QUADRATICS
MAKE SENSE
Quadratic Relations
and Transformations
First Differences
Calculating first and second differences helps you to figure out whether a set of data is linear or quadratic.
To do this you need to subtract the second y value from the first y value and continue doing so for each pair of y values. If the first differences are constant then this means that the relation is linear.
If the first differences are not the same, you find second differences by repeating the process with the first differences this time. If these values are constant, then the relation is quadratic.
Remember that you need to have all your x values in order.
This is an example of how to use first and second differences to find a quadratic relation.
Quadratic Relations
A quadratic relation is a relation that has an equation
y=ax^2 + bx + c
Here - a, b and c are real numbers and a can not be 0.
Transformations
Vertex form of a parabola is written as y = a(x-h)^2 + k
The value of a determines the orientation and the shape of the parabola.
If a is greater than 0 (positive), the parabola opens up.
If a is less than 0 (negative), the parabola opens down.
If a is greater than 1 (ex. -1, 3.5, 10), the proabola will be vertically stretched.
If a is less than 1 (ex. 0.5, -0,2), the parabola will be vertically compressed.
The value of k determines the vertical position of the parabola.
If k is greater than 0, the vertex moves up by k units.
If k is less than 0, the vertex moves down by k units.
The value of h determines the horizontal position of the parabola.
If h is greater than 0, the vertex moves to the right h units.
If h is less than 0, the vertex moves to the left h units.
*The value of h is always opposite to what you see in the bracket
Example:
y=2(x-3)^2 -4
a=2 k= -4 h=3
-the parabola opens up
-vertically stretched by a factor of 2
-translated 4 units left
-translated 3 units up
-vertex = (3, -4)