What is a parabola and what is itโs equation?
Understanding how to derive the equation of a quadratic function from a set of data points is a fundamental skill in algebra. Quadratic functions are represented by the general form
๐ฆ=๐๐ฅ^2+๐๐ฅ+๐
This equation uses 5 main components.
- Y - y is what the output will be when the the equation is in use.
- A - a is what dictates if the parabola will be opening upwards or downwards.
- x^2 - the squared symbol dictates that the parabola will be in a curve a number > 1 in front of the squared term results in a narrower, more curved parabola, while a number < 1 coefficient creates a wider, less curved parabola.
- B - b is how we know how much the parabola moves left or right.
- C - the C value is where the parabola intercepts with the y axis. Now that we have defined what a parabola is we can start explaining how we can make an equation from a table of values.
How can we make an equation using a table of values?
When you are given a table of values you should identify that you have the right type of graph. You can usually tell pretty quickly because a linear table of values increases or decreases by the same number the whole time. An exponential table of values will do the same as linear however it increases or decreases by multiplication or division instead of addtion or subtraction. The best way to identify a parabolic equation is to look for 2 key details.
- It doesnโt follow the patterns explained below - only caviot with this method is that it could be a different type of graph
- It repeats the x values for different y values.
Now to start with the table of values to equation transformation.
Using the information about what the key parts of a parabolaโs equation we can see that we only need to define 2 things. A and the X intercepts.
Dispite not being explicitly said above, The X intercepts will give you the factored form of the equation. The X intercepts is when the y = 0 the x value is what the x intercepts are.
For example, if the x intercepts are (0,5) and (0,-2) we have to use the zero product rule.
The rule dictates that if the product of 2 numbers is 0 then one of the numbers must be 0.
Using our example we can say that x = 5 and x = -2 so then to get 0 on one of the sides you have to subtract 5 from both sides for the first example and add 2 for the other example.
We are left with x - 5 = 0 and x + 2 = 0 we can then say that these two equations multiplied by each other must = 0
y = (x-5)(x+2)
If there are no other contraints then we can move on however if it is given in the table of values that it increases by 2 then 6 then 10 then we have to say that the equation is:
y = 2(x-5)(x+2)
This is because there was one key detail I forgot to mention before. When A = 1 parabolas increase or decrease by 1,3,5,7 etc. That means that when A = any number > 1 the parabolas increase or decrease is going to be 1 * the number, 3 * the number, 5 * the number, etc. The parabola will get steeper. Similarly, if A is 1/2 or <1 then the parabola will get wider. If you want to get an equation in y = ax^2+bx+c notation then you can just multiply the factored equation and you will end up with standard form.