04: Exponential Functions

Base: xᵐ

Exponent: xᵐ

Power: xᵐ

Exponent Rules

• xᵐ • xᵖ = xᵐ ⁺ ᵖ
• xᵐ ÷ xᵖ = xᵐ ⁺ ᵖ
• x⁰ = 1
• (xᵐ )ᵖ = xᵐᵖ
• x⁻ᵐ = 1/xᵐ
  • x⁻ᵐ / y⁻ᵐ = yᵐ / xᵐ
  • (x/y)⁻ᵐ = (y/x)ᵐ

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Rational Exponents

aᵐ/ᵖ

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If a has a negative fraction as an exponent: it stays with the m as a’s exponent, p (what we are “rooting” by) cannot be negative

• Make sure that you don’t put a 2 in for p, the 2 is assumed, and thus you put nothing extra there

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Simplifying Expressions Involving Exponents

• The exponent affects everything that is inside the brackets if it is outside of the brackets and if there is more than one term (it affects every term)
• You must convert radicals into fraction exponents to work with them
• Distributive property must be done sometimes

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Solving Exponential Equations

• To solve: express both sides as powers of the same base and then drop the bases
• You may use power of a power law here if needed: (xᵐ )ᵖ = xᵐᵖ

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Let a Represent

Sometimes you cannot just drop the bases

• Break things up as much as you can until left with one usable variable exponent (can be squared and/or have another number multiplied with it)
• Let a represent this usable value
• You will be left with something you can either factor or use quadratic formula for
• Solve for a and substitute back what you put in a for
• Solve (drop bases and get a final number)

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Exponential Functions in the Form y=bˣ

The parent function of exponential functions is:

y = bˣ

b is greater than 1

• As b gets larger, the curve moves closer to the axes
• D: {x E R}
• R: {y E R / 0 < y} because zero is the asymptote
• Increasing function
• Equation of asymptote: y=0
• y intercept: 1

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b is less than 1, but is greater than zero → b is a fraction

• Reflection in the y axis (it is flipped)
• As b gets larger, the curve moves away from the axes
• D: {x E R}
• R: {y E R / 0 < y}
• Decreasing function
• Equation of the asymptote: y=0
• y intercept: 1

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Transformations of Exponential Functions

Parent function: y=bˣ in the exponential function in the form:

g(x)=abᵏ⁽ˣ⁻ᵈ⁾ + c

• c is the asymptote
• Domain is always {x E R}
• Range depends on the location of the horizontal asymptote
• If it is below the horizontal asymptote, you have a reflection in the x axis

Mapping

• The number directly attached to the exponent of x (b) is part of the parent function and so it is taken for mapping notation → y values are found by plugging in the chosen x independent values into the parent function as the exponent of b
• If transformed, use the same @@((1/k)x + d, ay + c)@@ and transform your parent function as needed

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• Finding y intercept: Substitute zero into the equation in place of x

• Transformed parent function’s range (if it isn’t a parent function): {y E R / ==c== < y}

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Applications of Exponential Functions

In word problems…

• f(x): final amount
• a: initial amount
• b: growth/decay factor
• x: time/growth period

Exponential Growth

f(x) = abˣ

→ %%b = 1+ r%%

r is rate as a decimal

Exponential Decay

f(x) = abˣ

→ %%b = 1- r%%

r is rate as a decimal

Half Life

f(x) = a(1/2)ᵗ/ᴴ

→ b is always 1/2 for half-life

→ exponent of x becomes t/H

• t: total time
• H: half life

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