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How does the y-variable change in a linear function when the x-variable increases by 1? How does it change in an exponential function in the same circumstance? What characterizes a quadratic function's data pairs?
See solution.
Sometimes it is enough to plot a set of data pairs to identify which function could be a good model. If this is not enough, we can be more precise by analyzing the data since each type of function has distinctive characteristics. We will review these characteristics for each function and explain how to determine which model to use.
For an exponential function, each time the x-variable increases by 1 the function is multiplied by a constant factor. y=ab^x y=ab^(x+1) ⇕ y=ab^x(b) Therefore, for equally spaced x-values of a set of data pairs, if the y-values have a common ratio, then an exponential function fits the model. Similarly, if the ratio between pairs of y-values is approximately constant, then an exponential function is a good model as well.
Quadratic functions' data pairs have a characteristic feature as well. For equally spaced x-values, the y-values have constant second differences.
Similarly, if in the same conditions the second differences are approximately constant, then a quadratic function is a good model as well.