Quadratics - the Australian Mathematical Sciences Institute

A guide for teachers – Years 11 and 12 • {21}
The three cases are shown in the diagrams below.
y
0
x
y
0
x
y
0
x
There is also a special case when the line is parallel to the y-axis. In this case the line will
cut the parabola once. Having noted this case, we will not need to consider it further,
since the intersection point is easily found.
Suppose the line has the form y = mx +d and the parabola is y = ax 2 +bx +c. Equating,
we have
ax 2 + bx + c = mx + d =⇒ ax 2 + (b − m)x + (c − d) = 0.
This gives a quadratic equation. Hence we can conclude:
• A line can meet a parabola in at most two points.
• If the discriminant of the resulting quadratic is negative, the line does not meet the
parabola at all.
• If the discriminant of the resulting quadratic is positive, the line meets the parabola
in two distinct points.
• If the discriminant of the resulting quadratic is zero, then the line is tangent to the
parabola.