5. Transformations of Quadratic Functions
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Consider vertical translations, horizontal translations, and vertical stretches and shrinks.
Supposing that a, h, and k are positive numbers, g(x)=a(x-h)^2+k is a translation h units right, vertical stretch or shrink with factor a, followed by a translation k units up of the parent function f(x)=x^2. See solution for negative values of a, h, or k.
| Transformations of f(x)=x^2 | |
|---|---|
| Vertical Translations | Translation up k units, k>0 y=x^2+ k |
| Translation down k units, k>0 y=x^2- k | |
| Horizontal Translations | Translation right h units, h>0 y=(x- h)^2 |
| Translation left h units, h>0 y=(x+ h)^2 | |
| Vertical Stretch or Shrink | Vertical stretch, a>1 y= ax^2 |
| Vertical shrink, 0< a< 1 y= ax^2 | |
| Reflections | In the x-axis y=- x^2 |
| In the y-axis y=(- x)^2 | |
Let's suppose that a, h, and k are positive numbers. We conclude from the table that g(x)=a(x-h)^2+k is a translation h units right, vertical stretch or shrink with factor a, followed by a translation k units up of the parent function f(x)=x^2. What happens if a, h, or k are negative?