Understanding Vertical Stretches in Parabolas

When we study quadratic equations, we find a special type of graph called a parabola. Parabolas look like a U. One interesting change we can make to these U-shaped graphs is called a vertical stretch. This change makes the parabolas look different and more pronounced.

What is a Vertical Stretch?

1. Definition: A vertical stretch happens when we multiply the quadratic equation by a number greater than 1. For example, if we start with the basic parabola (y = x^2) and we stretch it by a factor of 3, it becomes (y = 3x^2).

2. Effects on the Shape:

   • Narrowing: The U shape becomes narrower. So, (y = 3x^2) is steeper than the original (y = x^2).
   • Increased Height: The points on the graph go higher more quickly away from the vertex (the bottom point of the U). For example, at (x = 1), in the stretched version (y = 3(1^2) = 3), while in the original, (y) is just 1.

Imagining the Change

You can think of a vertical stretch like pulling on a rubber band from the bottom. The more you pull, the sharper the U shape looks.

Summary

Vertical stretches make parabolas skinnier and raise their points higher. This shows how changes can greatly affect how parabolas look and behave.