What does the product (c/a) of a quadratic function tell you about the graph or the path of the parabola?

Expanding a quadratic aleady solved in the form

a(x – p)(x – q) = ax^2 – a(p + q)x + apq – 0, and then

comparing with .. ax^2 ..+... bx .....+ c, we have

Sum of roots p + q = -b/a, and

Product of roots pq = c/a

That is probably what the question meant. Compare these two quadratics.

y = 3x^2 – 3x – 18, product of roots pq = (-2)*(3) = c/a = -18/3 = -6

y = 2x^2 + 4x – 70, product of roots pq = (-7)*(5) = c/a = -70/2 = -35

See these graphs for comparison.

https://www.wolframalpha.com/input/?i=y+%3D+3x%5E2...

The larger value for c/a with the second quadratic tells us that the path of that parabola is wider, passing through more separated roots

A product is the result of multiplication. c/a is not a product; it is a quotient, the result of division.

A quadratic can be written in the form y = ax^2 + bx + c where a,b,c are constants.

a tells you whether the quadratic curve opens up or down. When a is positive, the curve has a minimum and opens up. When a is negative, the curve has a maximum and opens down.