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Quadratic Models
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Quadratic models in the form ax2+bx+c, including the parabola's vertex and axis of symmetry, maximum or minimum turning point, x-intercepts or roots, and fitting a quadratic from three data points.
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Quadratic Models
SL AI 2.5
A quadratic model has a turning point (vertex) at which its minimum or maximum value occurs. The general form of a quadratic equation is ax2+bx+c.
If a<0, the turning point of a quadratic is its maximum; if a>0, the turning point of a quadratic is its minimum.
Fitting a quadratic model
SL AI 2.5
Given 3 pieces of data, we can solve for a, b and c in a quadratic model ax2+bx+c.
Example
The points (1,−25), (−1,−1) and (−3,7) lie on a parabola with equation y=ax2+bx+c. Find a,b and c.
Plugging in the x coordinates and setting equal to the y-coordinates gives 3 equations:
⎩⎪⎨⎪⎧a⋅12+b⋅1+c=−25a⋅(−1)2+b⋅(−1)+c=−1a⋅(−3)2+b⋅(−3)+c=7⇒⎩⎪⎨⎪⎧a+b+c=−25a−b+c=−19a−3b+c=7
Solving this using a calculator gives a=−2,b=−12,c=−11. Thus the parabola has equation
y=−2x2−12x−11
Quadratic x-intercepts
SL AI 2.5
The roots of a quadratic correspond to the x-intercepts of its graph. When x=a or x=β, the entire expression equals zero, which is reflected on the graph.
The equation of the parabola below is −(x−α)(x−β):
Vertex and Axis of Symmetry
SL AI 2.5
The graph of a quadratic function has the general shape of a parabola.
It is symmetrical about the axis of symmetry and has a maxima or minima at the vertex, which lies on the axis of symmetry.
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