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Math Domain
Number/QuantityShape/Space(Function/PatternChance/DataArrangementMath Actions (possible weights: 0 through 4)
2Modeling/Formulating0Manipulating/Transforming2Inferring/Drawing Conclusions1CommunicatingMath Big Ideas
Scale(Reference Frame(RepresentationContinuityBoundednessInvariance/SymmetryEquivalence(General/ParticularContradictionUse of LimitsApproximation(OtherThe graph of a quadratic function is a parabola. This means that it has two distinct intervals where the function is increasing on one and decreasing on the other. They are connected at the vertex, where the graph is "cupped" and the curve appears to be horizontal. This does not mean that any interval taken out of the domain will contain the vertex. To the contrary, if the interval does not contain the vertex the graph appears to be monotonic.
The cubic graph has two possible shapes. It must contain an inflection point (road signs usually warn drivers of a "reverse curve"), that is, the curve bends in opposite directions on different sides of the point. Whether the curve appears horizontal at the center or not is irrelevant. On the other hand, the full curve may or may not contain a pair of vertices one pointing up and one down. If only the piece of a curve around such a vertex is selected, it is possible to distinguish it from a quadratic by the fact that the curve is not symmetric, while the parabola is symmetric with respect to the vertical line passing through the vertex. But it is also possible that if a selected piece of a graph is rotated slightly than the curve may appear to have a vertex (for example, if curve A is rotated counterclockwise by 45 it may appear to have a vertex in the middle). Unfortunately, this makes relying on vertices nearly impossible. However, the cubic curve can always be distinguished from the parabola if the cut-out piece contains the inflection point.
The exponential function is either always increasing or always decreasing (monotonic). Although it is never identical to a parabola or a graph of a cubic function, a piece of it may appear indistinguishable from the other two (with properly selected coefficients). Therefore, if the there are no symmetric vertices and no inflection points, there is no way to tell the curves apart.
In this case, curve C contains an obvious inflection point and could easily be marked as cubic. However, there is no easy way to distinguish between the other two and they must be redone.
An example of a difficult distinction can be found below. The graphs represent functions EMBED Equation.3 and EMBED Equation.3 which differ very little on the interval between 0.5 and 1. Under such circumstances the wire templates would be indistinguishable.
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