Week 16 KA: Identifying and Evaluating Exponential Function

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What is an exponential function?

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An exponential function is a mathematical function of the form f(x) = a * b^x, where 'a' is a constant, 'b' is a positive real number, and 'x' is the exponent. The function grows or decays at a constant percentage rate.

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How can you identify an exponential function from a table of values?

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You can identify an exponential function by checking if the ratio of consecutive y-values is constant as x increases by equal intervals. If the y-values are multiplied by a constant factor, it is exponential.

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What is a linear function?

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A linear function is a function that can be graphically represented as a straight line. It has the form f(x) = mx + b, where 'm' is the slope and 'b' is the y-intercept.

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What does it mean for a function to have a constant rate of change?

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A function has a constant rate of change if the difference in y-values divided by the difference in x-values is the same for any two points on the function. This indicates a linear relationship.

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What is a constant factor in the context of exponential functions?

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A constant factor in exponential functions refers to the fixed number by which the y-value is multiplied as x increases by 1. For example, if y is multiplied by 3 as x increases by 1, then 3 is the constant factor.

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What is the difference between exponential growth and exponential decay?

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Exponential growth occurs when the base 'b' in the function f(x) = a * b^x is greater than 1, leading to increasing values. Exponential decay occurs when 'b' is between 0 and 1, leading to decreasing values.

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Can a function be both linear and exponential?

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No, a function cannot be both linear and exponential. Linear functions have a constant rate of change, while exponential functions have a constant multiplicative factor.

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What is the significance of the base 'b' in an exponential function?

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The base 'b' determines the rate of growth or decay in an exponential function. If b > 1, the function grows; if 0 < b < 1, the function decays.

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How do you determine if a table represents a linear function?

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To determine if a table represents a linear function, check if the differences between consecutive y-values are constant as x increases by equal intervals.

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What is the y-intercept in a linear function?

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The y-intercept is the value of y when x = 0. It represents the point where the line crosses the y-axis.