Recall that to find the equation of the set of points equidistant from two given points, we let (x, y) represent an arbitrary point which fulfills this condition, then express the condition symbolically and simplify.

• (x,y) is equidistant from two points provided that the distance from the first point to (x,y) is equal to the distance from the second point to (x,y).
• For the points in the figure below the distances are as indicated.

Video Clip #01

The graph of the first equation below has x² and y² terms, both with coefficient 4, and so can be put into the form 4 ( x-h)² + 4 (y-k)² = c, indicated in red at the left of the figure.

• If c is a positive number this equation will turn out to be the equation of a circle.
• The graph will be the horizontal and vertical shift of the graph of 4 x² + 4 y² = c, which is identical to x² + y² = r for r = c /4.

The form of the second equation will the much like that of the first, except that the x and y terms will have different coefficients.

• The equation will be a shifted version of the graph of 2 x² + 8 y² = c.
• When this equation is divided by c, the result will be the standard form of an ellipse, as indicated at right.

The third equation differs from the second in that the quadratic terms have a difference signs.

• At right we see that the result is in the standard form of a hyperbola.

The fourth equation lacks the x² term of the equations seen so far.

• Solving the equation for x gives a shifted form of the standard form x = 1 / (4c) y² of a parabola opening in the x direction, vertex at the origin and focus at x = c.

The fifth equation lacks the y² term, so we be solve for y to obtain a shifted form of the standard form y = 1 / (4c) x² of a parabola opening in the y direction with vertex at your tangent focus at y = c.

Video Clip #02

The figure below has been previously explained as the way to sketch an ellipse or a hyperbola.

The figure below summarizes the shapes of the graphs of the two types of parabolas.

Video Clip #03

A set of parametric equations for x and y, in terms of parameter t, can often be put into the y = f(x) form by solving the first equation for t in substituting result into the second.

The figures below depict the parametric equations of a straight line with slope b/a through (x0, y0), and the parametric equations of and ellipse centered at (h,k) and having semimajor and semiminor axes a and b.

Video Clip #04

The histogram below depicts the probabilities of getting 0, 1 and 2 'heads' when flipping a coin twice.

• The equally likely outcomes HH, HT, TH and TT are listed.
• Since to these outcomes have one 'heads', the probability of one 'heads' is 2/4 = 1/2.
• The events '0 heads' and '2 heads' occur once each and therefore each has probability 1/4.

A similar histogram a sketch below for 3 flips of a coin.

• The eight equally likely outcomes of flipping three coins are listed.
• Three of these outcomes have 1 'heads' and three have 2 'heads', so the probabilities of these events are both 3/8.
• Only one outcome has 0 'heads' and only one has 3 'heads', so the probabilities of these events are both 1/8.

The table below depicts the number of ways of getting 0, 1, 2, 3, 4, ... heads on 0, 1, 2, 3 or 4 coin flips.

• The n = 0 row of the table corresponds to the 1 possible outcome when flipping no coins. That outcome is 0 'heads'.
• The n = 1 row corresponds to the possible outcomes of flipping one coin: the possible outcomes are 10 'heads' and 1 'heads'.
• The n = 2 row corresponds to the possible outcomes of flipping two coins, as previously seen.
• The n = 3 row corresponds to the possible outcomes of flipping three coins, as previously seen.

You should list the 16 possible outcomes for flipping four coins and verify the fourth row of the table.

The table below lists the probabilities associated with the table above.

The second and third rows depict the probabilities previously seen for 2 and 3 coin flips.

The common denominator of each row is the total number of possible outcomes for the corresponding number of flips. This number doubles with each new coin, and is equal to 2ⁿ.

The table below is a more symmetric rearrangement of the previous table.

• In this rearrangement every number in the interior of the table is equal to the sum of the two numbers above into the right and above and to the left of it.
• This figure is called Pascal's Triangle.
• Each row represents the number of ways to get 0, 1, 2, 3, ... heads on n flips.
• For example the fifth row corresponds to n = 4 flips and the third number in the row corresponds to the number ways are getting 2 'heads' on this number of flips.

• This number is symbolized in various ways .  Three of these forms are indicated in the figure. 
• We will use C(4, 2) to stand for this number.

• Note that the total of the number in each row is the total number of possible outcomes, and is thus equal to 2ⁿ.