Putting together our work from yesterday, can we answer the following?

Question 1. What is the equation of a parabola with focus $(p,q)$ and directrix $y=l$?

Example: What is the equation of the parabola with focus $(-2,1)$ and directrix $y=-5$?

Question 2. How do we find the focus and directrix of the parabola $y=ax^2+bx+c$?

Formula for parabola

Any parabola with horizontal directrix can be written in the form:

$$4p(y-k)=(x-h)^2$$

where $(h,k)$ is the vertex and $p$ is the distance from the vertex to the focus (also the distance from the vertex to the directrix).

Example: What is the focus and directrix of the parabola $12y=x^2-8x+24$?

Why quadratics?

[T]he study of quadratic equations is only a small part of the study of quadratic functions; namely, the former is about how to locate the zeros of quadratic functions. Moreover, TSM makes this topic more difficult than it needs to be, partly by presenting the technique of completing the square as a rote skill for one purpose only: getting the quadratic formula. Consequently, the quadratic formula ends up also being a rote skill and, likewise, the formula for the vertex of the graph of a quadratic function. In Section 2.1, however, we show that completing the square is the major idea that 

(a) leads to the proof of the quadratic formula and the formula of the vertex of the graph (see page 75),

(b) proves that the graph of f(x) = ax2+bx+c is congruent to the graph of fa(x) = ax2, 

(c) exhibits the commonality between the study of linear and quadratic functions, namely, the fact that both revolve around the shape of the graphs of the representative functions ax and ax2 (see page 73)

Wu, A&G, Preface pxiii

After such a detailed study of linear and quadratic functions, one would expect another long chapter on cubic functions (polynomial functions of degree 3) and yet another on quartic functions (polynomial functions of degree 4), etc. The fact that this does not happen is a consequence of the fact that the theory loses its simplicity when the degree of the polynomial exceeds 2. Recall that the study of quadratic functions was greatly facilitated by the method of completing the square. There is no tool of comparable power and simplicity for polynomial functions of degree exceeding 2. Analogs of the quadratic formula for quadratic equations continue to hold for cubic and quartic equations,  but they are unwieldy and therefore not particularly useful. It is a famous theorem of Abel and Galois that for equations of degrees exceeding 4, no analog of the quadratic formula exists. Consequently, we know far less about arbitrary polynomial functions. All we can do is to give a general discussion of the most basic properties of polynomial functions and rational functions.

Wu, A&G, p121

Completing the square

If we give ourselves the power to add a constant to our expression, then we can always turn the sum of an $x^2$ term and an $x$ term, $ax^2+bx$, into a perfect square.

Demonstration

Multiplication as area.

  1. $3\cdot5$  (rectangle)
  2. $a\cdot b$ (rectangle)
  3. $x^2$ (square)
  4. $(x+5)^2$ (square)
  5. $(x+k)^2$ (square – equivalent to sum of 4 rectangles)

Rewrite each expression using only a single instance of the variable $x$:

  1. $x^2+6x$
  2. $x^2-12x$
  3. $x^2+5x$
  4. $x^2+8x+11$
  5. $x^2-20x+16$
  6. $3x^2+12x-21$
  7. $7x^2-5x+12$