Answer:
Step-by-step explanation:
There are several ways you can find a quadratic from three points. The preferred method may depend on the points you are given, and their spacing.
Here are some choices.
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Put the three (x, y) values into the equation
y = ax² +bx +c
and solve the resulting system of equations for 'a', 'b', and 'c'.
There are a number of methods available for solving a system of three linear equations in three unknowns. Any of them will work for this purpose. Your graphing calculator and/or spreadsheet can generally solve these equations for you with little fuss.
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Graphing calculators and spreadsheets, along with a number of apps and web sites, will perform quadratic regression: fitting a quadratic curve to a set of data. It can be useful to become familiar with at least one of the available options in this area. Closest at hand may be your graphing calculator. It will generally work from tables or lists that you must create or fill in.
Spreadsheets can plot the three points in a "chart". Generally, one of the functions available for spreadsheet charts is the ability to display a regression formula. You would choose polynomial regression of degree 2 to have it display a quadratic equation for those points. (The displayed formula may only show a limited number of significant figures for the coefficients.)
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Sometimes the provided table will give the vertex of the function, and/or x- or y-intercepts. These can be used directly in the relevant variations of the quadratic equation. Only a little bit of work may be required to find any coefficients that are not shown directly.
If table points are evenly-spaced (for x=1, x=2, x=3), then you can write the equation based on the given values and their differences. If a1 = f(1), and d1 = f(2)-f(1), and d2 = (f(3) -f(2)) -(f(2) -f(1)) is the second difference, the equation will be of the form...
f(x) = a1 +(x -1)(d1 +(x -2)d2/2)
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The Lagrange interpolation formula can also be used to write the function. For points (x1, y1), (x2, y2), (x3, y3), it tells you the function is ...
[tex]f(x)=y_1\dfrac{(x-x_2)(x-x_3)}{(x_1-x_2)(x_1-x_3)}+y_2\dfrac{(x-x_1)(x-x_3)}{(x_2-x_1)(x_2-x_3)}+y_3\dfrac{(x-x_1)(x-x_2)}{(x_3-x_1)(x_3-x_2)}[/tex]
