis, meaning it mirrors itself on the left and right.
Understanding quadratic functions is crucial as they model numerous real-world situations such as projectile motion, and they serve as the foundation for more complex mathematical concepts.

The graph of a quadratic function is known as a parabola. The standard form for graphing a basic quadratic equation, like \( y = x^2 \), is straightforward due to its symmetry and consistent shape.
When sketching a parabola, it's helpful to identify a few key features:

• Vertex: This is the point where the parabola changes direction. For \( y = x^2 \), the vertex is at the origin \( (0,0) \).
• Axis of Symmetry: The line that splits the parabola into mirrored halves. This is vertical for \( y = x^2 \) and goes through the vertex, shown as \( x = 0 \).
• Intercepts: Points where the graph crosses the axes. Here, the y-intercept is \( (0,0) \), and it does not cross the x-axis because the range is positive.
• Direction: The parabola opens upwards since \( a = 1 \), indicating a U shape.

To graph \( y = x^2 \) on the interval \([-10, 10]\), calculate the y-values for several key x-values within this range to create an accurate parabola representation. The symmetry ensures a smooth curve connecting these points, emphasizing the consistent nature of the parabola's shape.

The range of a function describes the set of all possible y-values that can be output by the function. For a quadratic function like \( y = x^2 \), understanding the range is important to grasp how the parabola behaves.
When graphing \( y = x^2 \) over the interval \([-10, 10]\), we compute some specific y-values:

• At \( x = 0 \), \( y \) is \( 0 \).
• This is the minimum y-value, as the vertex is the bottom point of the parabola.
• As \( x \) moves to \(-10 \) or \( 10 \), \( y \) reaches its peak value of \( 100 \).

Hence, for the given viewing window, the range of the function is \([0, 100]\).
This outcome stems from the fact that squaring any real number results in a non-negative number, ensuring that the vertex \( y = 0 \) is the minimum point, while y-values increase symmetrically as x moves away from zero, peaking at the window's boundaries.