Simple Quadratic function exercise

Question

I am a math teacher and I want to have some other opinions regarding an exercise made from one of my colleagues because I think that she was wrong when correcting the solutions from her students.

The exercise is the following - it's rather easy:

Which two of the following answers are correct when considering the following function

f(x)=a*x^2+b with a > 0 and b < 0:

(A) f intersects with the y-axis at the point P(0|b).
(B) f has two roots.
(C) The bigger b is the steeper is the graph of f.
(D) The smaller a is the more flat is the graph of f.
(E) f has a maximum.

Answer 1

(A) and (B) are true. For a given value $x$ the slope is smaller there in absolute value as $a$ becomes smaller (but remains positive), so (D) is also true in that sense. (C) and (E) are false.

I suppose one can argue about which of the statements are "most true". The interpretation of (D) is the most subjective.

Answer 2

• What does f(x)=0 tell us? Indeed a>0, b<0 makes x to be x_0=+\sqrt{b/a} and x_1=-\sqrt{b/a} so the function intersect $x$ axis at two reals.

• If x=0 then f(0)=b so the intersection point with $y$ axis is (0,b).

• a>0 so the curve is upward, so the last option is wrong.