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## How to determine the quadratic function from the given graph?

Question:

Given the graph of the quadratic function y = ax^{2} + bx + c, find the value of a, b c, and write the function of this graph.

Solution:

The graph intersects the x-axis on two points, one is negative 3 and another one is positive 2, so we use the intersection form of the quadratic function
y = a(x - x_{1})(x - x_{2}), in which a is the coefficient of the x square term. Apply x_{1} = -3 and x_{2} = 2 into the equation,
we get y = a (x + 3) (x - 2). Now we have the third point (1, -2), because this point lies on the graph, so this point will satisfy the quadratic equation.
So, substitute the x-coordinate and y-coordinate of this point into the equation, we will find the value of a, which is the coefficient of the x square term.

- substitute y = -2 and x = 1 into equation y = a (x + 3)(x - 2)
- - 2 = a (1 + 3) (1 - 2)
- - 2 = a (4) (- 1)
- - 2 = - 4a
- a = 1/2
- y = (1/2) (x + 3) (x - 2)
- = (1/2) (x
^{2}- 2x + 3x - 6) - = (1/2) (x
^{2}+ x - 6) - = (1/2)x
^{2}+ (1/2) x - 3 - so, a = 1/2, b = 1/2, and c = -3