Algebra

Definition. A quadratic function is a function of the form $f(x)=ax^2+bx+c$.

Discuss: What does the graph of a quadratic function look like? Major features and how to find them. (Opens up/down, vertex, “width,” intercepts)

Geometry

Definition. A parabola is the set of points equidistant from a given point $P$ (the focus) and a given line $L$ (the directrix) not containing the point $P$.

Goal: Let’s connect these two ideas.

We know the meaning of distance between two points. what is the meaning of “distance between a point and a line?”

Distance to a line

Definition. The distance from a point $P$ to a line $L$ is the length of the line segment $PQ$, where $Q$ is a point on $L$ and $PQ$ is perpendicular to $L$.

Group Activity 1: A parabola with focus $(2,5)$ and directrix $y=-3$. Suppose $(x,y)$ is any point on the plane:

  • Find a formula for $d_1$, the distance from $(x,y)$ to the point $(2,5)$.
  • Now find a formula for $d_2$, the distance from $(x,y)$ to the line $y=-3$.
  • Now, suppose we only want to consider points $(x,y)$ for which these two distances are equal. Write an equation expressing this idea. Simplify and solve for $y$.
  • Find the coordinates of the vertex of the parabola.
  • Find the x-intercepts of the parabola.

Group Activity 2: Consider the parabola $x^2-6x-4y+45=0$.

  • Find the coordinates of the vertex.
  • This parabola has focus $(3,10)$. What is the equation of the directrix?
  • Verify your work by setting equal the formulas for:
    • the distance from a point $(x,y)$ to the focus $(3,10)$, and
    • the distance from $(x,y)$ to the directrix found above.
  • Simplify. Do you get the same equation for a parabola?