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
Rational Exponents
aᵐ/ᵖ <----------> ᵖ√ aᵐ
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
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
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
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)
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
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
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
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}
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
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
Rational Exponents
aᵐ/ᵖ <----------> ᵖ√ aᵐ
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
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
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
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)
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
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
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
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}
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