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)=x2. See solution for negative values of a, h, or k.
We want to determine how the constants a, h, and k affect the graph of the quadratic function g(x)=a(x−h)2+k. To do so, let's look at the possible transformations. Then we can more clearly identify the ones being applied to the parent function f(x)=x2.
| Transformations of f(x)=x2 | |
|---|---|
| Vertical Translations | Translation up k units, k>0y=x2+k
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| Translation down k units, k>0y=x2−k
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| Horizontal Translations | Translation right h units, h>0y=(x−h)2
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| Translation left h units, h>0y=(x+h)2
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| Vertical Stretch or Shrink | Vertical stretch, a>1y=ax2
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| Vertical shrink, 0<a<1y=ax2
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| Reflections | In the x-axisy=-x2
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| In the y-axisy=(-x)2
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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)=x2. What happens if a, h, or k are negative?