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A quadratic function typically follows the form of \( y = ax^2 + bx + c \). However, we can apply transformations to shift, stretch, or reflect the graph. These transformations are guided by the general form \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola and \( a \) indicates the direction and the vertical stretch or compression of the graph.

For example, adding a positive \( h \) value shifts the parabola to the right, while a negative \( h \) shifts it to the left. Similarly, adding \( k \) lifts the parabola up for positive values and down for negative values. Applying a negative \( a \) reflects the graph over the x-axis, while its absolute value controls the steepness—less than 1 for a shrink, greater than 1 for a stretch.

The vertex of a parabola is the highest or lowest point on the graph, depending on the direction of its opening. It serves as a pivotal reference for the parabola's shape and is found at coordinate \( (h, k) \) in the vertex form of a quadratic function, \( y = a(x - h)^2 + k \). The vertex is crucial when graphing because all other points on the parabola are symmetrically arranged around it.

For instance, in \( f(x) = -\frac{1}{2} (x - 2)^2 + 1 \), the vertex is at \( (2, 1) \) showing us not only the peak of the graph but also helping us understand the shifts applied to the parent function \( y = x^2 \) to achieve this particular curve.

Reflection of a function occurs when the graph of the function is mirrored over a specific line, such as the x-axis or y-axis. This is often the result of a negative coefficient in front of the leading term in a quadratic function.

For example, \( f(x) = -\frac{1}{2} (x - 2)^2+1 \) is a reflection of the function \( y = x^2 \) over the x-axis because of the negative sign before \( \frac{1}{2} \) . This negative causes the parabola that normally opens upwards to open downwards instead. Reflections are fundamental when analyzing symmetries and predicting the behavior of a graph.

In the context of quadratic functions, a vertical stretch makes the parabola steeper while a vertical shrink makes it wider. This effect is controlled by the absolute value of the coefficient \( a \) in the quadratic function \( y = a(x - h)^2 + k \). A vertical stretch corresponds to \( |a| > 1 \) and a vertical shrink corresponds to \( 0 < |a| < 1 \).

For example, \( k(x) = [2(x + 1)]^2 + 4 \) exhibits a vertical stretch because the absolute value of the coefficient \( a \) is greater than 1. This dramatically changes the graph's appearance compared to the parent function, causing it to become narrower and highlighting the importance of \( a \) in graph analysis.