Algebra 1 (HS pathway)

Exponential Functions: Growth, Decay, and Exponent Rules

Algebra 1

✓By the end of this lesson students will be able to identify and define exponential functions.
✓By the end of this lesson students will be able to distinguish between exponential growth and exponential decay.
✓By the end of this lesson students will be able to apply exponent rules to simplify algebraic expressions.
✓By the end of this lesson students will be able to model and solve real-world problems involving exponential growth and decay.

Key concepts

Exponential Function

An exponential function is a function of the form y = ab^x, where 'a' is a non-zero constant, 'b' is a positive constant not equal to 1, and 'x' is the independent variable. The base 'b' is raised to the power of the variable 'x'. The value 'a' represents the initial value or y-intercept (when x=0).

y = ab^x

Exponential Growth

Exponential growth occurs when a quantity increases by a constant percentage over a period of time. In the function y = ab^x, exponential growth happens when the base 'b' is greater than 1 (b > 1). Often, 'b' can be expressed as (1 + r), where 'r' is the growth rate as a decimal.

y = a(1 + r)^x

Exponential Decay

Exponential decay occurs when a quantity decreases by a constant percentage over a period of time. In the function y = ab^x, exponential decay happens when the base 'b' is between 0 and 1 (0 < b < 1). Often, 'b' can be expressed as (1 - r), where 'r' is the decay rate as a decimal.

y = a(1 - r)^x

Product Rule for Exponents

To multiply two powers with the same base, add the exponents.

x^a · x^b = x^(a+b)

Quotient Rule for Exponents

To divide two powers with the same base, subtract the exponents.

x^a / x^b = x^(a-b)

Power of a Power Rule

To raise a power to another power, multiply the exponents.

(x^a)^b = x^(ab)

Power of a Product Rule

To raise a product to a power, raise each factor to that power.

(xy)^a = x^a · y^a

Power of a Quotient Rule

To raise a quotient to a power, raise both the numerator and the denominator to that power.

(x/y)^a = x^a / y^a

Zero Exponent Rule

Any non-zero number raised to the power of zero is 1.

x^0 = 1 (where x ≠ 0)

Negative Exponent Rule

Any non-zero number raised to a negative exponent is equal to its reciprocal raised to the positive exponent.

x^(-a) = 1/x^a (where x ≠ 0)

Key facts to remember

1The general form of an exponential function is y = ab^x, where 'a' is the initial value and 'b' is the base or growth/decay factor.
2For exponential growth, the base 'b' is greater than 1 (b > 1). This often means b = 1 + r, where 'r' is the growth rate.
3For exponential decay, the base 'b' is between 0 and 1 (0 < b < 1). This often means b = 1 - r, where 'r' is the decay rate.
4Remember the exponent rules: Product (add exponents), Quotient (subtract exponents), Power of a Power (multiply exponents).
5Any non-zero number raised to the power of zero is 1 (x^0 = 1).
6A negative exponent indicates a reciprocal (x^-a = 1/x^a).

Worked examples

Example 1

Simplify the expression: (2x^3 y^-2)^3 · (3x^-1 y^4)

IApply the Power of a Product Rule and Power of a Power Rule to the first term: (2^3)(x^(3*3))(y^(-2*3)) = 8x^9y^-6.

IIThe expression becomes: 8x^9y^-6 · 3x^-1y^4.

IIIRearrange terms to group coefficients and like bases: (8 · 3) · (x^9 · x^-1) · (y^-6 · y^4).

IVMultiply the coefficients: 8 · 3 = 24.

VApply the Product Rule for Exponents for x: x^9 · x^-1 = x^(9 + (-1)) = x^8.

VIApply the Product Rule for Exponents for y: y^-6 · y^4 = y^(-6 + 4) = y^-2.

VIICombine the simplified terms: 24x^8y^-2.

VIIIApply the Negative Exponent Rule to y^-2: y^-2 = 1/y^2.

9Write the final expression with positive exponents.

Answer

24x^8 / y^2

Remember to apply the exponent to all factors inside the parentheses, including numerical coefficients.

Example 2

A population of 500 bacteria doubles every hour. Write an exponential function to model this growth and find the population after 4 hours.

IIdentify the initial value (a): The initial population is 500, so a = 500.

IIIdentify the growth factor (b): The population 'doubles' every hour, meaning it increases by 100%. So, the growth rate r = 1 (as a decimal). The growth factor b = 1 + r = 1 + 1 = 2.

IIIWrite the exponential function: Substitute a and b into y = ab^x. So, y = 500(2)^x.

IVTo find the population after 4 hours, substitute x = 4 into the function: y = 500(2)^4.

VCalculate the value of 2^4: 2^4 = 2 · 2 · 2 · 2 = 16.

VIMultiply the initial value by the calculated power: y = 500 · 16.

VIIPerform the multiplication.

Answer

The exponential function is y = 500(2)^x. After 4 hours, the population will be 8000 bacteria.

When a quantity 'doubles', the growth factor is 2. If it 'triples', the growth factor is 3.

Example 3

A new car costs $25,000 and depreciates at a rate of 15% per year. Write an exponential function to model the car's value and find its value after 3 years.

IIdentify the initial value (a): The initial cost of the car is $25,000, so a = 25000.

IIIdentify the decay rate (r): The car depreciates at 15% per year, so r = 0.15 (as a decimal).

IIICalculate the decay factor (b): For decay, b = 1 - r = 1 - 0.15 = 0.85.

IVWrite the exponential function: Substitute a and b into y = ab^x. So, y = 25000(0.85)^x.

VTo find the car's value after 3 years, substitute x = 3 into the function: y = 25000(0.85)^3.

VICalculate the value of (0.85)^3: (0.85)^3 = 0.85 · 0.85 · 0.85 ≈ 0.614125.

VIIMultiply the initial value by the calculated power: y = 25000 · 0.614125.

VIIIPerform the multiplication and round to two decimal places for currency.

Answer

The exponential function is y = 25000(0.85)^x. After 3 years, the car's value will be approximately $15,353.13.

Depreciation means the value decreases, so use the decay formula (1 - r).

Common mistakes

✗Confusing exponential functions with linear functions. Exponential functions have the variable in the exponent, while linear functions have a constant rate of change.
✗Incorrectly applying exponent rules, such as adding exponents when multiplying different bases or multiplying exponents when adding/subtracting powers.
✗Misidentifying the initial value ('a') or the growth/decay factor ('b') in word problems, especially confusing the rate 'r' with the factor 'b'.
✗Forgetting that a negative exponent means taking the reciprocal, not making the number negative.
✗Assuming x^0 = 0 instead of x^0 = 1.

Exam tips

★Memorize all the exponent rules and practice applying them to various expressions. Pay close attention to signs and order of operations.
★When solving word problems, clearly identify the initial amount ('a'), the rate of change ('r'), and whether it's growth or decay to correctly determine the base 'b'.
★Always check if your calculated base 'b' makes sense: b > 1 for growth, 0 < b < 1 for decay.
★Show all steps when simplifying expressions with exponents to avoid errors and to receive partial credit if a mistake is made.

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