In this lesson, we will determine the different types of **Transformations** that can be applied to the parent function \(f(x) = x^2\) by expressing it in the **Vertex Form** \(f(x) = a(x-h)^2 + k\).
## Reflection, Stretching and Compression

The \(a\) value will both determine whether the quadratic function will get either stretched or compressed and which direction it will open:

## Horizontal Shifts

The \(h\) value will determine which direction horizontally the graph will get shifted. This change can be represented algebraically by the equation \(f(x) = (x-h)^2\):

## Vertical Shifts

The \(k\) value will determine which direction vertically the graph will get shifted. This change can be represented algebraically by the equation \(f(x) = x^2 + k\):

- If \(|a| > 1\), the graph will get stretched by a factor of \(a\). This will make the graph appear narrower, as the x value will have a greater effect on the y value
- If \(|a| < 1\), the graph will get compressed by a factor of \(a\). This will make the graph appear wider, as the x value won't have as greate of an effect on the y value
- If \(a > 0\), the function will be positive, making the parabola open upward
- If \(a < 0\), the function will be negative, reflecting the graph and making the parabola open downward

State the transformation(s) in the quadratic equation \(f(x) = -6x^2\)

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- If \(h > 0\), the graph will get shifted to the right by \(h\) units
- If \(h < 0\), the graph will get shifted to the left by \(h\) units

Identify the quadratic equation for a graph that is shifted \(5\) units left

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- If \(k > 0\), the graph will get shifted upwards \(k\) units
- If \(k < 0\), the graph will get shifted downwards \(k\) units

Identify the quadratic equation for a graph that is shifted \(7\) units upward

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Identify and graph a qudratic equation that is reflected, stretched by a factor of \(4\), shifted \(3\) units right and shifted \(2\) units downward

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