A parabola has equation (y − 2)² = −8(x + 1). In which direction does it open?
AUpward, because 4p = −8 is negative
BDownward, because the y-term is squared
CLeft, because y is squared and p < 0
DRight, because y is squared
When y is squared, the axis of symmetry is horizontal and the equation has the form (y−k)² = 4p(x−h). The parabola opens right when p > 0 and left when p < 0. Here 4p = −8, so p = −2 < 0, giving a leftward-opening parabola. Options A and B are the typical errors: assuming all parabolas open up or down and reading the sign incorrectly.
Question 2 Multiple Choice
A point P lies on a parabola with focus F(0, 3) and directrix y = −3. The distance from P to the focus is 7. What is the distance from P to the directrix?
A3 — the distance from the vertex to the directrix
B4 — the difference between the focus distance and the focal length
C7 — equal to the distance to the focus by definition
D10 — the sum of the focus distance and the focal length
The geometric definition of a parabola states that every point on it is equidistant from the focus and the directrix. If |PF| = 7, then the distance from P to the directrix must also equal 7. This is not a coincidence — it is the defining property. Students who confuse the focus with the directrix distance, or use the focal length (p = 3) instead, arrive at wrong answers.
Question 3 True / False
A horizontal-axis parabola of the form (y − k)² = 4p(x − h) is not a function because it fails the vertical line test.
TTrue
FFalse
Answer: True
A horizontal parabola opens left or right, so for a single x-value there are typically two y-values (one above and one below the axis of symmetry). This fails the vertical line test — the definition of a function. This is why algebra courses introduce only vertical parabolas: they are functions. As conic sections, both orientations are equally valid geometric objects.
Question 4 True / False
The vertex of a parabola and its focus are the same point.
TTrue
FFalse
Answer: False
The vertex is the turning point of the parabola; the focus is a distinct point located p units from the vertex along the axis of symmetry, inside the curve. They coincide only in the degenerate case p = 0, which is not a real parabola. Confusing vertex and focus is one of the most common errors in this topic — the vertex sits on the parabola, while the focus is interior to it.
Question 5 Short Answer
Satellite dishes and headlight reflectors use a paraboloid shape to concentrate signals or light. What property of the parabola — derived from its geometric definition — makes this work?
Think about your answer, then reveal below.
Model answer: Any ray traveling parallel to the axis of symmetry reflects off the parabola and passes through the focus. This reflective property follows from the equidistance definition: the tangent at any point bisects the angle between the line to the focus and the perpendicular to the directrix, causing parallel rays to converge at the focus.
This is not an incidental engineering trick — it is a geometric theorem. A satellite dish collects parallel rays from a distant source and focuses them at the receiver at the focus point. A car headlight reverses the process: a bulb at the focus emits rays that reflect off the paraboloid surface as parallel beams. Understanding that the reflective property is a consequence of the geometric definition (not a separate fact) shows mastery of what the definition is actually doing.