Question

On a coordinate plane, a hyperbola and parabola are shown. The hyperbola has a curve in quadrants 1 and 2. The parabola opens down and has a vertex at (0, 0).
On a coordinate plane, a hyperbola and a cubic functions are shown. They hyperbola has curves in quadrants 1 and 3, and the cubic function goes through (0, 0).
On a coordinate plane, a hyperbola is shown. One curve is in quadrant 1 and the other curves goes through quadrants 2 and 4.
On a coordinate plane, a hyperbola is shown. Both curves open downward in quadrants 3 and 4.

Answer 1

I’m missing what you want done with these descriptions — do you want equations that match each graph, identification of which family each curve belongs to, sketches, or something else?

If you want example equations that fit the described shapes, here are simple choices and the reasoning for each:

1) "Hyperbola has a curve in quadrants 1 and 2. Parabola opens down and has vertex at (0,0)."
- Parabola: y = −x^2 (opens down, vertex at (0,0)).
- Hyperbola with branches above and below the x-axis (top branch lies in Q1 and Q2): y^2/a^2 − x^2/b^2 = 1. Example: y^2 − x^2 = 1 (top branch is in Q1 and Q2, bottom in Q3 and Q4).

2) "Hyperbola has curves in quadrants 1 and 3, and the cubic goes through (0,0)."
- Hyperbola with opposite branches in Q1 and Q3: xy = c with c > 0. Example: y = 1/x.
- Cubic through the origin: y = x^3 (or y = ax^3).

3) "Hyperbola shown. One curve is in quadrant 1 and the other curve goes through quadrants 2 and 4."
- This indicates a hyperbola whose center is shifted off the origin so the two branches lie in different combinations of the coordinate quadrants. A generic shifted rectangular hyperbola is y = a/(x − h) + k; by choosing (h,k) you can place one branch mostly in Q1 while the other occupies Q2 and Q4 relative to the axes. Example (one possible choice): y = 1/(x − 1) + 0.5 — the branches are displaced so one branch may appear in Q1 while the other spans left-side quadrants.

4) "Hyperbola shown. Both curves open downward in quadrants 3 and 4."
- That describes a vertical-transverse hyperbola whose center is below the x-axis so both branches lie below the x-axis. General form: (y − k)^2/a^2 − (x − h)^2/b^2 = 1 with k negative and a large enough so both branches fall below y = 0. Example: (y + 2)^2/4 − x^2 = 1 (both branches lie below the x‑axis, occupying Q3 and Q4 regions).

If you want me to match these descriptions to a specific list of candidate equations, or to sketch each graph, or to find asymptotes/intersections, tell me which and I’ll do that.