**Outcome C10 (adv): **determine the equation of a quadratic function using finite differences

**Originators: **Hanna + Erika

**Explanation:** A *quadratic function* is a function that forms a parabola when graphed. The equation of a quadratic function can be written in many different forms. The most common form is called *general form*, and is written as "y = ax^{2} + bx + c" where the "x^{2"} is the maximum number of x-intercepts.

In a quadratic function, every co-ordinate has one x and one y value. When a list of ordered co-ordinates is compiled, with the y value being the number listed in the brackets and the x value being what number the y value has in the sequence, it is called a quadratic sequence. An example of a quadratic sequence is {1, 4, 8, 13, 19, 26}. The co-ordinate for the second value would be (2,4). To obtain the equation of a quadratic function using a quadratic sequence, the second level of differences between the y value of the co-ordinates must be determined.

The value of D^{2} is equal to 2a, in the equation y = ax^{2} + bx + c. First the a value must be plugged into the equation. In this case the equation would now be "y = 1/2x ^{2} + bx + c". From here, two (x,y) co-ordinates can be plugged into two seperate instances of the equation. next these two equations should be reduced down into lowest form and using the elimination method the b value should be isolated. After this, the b value can be put into one of the previous two equations and solved through, revealing the value of c.

ex.

So the final equation is "y = 1/2x^{2} + 3/2x -1".

**Sample Question:**

Alan was watering his garden, when he noticed that the water from the hose formed a parabola. He remembered a physics project that involved graphing a parabola. He quickly asked his twin sister, Alanna, to help him record the data so he could graph the parabola later. The following sequence was recorded to show the height the water reaches in meters above the ground after 1, 2, 3, 4, 5, 6 seconds: {2, 3, 6, 11, 18, 27}. Determine the function of this quadratic sequence using sequence of differences.

**Solution:**

Step 1: Determine the common difference of D2

Sequence: {2, 3, 6, 11, 18, 27}

D1: {1, 3, 5, 7, 9}

D2: {2, 2, 2, 2}

Step 2: Plug in the value of D2 into equation D2 = 2a and solve for a

D2=2, D2=2a

2=2a

2/2=a

1=a

Step 3: Plug in a value and 2 different (x, y) coordinates into general form of quadratic function and solve for the value b and c within the quadratic function.

y = ax^{2} + bx + c

(1, 2) 2 = 1(1)^{2} + b(1) + c 2 = 1 + b + c -16 = -24 - 4b

(5, 18) 18 = 1(5)^{2} + b(5) + c - 18 = 25 + 5b + c -16 + 24 = -24 -4b + 24

-16 = -24 - 4b 8 /-4 = -4b/-4

** -2 = b**

2 = 1(1)^{2} - 2(1) + c

2 + 1 = -1 + c + 1

** 3 = c**

Step 4: plug in the determined a, b, and c values and write the resulting quadratic function

y = x ^{2}- 2x + 3

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