The parent function of a parabola is y = x²

Vertical translation is shown when there is a term added to x². So, if y = x² + 5, the parent function would be moved 5 units up.

If y = x² – 7, the parent function would be moved down 7 units.

Horizontal translations are shown when y = (x – p)², p being the translation. If the parent function moves 5 to the right, we replace p with 5, y = (x – (5))²

If p = -4, we replace p and the graph moves 4 to the left, y = (x – (-4))² or y = (x +4)²

Now, if we add this together, we can understand how a parent function changes.

y = (x +4)² – 5

This means that the horizontal translation is 4 units to the left, then 5 units down.

The scale can affect whether the parabola is stretched or compressed. If there are no terms before the brackets, the scale is as original, 1, 3, 5, etc. If there is a number higher than 0 in front of the brackets, y = 2(x +4)² – 5, the parabola becomes stretched.

If the number is less than 1 and greater than 0, y = .2(x +4)² – 5, the parabola is compressed.

Lastly, if this term before the brackets is negative, the parabola becomes down facing, y = -(x +4)² – 5

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