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Quadratic functions are a cornerstone of algebra, commonly represented by the equation \(y = ax^2 + bx + c\). Here, \(a\), \(b\), and \(c\) are constants, and the graph of a quadratic function is a parabola. The simplest quadratic function is \(y = x^2\), which opens upwards with its vertex at the origin (0,0).

Key characteristics of quadratic functions include:

• Symmetry: Quadratic graphs are symmetric about a vertical line called the axis of symmetry.
• Vertex: This is the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards.
• Shape: The graph is "U" shaped for a positive \(a\) and "abla" shaped for a negative \(a\).

Understanding these features helps in graphing quadratic functions and applying transformations effectively.

Translations are a type of transformation that shift a graph horizontally and/or vertically without altering its shape. For quadratic functions like \(y = x^2\), translations can move the graph along the \(x\)-axis or \(y\)-axis.

To translate the graph vertically, simply add or subtract a constant \(k\). For example, \(y = x^2 + 3\) translates the parabola up by 3 units. To translate the graph horizontally, replace \(x\) with \(x - h\), where \(h\) is the number of units to move right (or left if \(h\) is negative). Thus, \(y = (x - 2)^2\) shifts the graph 2 units to the right.

Combining these transformations lets us shift the graph in both directions simultaneously, such that \(y = (x - 2)^2 + 3\) translates the graph upwards by 3 units and rightwards by 2 units.

Reflections in graph transformations invert the graph across a specified axis, changing the direction in which it opens. For a quadratic function like \(y = x^2\), reflecting across the \(x\)-axis involves multiplying the entire function by \(-1\).

This results in \(y = -x^2\), which flips the parabola downwards, creating a maximum point instead of a minimum point. This transformation does not affect the position of the vertex or the shape of the graph, only its direction.

Combining a reflection with other transformations, such as a translation, allows further modification of the graph. For instance, \(y = -x^2 + 4\) not only reflects the graph but also moves it up by 4 units, shifting the vertex to a new position.

Stretches adjust the size of a graph by compressing or expanding it vertically or horizontally. For quadratic functions, a vertical stretch affects the "steepness" of the parabola.

To perform a vertical stretch on \(y = x^2\), multiply the function by a constant factor. For example, \(y = 3x^2\) stretches the graph vertically by a factor of 3, making the parabola narrower. This does not change the parabola's vertex but alters its sharpness.

By combining stretches with other transformations like translations, you can create a variety of effects. The equation \(y = 3(x - 1)^2\) shows a vertical stretch by a factor of 3, alongside a translation of the graph 1 unit to the right, demonstrating how different transformations can work together.