Algebra Unit Test

1.
  The length between one and nine is eight. It would take one division to get the ratio, 3:5.
  The point that divides the line segment into the ratio is the point at four.

2.
      ( x1 + x2  y1 + y2 )
  M = ( -------, ------- )
      (    2        2    )

      ( 4 - 2  2 + 8 )
    = ( -----, ----- )
      (   2      2   )

      (2  1 0)
    = (-, ---)
      (2   2 )

    = (1,  5)

3.
  d = sqrt((y2 - y1)^2 + (x2 - x1)^2)
    = sqrt((-1 - 3)^2 + (2 + 4)^2)
    = sqrt((-4)^2 + (6)^2)
    = sqrt(16 + 36)
    = sqrt(52)
    = sqrt(4 * 13)
    = sqrt(4) * sqrt(13)
    = 2 sqrt(13)
    ~ 7.211

4.
  First, let's find the x and y intercepts of the equation.

  y = -2x + 3
    = -2(0) + 3
    = 3
  (0, 3)

   y = -2x + 3
   0 = -2x + 3
  -3 = -2x
  -(3/2) = -x
  (3/2) = x
  (3/2, 0)

  Graph the line between these ordered pairs:
  https://rawgit.com/caffinatedmonkey/f77bd6a22cf1fe89a5da/raw/625b74bf38c3af9e09202f9ea0a4e774906688e7/graph.svg

5.
  First we need to write an equation for the distance between the point on the parabola and
  its focus.

  d = sqrt((y2 - y1)^2 + (x2 - x1)^2)
    = sqrt((y - 5)^2 + (x - 2)^2)

  This needs to be equal to the distance between a point on the parabola and its
  directrix.

  d = sqrt((y2 - y1)^2 + (x2 - x1)^2)
    = sqrt((y - 3)^2)
    = |y - 3|

  All points on this parabola need to be equal to both the equations above.

       |y - 3| = sqrt((y - 5)^2 + (x - 2)^2)
     (y - 3)^2 = (y - 5)^2 + (x - 2)^2
  y^2 - 6y + 9 = y^2 + x^2 - 10y - 4x + 4 + 25
      y^2 - 6y = y^2 + x^2 - 10y - 4x + 4 + 25 - 9
               = y^2 + x^2 - 10y - 4x + 20
     -6y + 10y = x^2 - 4x + 20
            4y = x^2 - 4x + 20
             y = (x^2)/4 - x + 5

graph.svg

<svg xmlns="http://www.w3.org/2000/svg" width="100" height="100" viewBox="0 0 100 100">
  <defs>
    <pattern id="space" width="5" height="5" patternUnits="userSpaceOnUse">
      <path d="M 5 0 L 0 0 0 5" fill="none" stroke="gray" stroke-width="0.5"/>
    </pattern>
    <pattern id="grid" width="50" height="50" patternUnits="userSpaceOnUse">
      <rect width="50" height="50" fill="url(#space)"/>
      <path d="M 50 0 L 0 0 0 50" fill="none" stroke="black" stroke-width="1"/>
    </pattern>
  </defs>

  <rect width="100%" height="100%" fill="url(#grid)"/>
  <g transform="translate(50, 50)">
   <line x1="7.5" y1="0" x2="0" y2="-15" stroke="orange"/>
  </g>
  <path d="m22.5-15,70,130z" stroke-width="0.1" fill="none" stroke="#000"/>
</svg>