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A parabola is a symmetrical, U-shaped curve that represents the graph of a quadratic function. The general form of a quadratic function is given by the equation \( y = ax^2 + bx + c \). In this exercise, all the quadratic functions have no linear term or constant term, so they simplify to \( y = ax^2 \). The vertex of these parabolas is at the origin (0,0). The direction in which the parabola opens depends on the coefficient \( a \). If \( a \) is positive, the parabola opens upwards, forming a minimum point at the vertex. If \( a \) is negative, it opens downwards, forming a maximum point at the vertex. Parabolas are useful in many fields such as physics, engineering, and finance because they describe naturally occurring phenomena like projectile motion and optimization problems.

Reflection across the x-axis involves flipping a graph over the x-axis. When we reflect a parabola across the x-axis, its orientation changes from up to down or vice versa. For instance, in our problem:

• The function \( y_1 = x^2 \) gives an upward-opening parabola.
• The function \( y_2 = -x^2 \) which reflects \( y_1 \) across the x-axis, results in a downward-opening parabola.

The reflection shows that multiplying the quadratic term by -1 changes the direction in which the graph opens but does not alter the shape or the width of the parabola. This is a crucial concept for understanding the symmetry in quadratic functions.

The coefficient \( a \) in the quadratic function \( y = ax^2 \) controls the width and the steepness of the parabola. Its absolute value determines whether the parabola will be steeper (narrower) or shallower (wider).

• For \( y_3 = 2x^2 \), the positive coefficient 2 makes the parabola narrower than the basic parabola \( y_1 = x^2 \).
• Conversely, for \( y_5 = \frac{1}{2}x^2 \), the coefficient \( \frac{1}{2} \) makes the parabola wider.
• A negative coefficient, as seen in \( y_4 = -2x^2 \) and \( y_6 = -\frac{1}{2}x^2 \), still affects the width in the same way but inverts the direction of the parabola.

Graphs with larger coefficients open more narrowly because the vertical stretching effect is more pronounced. Meanwhile, graphs with smaller coefficients open more widely due to vertical compression. This is essential in understanding how different quadratic equations can represent various ranges of values.

When comparing the graphs of different quadratic functions, we consider both their shape and orientation. Looking at our functions:

• \( y_1 = x^2 \) is the standard upward-opening parabola, and its negative counterpart, \( y_2 = -x^2 \), opens downward.
• \( y_3 = 2x^2 \) and \( y_4 = -2x^2 \) are both steeper than \( y_1 = x^2 \) and \( y_2 = -x^2 \), respectively.
• \( y_5 = \frac{1}{2}x^2 \) and \( y_6 = -\frac{1}{2}x^2 \) are broader than \( y_1 = x^2 \) and \( y_2 = -x^2 \), respectively.

By analyzing these comparisons, one can observe how quadratics with positive coefficients form parabolas that open upward, and those with negative coefficients open downward. Furthermore, larger coefficients lead to narrower graphs, whereas smaller coefficients result in wider graphs. Understanding these differences helps in graphing and solving quadratic equations effectively.