Week 8: General Form, Quadratics

This week I learned many cool and fascinating facts about the general form of a quadratic equation ( ax^2 + bx +c ) as well as analyzing y=a (x-p)^2 + q

General form

x^2 :

  • Positive – Open upward
  • Negative – Opens downwards


  • is a mixture of both ax^2 and bx so it can differ


  • Is the y axis

Analyzing equation:  y=a (x-p)^2 + q

using this formula can tell us a lot of things that will help us graph the equation. In fact it will tell us 8 helpful clues.

ex. $latex -2(x+2)^2 – 1

  1. Vertex: the vertex will be p and q … (-2,-1) * p is always backwards from the formula because in the original equation it is (x – p) therefore it has to be a negative number for it to become positive in the new equation.
  2. Axis of symmetry: This will be p because it is the x value on a graph …-2
  3. Opens up or down: This is determined if x^2 is negative or positive. If it is negative it will be opening down, and if it is positive it will be opening upwards… Opens down because it is -2
  4. If it is congruent to y = x^2. This means that the pattern of the graph would be 1, 3, 5. Meaning 1 over 1 up, 1 over 3 up, 1 over 5 up etc. If it does not follow the 1,3,5 rule it means that a has a value different from -1 or 1. Say a = 4, multiply 4 to each number from the original pattern. ex/ 1,3,5 is now equal to 4, 12 , 20 etc. This pattern will also tell us if the parabola will be stretched or compressed. If 1> a < 0 ( fraction ) it means that the parabola will be compressed. If  1 < a > 1 it means that the parabola will be stretched…. this equation is congruent to y = -2x^2, it is being stretched because 2 > 1, it is negative so it will be flipped upside down but the structure is the same. (congruent)
  5. Minimum or maximum: If a is positive, it will have a minimum. If a is negative it will have a maximum… It has a maximum of -1
  6. Domain. We know the XER because the parabola never ends.
  7. Range: If a is positive y will be greater than the minimum. If a is negative, y will be smaller than the maximum… y < -1, yER


$latex y = 3x^2 (x – 3)^2 + 4

The horizontal translation will be 3 units to the right even though it says -3.



I also learned that general form can be turned into the analyzing equation of  y=a (x-p)^2 + q by completing the square.

Here is a video that helped me understand how to complete the square


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