This week I learned many cool and fascinating facts about the general form of a quadratic equation ( ) as well as analyzing
- is a mixture of both and bx so it can differ
- Is the y axis
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
- 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.
- Axis of symmetry: This will be p because it is the x value on a graph …-2
- Opens up or down: This is determined if 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
- If it is congruent to . 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 = , it is being stretched because 2 > 1, it is negative so it will be flipped upside down but the structure is the same. (congruent)
- 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
- Domain. We know the XER because the parabola never ends.
- 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 by completing the square.
Here is a video that helped me understand how to complete the square