Question

What is the minimum number of intersections between two quadratic functions graphed in the same coordinate plane?

Answer 1

To determine the minimum number of intersections between two quadratic functions graphed in the same coordinate plane, we first recall the general form of a quadratic function, which can be expressed as:

\[
f(x) = ax^2 + bx + c
\]

and

\[
g(x) = mx^2 + nx + p
\]

where \(a, b, c, m, n,\) and \(p\) are coefficients.

To find the intersections between these two functions, we set them equal to each other:

\[
ax^2 + bx + c = mx^2 + nx + p
\]

Rearranging this gives:

\[
(ax^2 - mx^2) + (bx - nx) + (c - p) = 0
\]

which simplifies to:

\[
(a - m)x^2 + (b - n)x + (c - p) = 0
\]

This is a standard quadratic equation of the form:

\[
Ax^2 + Bx + C = 0
\]

where:
- \(A = a - m\)
- \(B = b - n\)
- \(C = c - p\)

The number of intersections (real solutions) of the two quadratic functions corresponds to the number of real solutions of this quadratic equation, which is determined by the discriminant:

\[
D = B^2 - 4AC
\]

The discriminant \(D\) tells us about the nature of the roots:
- If \(D > 0\), there are 2 distinct real solutions, meaning the functions intersect at 2 points.
- If \(D = 0\), there is 1 real solution (a repeated root), meaning the functions intersect at exactly 1 point (they touch).
- If \(D < 0\), there are no real solutions, meaning the functions do not intersect at any point.

To find the minimum number of intersections, we observe that the minimum value occurs when the discriminant is negative, which corresponds to no real solutions. Therefore, the two quadratic functions can graphically not intersect at all.

Thus, the minimum number of intersections between two quadratic functions graphed in the same coordinate plane is:

\[
\boxed{0}
\]