Transformers: More Than Meets the y’s

Learning Focus

Describe vertical stretches, compressions, and reflections of quadratics.

Big idea

$y=a(x-h)^2+k$

• $|a|>1$: vertical stretch (narrower).
• $0<|a|<1$: vertical compression (wider).
• $a<0$: reflection across the horizontal line through the vertex (opens down).

The vertex $(h,k)$ is unchanged by $a$.

Worked example

$y=-2(x-1)^2+3$: opens down, stretched by factor 2, vertex $(1,3)$ is the maximum. At $x=2$: $y=-2(1)^2+3=1$.

Takeaways

• Sign of $a$ gives direction; size of $|a|$ gives width.
• $a$ multiplies every $y$-distance from the vertex line.
• Combine with shifts to graph any $a(x-h)^2+k$.