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Exercises on Distance, Midpoint Formulae, Standard Equation of a Circle, and Concepts of Functions

Let $P=(x_{1},y_{1}),$ $Q=(x_{2},y_{2}).$ Then we have the followings.
1. Distance
2. Midpoint of and
Standard Equation of circle, center at and radius
Determine if a graph represents a function.
1. Method 1: Recall that we can use the vertical line test to check if a graph represents a function.
2. Method 2: Given an equation, if can be expressed in terms of then the graph represents a function. The graphs of the following examples represent functions.
  1. $y=3x^{2}+1,$
  2. $y=\sqrt{x-1}+1$
  3. $y=\frac{1}{x}$
Exercises:

Given the following points and find (a) the distance of and (b) midpoint of (c) the standard equation of the circle which uses as its diameter.

Given the following equations, if it is circle, find its center and radius. If it is not a circle, explains why.
1. [center and radius $\sqrt{5}]$
2. [center radius $\sqrt{7}]$
3. [this is not a circle, why?]
4. [ this is a point, (3,2) which is the center.]
5. [ this does not represent a graph, the left hand side of the equation is positive, but the right hand side is negative.]
Given and Find the corresponding points $P^{\prime }$ and $Q^{\prime }$ if and are being shifted left units and down units. Repeat the process by using the points and from Questions 1 to 4 above.

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