Exponential Functions and Applications

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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. It describes growth or decay processes.

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What is the formula for compound interest?

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The formula for compound interest is A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n years, P is the principal amount (the initial amount), r is the annual interest rate (decimal), n is the number of times that interest is compounded per year, and t is the number of years.

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How do you convert a radical expression to a rational exponent?

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Front

What is the half-life of a substance?

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The half-life of a substance is the time required for half of the substance to decay or be eliminated. It is a key concept in exponential decay.

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How do you calculate the remaining amount of a substance after a certain time using half-life?

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To calculate the remaining amount, use the formula: Remaining Amount = Initial Amount * (1/2)^(t/T), where t is the elapsed time and T is the half-life.

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What is a horizontal asymptote?

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A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches infinity or negative infinity. It indicates the behavior of the function at extreme values.

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What does it mean to translate a function horizontally?

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Translating a function horizontally means shifting the graph left or right. For example, f(x - h) translates the graph of f(x) to the right by h units.

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

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

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How do you find the vertical asymptote of a function?

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To find the vertical asymptote, set the denominator of a rational function to zero and solve for x. The values of x that make the denominator zero are the vertical asymptotes.

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What is the relationship between exponential functions and logarithms?

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Exponential functions and logarithms are inverse functions. If y = b^x, then x = log_b(y). This means that logarithms can be used to solve for exponents.