Question

Find the vertex, axis, focus, directrix, latus rectum of the parabola $y = x^2 - 2x + 3$.

Answer 1

• Vertex: $(1, 2)$
• Axis: $x = 1$
• Focus: $(1, 9/4)$
• Directrix: $y = 7/4$
• Latus Rectum Length: $1$

Answer 2

The equation $y = x^2 - 2x + 3$ is rewritten by completing the square as $(x-1)^2 = y-2$. This matches the standard form $(x-h)^2 = 4a(y-k)$ for a parabola opening upwards. By comparing, we identify the vertex $(h,k)=(1,2)$ and $4a=1$, so $a=1/4$. The axis of symmetry is $x=h$, which is $x=1$. The focus is at $(h, k+a)$, yielding $(1, 9/4)$. The directrix is $y=k-a$, which is $y=7/4$. The length of the latus rectum is $|4a|$, which is 1.