Question

Sketch each parabola, labeling its focus and directrix. y=(1/10)(x-1)²-2

Answer 1

The equation of parabola is $y=\frac{1}{10}\left(x-1\right)^{2}-2$

Comparing with the vertex form of the parabola: $y=a(x-h)^2+k$

$a=\frac{1}{10},h=1,k=-2$

Hence, the vertex of the parabola is given by (h,k) = (1, -2)

Now, vertex is the midpoint of the focus and the point on the directrix.

Distance, between vertex and focus is p and that of point on the directrix is p.

Now, let us find p

$p=\frac{1}{4a}\\\\p=\frac{1}{4\cdot1/10}\\\\p=\frac{5}{2}$

Thus, the focus is given by

$(h,k+p)\\\\=(1,-2+5/2)\\\\=(1,1/2)=(1,0.5)$

And the directrix is given by

$y=k-p\\\\y=-2-5/2\\\\y=-\frac{9}{2}$

Since, a >0 hence, it is an upward parabola.

The graph is shown in the attached file.