The parabola \( y=x^{2} \) is scaled vertically by a factor of \( \frac{2}{3} \). What is the equation of the new parabola? \( y=\square \)

When the parabola \( y = x^2 \) is scaled vertically by a factor of \( \frac{2}{3} \), you simply multiply the output of the function by \( \frac{2}{3} \). Therefore, the new equation becomes: \[ y = \frac{2}{3} x^2 \] This new equation represents a parabola that is narrower or "squished" compared to the original \( y = x^2 \). Scaling a parabola vertically affects how quickly it opens up or down, allowing for fascinating changes in its graph. If you think about real-life applications, this can relate to optimizing the shape of a bridge or a projectile's trajectory, where the curve's steepness affects structural integrity and performance.