# Given the graph with the points (1,5) (2,0) (3,-3) (4,-4) (5,-3) (6,0) (7,5) write the function for f(x) in standard form.

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First note that points on a parabola have mirror points across the axis of symmetry. (This is the line x=h where h is the x-coordinate of the vertex.)

Since (3,-3) and (5,-3) lie on the parabola, the line of symmetry is x=h where h is the arithmetic mean of the x-coordinates. Thus `x=(3+5)/2==>x=4` is the line of symmetry.

The vertex is the only point on the parabola to lie on the axis. Since (4,-4) is on the axis of symmetry, the vertex is (4,-4).

The vertex form of a quadratic is `y=a(x-h)^2+k` where the vertex is at (h,k) and a determines the direction the parabola opens (up/down) and how wide it opens.

Since the vertex is (4,-4) we have `y=a(x-4)^2-4` . We can substitute one of the know points (besides the vertex) to find a. Arbitrarily choose (3,-3).

`-3=a(3-4)^2-4`

`-3=a-4`

a=1

So `y=(x-4)^2-4`

To put in standard form expand to get:

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The equation in standard form is `x^2-8x+12`

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The graph:

There are a number of other ways to find this equation -- use technology; solve a system of linear equations for the parameters a,b, and c; draw a smooth curve through the given points to find the vertex -- a can be found by noticing the vertical distance between consecutive points, etc...