Writing Exponential Equations

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What is an exponential function?

Back

An exponential function is a mathematical function of the form f(x) = a(b)^x, where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent.

Front

What does the base of an exponential function represent?

Back

The base of an exponential function determines the rate of growth or decay. If the base is greater than 1, the function represents exponential growth; if the base is between 0 and 1, it represents exponential decay.

Front

How do you identify the y-intercept of an exponential function?

Back

The y-intercept of an exponential function f(x) = a(b)^x is found by evaluating f(0), which equals a. This is the point where the graph intersects the y-axis.

Front

What is the general form of an exponential equation?

Back

The general form of an exponential equation is f(x) = a(b)^x, where 'a' is the initial value and 'b' is the growth factor.

Front

How do you write an exponential equation from a graph?

Back

To write an exponential equation from a graph, identify the y-intercept (a) and the growth factor (b) by analyzing the rate of increase between points on the graph.

Front

What is the significance of the initial value 'a' in an exponential function?

Back

The initial value 'a' represents the starting amount or value of the function when x = 0.

Front

How can you determine if a function is exponential from a table of values?

Back

A function is exponential if the ratio of consecutive outputs (y-values) is constant when the inputs (x-values) increase by the same amount.

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What is the difference between exponential growth and exponential decay?

Back

Exponential growth occurs when the base of the exponential function is greater than 1, leading to increasing values. Exponential decay occurs when the base is between 0 and 1, leading to decreasing values.

Front

How do you convert an exponential equation to logarithmic form?

Back

To convert an exponential equation a^b = c to logarithmic form, write it as log_a(c) = b.

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What is the formula for exponential growth?

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The formula for exponential growth is f(t) = a(1 + r)^t, where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.