Answer
(a) See the graph.
$y=\frac{1}{2}x^2~~$ (Red)
$y=x^2~~~~~$ (Green)
$y=2x^2~~~$ (Blue)
$y=4x^2~~~$ (Black)
(b)
$y=\frac{1}{2}x^2~~$
Focus: $F(0,\frac{1}{8})$
$y=x^2~~~~~$
Focus: $F(0,\frac{1}{4})$
$y=2x^2~~~$
Focus: $F(0,\frac{1}{2})$
$y=4x^2~~~$
Focus: $F(0,1)$
(c) The focus is at $F(0,\frac{k}{4})$. So, as $k$ increases the foucus moves upward along the y-axis.
Work Step by Step
(b) Parabola with vertical axis and vertex at origin: $x^2=4py$
$y=kx^2$
$4p=k$
$p=\frac{k}{4}$
Focus: $F(0,p)=F(0,\frac{k}{4})$
$y=\frac{1}{2}x^2~~$
Focus: $F(0,\frac{\frac{1}{2}}{4})=F(0,\frac{1}{8})$
$y=x^2~~~~~$
Focus: $F(0,\frac{1}{4})$
$y=2x^2~~~$
Focus: $F(0,\frac{2}{4})=F(0,\frac{1}{2})$
$y=4x^2~~~$
Focus: $F(0,\frac{4}{4})=F(0,1)$
