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How to determine the quadratic function from the given graph?

Question:

Given the graph of the quadratic function y = ax2 + 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 - x1)(x - x2), in which a is the coefficient of the x square term. Apply x1 = -3 and x2 = 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) (x2 - 2x + 3x - 6)
= (1/2) (x2 + x - 6)
= (1/2)x2 + (1/2) x - 3
so, a = 1/2, b = 1/2, and c = -3