Exponential functions
53 flashcards covering Exponential functions for the SAT Math section.
Exponential functions describe how quantities change by multiplying by a constant factor over equal intervals, rather than adding a fixed amount like in linear growth. For instance, if you start with 2 bacteria and they double every hour, the population follows an exponential pattern: after one hour, you have 4; after two, 8; and so on. This is represented by equations like f(x) = a * b^x, where 'a' is the starting value and 'b' is the growth factor. These functions are key for modeling real-world scenarios such as population growth, radioactive decay, or compound interest, making them essential for understanding rapid changes.
On the SAT Math section, exponential functions often appear in questions about graphing curves, solving equations, or interpreting word problems involving growth and decay. Common traps include confusing them with linear functions, misapplying exponent rules, or overlooking asymptotes in graphs. Focus on mastering properties like the base (greater than 1 for growth, between 0 and 1 for decay) and how to manipulate equations quickly. For success, practice identifying exponential patterns in context.
Terms (53)
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Exponential function
An exponential function is a function of the form f(x) = a b^x, where a is a constant, b is the base greater than 0 and not equal to 1, and x is the exponent.
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Base of an exponential function
The base is the number b in the function f(x) = a b^x, which determines the growth or decay rate and must be positive and not equal to 1.
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Exponent in an exponential function
The exponent is the variable or expression, like x in f(x) = a b^x, that indicates the power to which the base is raised.
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Exponential growth
Exponential growth occurs when a quantity increases by a fixed percentage over equal time intervals, modeled by functions like f(x) = a b^x where b > 1.
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Exponential decay
Exponential decay happens when a quantity decreases by a fixed percentage over equal time intervals, modeled by functions like f(x) = a b^x where 0 < b < 1.
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Asymptote of an exponential function
An asymptote is a line that the graph of an exponential function approaches but never touches, typically a horizontal line like y = 0 for basic forms.
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Horizontal asymptote
In an exponential function, the horizontal asymptote is the line that the graph approaches as x goes to positive or negative infinity, such as y = 0 for f(x) = b^x.
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Y-intercept of an exponential function
The y-intercept is the point where the graph crosses the y-axis, which for f(x) = a b^x is the value f(0) = a.
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Domain of an exponential function
The domain of an exponential function like f(x) = a b^x is all real numbers, meaning x can be any value.
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Range of an exponential function
The range of an exponential function f(x) = a b^x, where a > 0 and b > 0, b ≠ 1, is all positive real numbers if a > 0.
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Compound interest formula
The compound interest formula is A = P(1 + r/n)^(nt), where A is the amount after time t, P is the principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is time in years.
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Continuous compounding
Continuous compounding uses the formula A = Pe^(rt), where A is the final amount, P is the principal, r is the annual interest rate, t is time, and e is approximately 2.718, for interest compounded infinitely often.
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Half-life
Half-life is the time required for a quantity to decay to half its initial value, often used in exponential decay models like radioactive substances.
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Doubling time
Doubling time is the period it takes for a quantity to double in an exponential growth model, calculated using the formula t = ln(2) / k, where k is the growth rate.
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Solving exponential equations
Solving exponential equations involves finding the value of x in equations like b^x = c, often by taking logarithms if the bases are the same or by rewriting them.
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Logarithm
A logarithm is the inverse operation of exponentiation, such as logb(a) = c meaning b^c = a, used to solve exponential equations.
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Common logarithm
The common logarithm is the logarithm with base 10, written as log(x), and is used to solve equations like 10^x = y by finding x = log(y).
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Natural logarithm
The natural logarithm is the logarithm with base e (approximately 2.718), written as ln(x), and is used in continuous growth models to solve for time or rates.
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Properties of exponents: product rule
The product rule for exponents states that when multiplying powers with the same base, you add the exponents: a^m a^n = a^(m+n).
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Properties of exponents: quotient rule
The quotient rule for exponents states that when dividing powers with the same base, you subtract the exponents: a^m / a^n = a^(m-n).
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Properties of exponents: power rule
The power rule for exponents states that when raising a power to another power, you multiply the exponents: (a^m)^n = a^(mn).
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Graph of an exponential function
The graph of an exponential function like f(x) = a b^x starts at the y-intercept and either increases or decreases rapidly, approaching a horizontal asymptote.
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Increasing exponential function
An increasing exponential function has a base b > 1, so as x increases, f(x) increases rapidly.
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Decreasing exponential function
A decreasing exponential function has a base 0 < b < 1, so as x increases, f(x) approaches the horizontal asymptote and decreases.
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Vertical shift of an exponential function
A vertical shift moves the graph of an exponential function up or down, as in f(x) = a b^x + k, where k is the shift amount.
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Horizontal shift of an exponential function
A horizontal shift moves the graph left or right, as in f(x) = a b^(x-h), where h is the shift amount; positive h shifts right.
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Reflection of an exponential function
A reflection flips the graph over the x-axis or y-axis, such as f(x) = -a b^x for reflection over the x-axis.
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Stretch of an exponential function
A stretch vertically multiplies the function by a factor greater than 1, like f(x) = a k b^x where k > 1, making the graph taller.
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Compression of an exponential function
A compression vertically multiplies the function by a factor between 0 and 1, like f(x) = a k b^x where 0 < k < 1, making the graph shorter.
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Initial value
The initial value in an exponential function f(x) = a b^x is a, which is the value of the function when x = 0.
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Growth factor
The growth factor is the base b in an exponential growth function where b > 1, representing the multiplier per unit increase in x.
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Decay factor
The decay factor is the base b in an exponential decay function where 0 < b < 1, representing the multiplier per unit increase in x.
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Rate of growth
The rate of growth is the percentage increase per unit time in an exponential function, often derived from the base as r = b - 1 for discrete growth.
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Rate of decay
The rate of decay is the percentage decrease per unit time in an exponential function, calculated as r = 1 - b for 0 < b < 1.
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Example: Solve 3^x = 27
To solve 3^x = 27, recognize that 27 = 3^3, so x = 3.
If 3^x = 27, then x = log3(27) = 3.
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Example: Solve 2^x = 8
To solve 2^x = 8, note that 8 = 2^3, so x = 3.
Using logarithms: x = log2(8) = 3.
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Example: Compound interest calculation
For an investment of $1000 at 5% annual interest compounded annually for 3 years, use A = 1000(1 + 0.05)^3 to get A = 1000(1.05)^3 = 1157.63.
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Example: Population growth model
A population starting at 1000 growing at 2% per year is modeled by P(t) = 1000 (1.02)^t, where t is years.
After 5 years, P(5) = 1000 (1.02)^5 ≈ 1104.
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Example: Radioactive decay
A substance with a half-life of 10 years starting at 100 grams decays via A = 100 (0.5)^(t/10), where t is years.
After 20 years, A = 100 (0.5)^2 = 25 grams.
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Trap: Mistaking exponential for linear growth
A common trap is assuming a quantity grows linearly when it grows exponentially, leading to underestimating rapid increases over time.
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Trap: Incorrectly applying exponent rules
Students often err by adding exponents when multiplying different bases or forgetting to distribute exponents in products, like mistakenly calculating (23)^2 as 2^2 3^2.
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Identifying exponential functions from tables
To identify an exponential function from a table, check if the ratios of consecutive y-values are constant, indicating multiplicative growth or decay.
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Identifying exponential functions from graphs
From a graph, an exponential function shows rapid increase or decrease and approaches a horizontal asymptote, unlike linear functions which have constant slopes.
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Applications in finance
Exponential functions model financial growth, such as compound interest or investments, to calculate future values based on interest rates.
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Applications in science
Exponential functions describe phenomena like population growth, bacterial reproduction, or radioactive decay in scientific contexts.
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Formula for exponential growth: A = P(1+r)^t
This formula models discrete exponential growth, where A is the final amount, P is the initial amount, r is the growth rate, and t is time.
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Formula for exponential decay: A = P(1-r)^t
This formula models discrete exponential decay, where A is the final amount, P is the initial amount, r is the decay rate, and t is time.
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e (Euler's number)
e is approximately 2.718 and is the base of the natural exponential function, used in continuous growth and decay models.
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Base e exponential function
A base e exponential function is of the form f(x) = a e^(kx), used for continuous processes like population growth at a constant rate.
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Percent growth rate
The percent growth rate is the annual percentage increase in an exponential growth model, such as 5% meaning the quantity multiplies by 1.05 each year.
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Percent decay rate
The percent decay rate is the annual percentage decrease in an exponential decay model, such as 3% meaning the quantity multiplies by 0.97 each year.
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Asymptotic behavior
In exponential functions, asymptotic behavior refers to the graph approaching but never reaching the horizontal asymptote as x increases or decreases.
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Intercepts in exponential graphs
Exponential graphs typically have a y-intercept at (0, a) and no x-intercept if a > 0, since the function never crosses the x-axis.