Additive pattern in tables indicating linear behavior: When the values in a data table increase or decrease by a constant amount between successive entries, the pattern is additive, which signifies a linear function.
Multiplicative pattern in tables indicating exponential behavior: When the ratio of successive values in a data table remains constant, the pattern is multiplicative, indicating an exponential function.
Constant rate of change for linear functions: A property where the change in the function's output is the same for equal increases in input, reflected as a constant difference in table values.
Constant growth factor for exponential functions: The fixed ratio by which the function's value multiplies over equal intervals of input, observed as a consistent ratio between successive table entries.
1. What is the primary purpose of linear and exponential functions in data modeling?
2. A company models its population growth with the function P(t) = 500 * 1.2^t, where t is in years. If the population after some years is 1,000, how can you find the number of years t?
3. What is the primary effect of converting multi-year growth factors into per-year growth factors when developing exponential models?
Recognizing linear pattern
Constant difference indicates linear behavior.
Recognizing exponential pattern
Constant ratio indicates exponential behavior.
Inverse function — definition?
Reverses input-output relationship of original function.
Finding inverse of linear
Solve for x, then swap x and y.
Inverse of exponential — method?
Use logarithms to solve for x.
Graph of inverse — reflection?
Reflection across y = x line.
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