To sketch the graphs of the given equations $y = x^2$, $y = 2x^2$, and $y = \frac{1}{2}x^2$ on the same axes, follow these steps:

Steps to sketch:

1. Start with the basic parabola: $y = x^2$

   • This is a standard parabola with the vertex at the origin (0, 0).
   • The graph opens upwards and is symmetric about the y-axis.
   • As $x$ moves away from 0, $y = x^2$ grows quadratically.

2. Next, sketch $y = 2x^2$

   • This is also a parabola opening upwards, but it's "narrower" than $y = x^2$ because the coefficient 2 makes the graph stretch more steeply.
   • It has the same vertex at (0, 0), but for any given $x$, $y = 2x^2$ will be twice as large as $y = x^2$.

3. Finally, sketch $y = \frac{1}{2}x^2$

   • This is also a parabola opening upwards, but it is "wider" than $y = x^2$ because the coefficient $\frac{1}{2}$ makes the graph grow slower.
   • The vertex is at (0, 0), and for any given $x$, $y = \frac{1}{2}x^2$ will be half as large as $y = x^2$.

Key differences between the graphs:

• ( y = 2x^2 \