Answer:
y = x² +9x +4
Step-by-step explanation:
You want the quadratic equation that describes the values in the table.
- x = -2, -1, 0
- y = -10, -4, 4
Regression
We like to use the quadratic regression function of a calculator for problems like this. As the attachment shows, using the first three points in the table gives the quadratic function ...
y = x² +9x +4
Differences
There are some relatively simple steps you can take to determine the coefficients of the quadratic function y = ax²+bx+c "by hand."
- Form first differences and second differences of sequential values in the table.
Here, the x-values are spaced 1 unit apart. This is ideal for using differences to find the quadratic coefficients. The first differences are ...
-4-(-10) = 6, 4-(-4) = 8, 10, 12
The second differences are the differences of these:
8-6 = 2, 2, 2
The "a" coefficient in the equation is half the second difference: 2/2 = 1. - Using the coefficient just found, subtract ax² from each table value, and find the first differences of the resulting values.
-10-(-2)² = -14, -4-(-1)² = -5, 4-0² = 4, ...
The differences of these are ...
-5-(-14) = 9, 4-(-5) = 9, ...
The "b" coefficient is this first difference: 9. - Determine the "c" coefficient by subtracting ax²+bx from the table values:
-10 -((-2)²+9(-2)) = -10-(-14) = 4
The "c" coefficient is this value. (It is also the value of y where x=0, which happens to be in this table.)
The quadratic equation found this way is ...
y = x² +9x +4
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Additional comment
You may notice that this method using differences can be automated by a spreadsheet. It only needs 3 ordered pairs with sequential x-values to determine the coefficients of the quadratic. The regression calculator can use non-sequential x-values.
In general, you can use any three (x, y) pairs to compute the coefficients a, b, c. Put the x- and y-values into the equation ax² +bx +c = y, and you will get three linear equations in a, b, c that can be solved in any of the usual ways.
If the second differences are not constant, the relation is not quadratic.
