Question

Which of the following could be the graph of an equation of the form
y
=
−
x
3
+
a
x
2
+
b
x
+
c
for some real numbers a, b, and c?

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Answer 1

I can’t see the pictures you attached. Give me the images again or describe each graph briefly (end behavior and number/location of turning points and x‑intercepts). Meanwhile, here’s how to tell which graphs could come from y = −x^3 + ax^2 + bx + c:

Key facts for y = −x^3 + ax^2 + bx + c
- End behavior: as x → −∞, y → +∞ (left end up); as x → +∞, y → −∞ (right end down). Any candidate graph must have left up, right down.
- Shape: a cubic has at most two turning points; for a cubic with negative leading coefficient the left turning point (if present) is a local maximum and the right turning point (if present) is a local minimum. So left-to-right you should see a peak then a trough (or just a monotone decrease with an inflection if there are no real turning points).
- x-intercepts: a real cubic has 1 or 3 real roots (counting multiplicity). It may cross the x-axis 1 time or 3 times; it can also touch the axis at a double root (tangent) and cross elsewhere.
- No horizontal asymptotes (polynomial).

Use these tests:
1. If the right end goes up (instead of down) the graph cannot be this cubic.
2. If the left end goes down (instead of up) the graph cannot be this cubic.
3. If the left-to-right turning points are a minimum then a maximum (i.e., trough then peak) it cannot be this cubic (order must be max then min).
4. If it has more than two turning points, it cannot be this cubic.

If you reupload or describe each option I’ll pick which one(s) match.