Exponential Functions
Exponential Functions
Identifying Exponential Functions We have learned to solve for x in many different types of equations.
4 x 20 x3 0 x 8 x 8 7 2 x 81
x5 x –3 x 64 x 56 x 9
Identifying Exponential Functions
In exponential functions the variable (x) is the exponent.
Exponent
yb
x
Base
b must be greater than 0.
Graphing Exponential Functions The graph below shows the equations y = 2x and y = 3x.
y = 2x y = 3x
The red line along the x axis is the horizontal asymptote. The graphed equations will never intersect the asymptote.
Graphing Exponential Functions The graph below shows the equations y = 2x and y = 3x.
y = 2x y = 3x
The graphed equations will get exponentially closer to, and appear to touch, the asymptote but they will NEVER intersect the asymptote.
Graphing Exponential Functions The graph below shows the equations y = 0.1x and y = 0.5x.
y = 0.1x y = 0.5x
The horizontal asymptote is still along the x axis. As before, the graphed equations will NEVER intersect the asymptote.
Graphing Exponential Functions Exponential functions with bases > 1 are increasing functions.
y = 0.1x
y = 0.5x
y = 2x
y = 3x
Graphing Exponential Functions With increasing functions, larger bases (> 1) indicate a faster increase.
y = 0.1x
y = 0.5x
y = 2x
y = 3x
Graphing Exponential Functions Exponential functions with bases > 0 but < 1 are decreasing functions.
y = 0.1x
y = 0.5x
y = 2x
y = 3x
Graphing Exponential Functions With decreasing functions, bases (> 0, < 1) closer to 0 indicate a faster decrease.
y = 0.1x
y = 0.5x
y = 2x
y = 3x
Applications of Exponential Functions Exponential functions are used in many applications.
Population
Radioactive Decay Compound Interest
P Po e
kt
N Noe
t
i S a 1 n
nt
Applications of Exponential Functions Exponential functions are used in many applications.
Population
P Po e
kt
e is a constant NOT a variable e is Euler’s number It is an irrational number equal to… 2.718281828… or
1 1 1 2 ... 2! 3! 4!
Evaluating Exponential Functions Exponential functions can be used to calculate compounding interest.
A total of $1250 is deposited with a compounding interest rate of 4.6%. If interest is compounded monthly, what will the total amount accumulate to in 3 years?
Evaluating Exponential Functions Step 1: Locate and input the values for variables a, i, n and t.
A total of $1250 is deposited with a compounding interest rate of 4.6%. If interest is compounded monthly, what will the total amount accumulate to in 3 years?
i S = a 1 n
nt
a = initial amount (principal) i = interest rate (decimal) n = number of compounding periods per year t = time in years
Evaluating Exponential Functions Step 1: Locate and input the values for variables a, i, n and t.
A total of $1250 is deposited with a compounding interest rate of 4.6%. If interest is compounded monthly, what will the total amount accumulate to in 3 years?
i S = a 1 n
nt
a = initial amount (principal) i = interest rate (decimal) n = number of compounding periods per year t = time in years
Evaluating Exponential Functions Step 1: Locate and input the values for variables a, i, n and t.
A total of $1250 is deposited with a compounding interest rate of 4.6%. If interest is compounded monthly, what will the total amount accumulate to in 3 years?
(12)(3)
0.046 S = (1250) 1 12
a = initial amount (principal) i = interest rate (decimal) n = number of compounding periods per year t = time in years
Evaluating Exponential Functions Step 2: Evaluate the function.
A total of $1250 is deposited with a compounding interest rate of 4.6%. If interest is compounded monthly, what will the total amount accumulate to in 3 years?
(12)(3)
0.046 S = (1250) 1 12
a = initial amount (principal) i = interest rate (decimal) n = number of compounding periods per year t = time in years
Evaluating Exponential Functions Step 2: Evaluate the function.
(0.046) S (1250)1 (12)
(12 )( 3)
S (1250)1 0.003833...
36
S (1250)1.003833...
36
S (1250)1.147672...
S 1434.590905... $1434.59 The total amount accumulated in 3 years would be $1434.59.
Evaluating Exponential Functions Continuous exponential growth can be calculated using a formula.
2000 units grow at an exponential rate of 3.5% a day for 19 days. What is the final amount at the end of this period?
Evaluating Exponential Functions Step 1: Locate and input the values for variables a, n and t.
2000 units grow at an exponential rate of 3.5% a day for 19 days. What is the final amount at the end of this period?
a = initial amount (principal)
y = aent
n = rate of growth t = time period
Evaluating Exponential Functions Step 1: Locate and input the values for variables a, n and t.
2000 units grow at an exponential rate of 3.5% a day for 19 days. What is the final amount at the end of this period?
a = initial amount (principal)
y = aent
n = rate of growth t = time period
Evaluating Exponential Functions Step 1: Locate and input the values for variables a, n and t.
2000 units grow at an exponential rate of 3.5% a day for 19 days. What is the final amount at the end of this period?
a = initial amount (principal)
y = (2000)e(0.035)(19)
n = rate of growth t = time period
Evaluating Exponential Functions Step 2: Evaluate the function.
2000 units grow at an exponential rate of 3.5% a day for 19 days. What is the final amount at the end of this period?
a = initial amount (principal)
y = (2000)e(0.035)(19)
n = rate of growth t = time period
Evaluating Exponential Functions Step 2: Evaluate the function.
y = (2000)e(0.035)(19) y = (2000)e(0.665) y = (2000)(2.718281…)(0.665)
y = (2000)(1.944490…)
Remember, e is Euler’s number – a known constant
y = 3888.981042… = 3889 The final amount of units after 19 days is 3889.
Evaluating Exponential Functions Continuous exponential decay proportional to the present amount can be calculated using a formula.
A population of bacteria colonies is exposed to antibiotics and is decreasing at a rate of 8% an hour. What is the final number of colonies after 1 day if there was 271 colonies to begin with?
Evaluating Exponential Functions Step 1: Locate and input the values for variables a, n and t.
A population of bacteria colonies is exposed to antibiotics and is decreasing at a rate of 8% an hour. What is the final number of colonies after 1 day if there was 271 colonies to begin with?
a = initial amount (principal)
y = ae –nt
n = rate of decay t = time period
Evaluating Exponential Functions Step 1: Locate and input the values for variables a, n and t.
A population of bacteria colonies is exposed to antibiotics and is decreasing at a rate of 8% an hour. What is the final number of colonies after 1 day if there was 271 colonies to begin with?
a = initial amount (principal)
y = ae –nt
n = rate of decay t = time period
Evaluating Exponential Functions Step 1: Locate and input the values for variables a, n and t.
A population of bacteria colonies is exposed to antibiotics and is decreasing at a rate of 8% an hour. What is the final number of colonies after 1 day if there was 271 colonies to begin with?
a = initial amount (principal)
y = (271)e –(0.08)(24)
n = rate of decay t = time period
Note: ‘1 day’ becomes ‘24 hours’ because n and t must be in the same unit.
Evaluating Exponential Functions Step 2: Evaluate the function.
A population of bacteria colonies is exposed to antibiotics and is decreasing at a rate of 8% an hour. What is the final number of colonies after 1 day if there was 271 colonies to begin with?
a = initial amount (principal)
y = (271)e –(0.08)(24)
n = rate of decay t = time period
Note: ‘1 day’ becomes ‘24 hours’ because n and t must be in the same unit.
Evaluating Exponential Functions Step 2: Evaluate the function.
y = (271)e –(0.08)(24) y = (271)e –(1.92)
y = (271)(2.718281…) –(1.92) y = (271)(0.146606…)
Remember, e is Euler’s number – a known constant
y = 39.730486… = 40 The final number of colonies after 1 day is 40.
Evaluate the following equation:
𝒚 = 𝟏𝟖𝒙
6 x 4
x 0.7
Identifying Exponential Functions We have learned to solve for x in many different types of equations.
4 x 20 x3 0 x 8 x 8 7 2 x 81
x5 x –3 x 64 x 56 x 9
Identifying Exponential Functions
In exponential functions the variable (x) is the exponent.
Exponent
yb
x
Base
b must be greater than 0.
Graphing Exponential Functions The graph below shows the equations y = 2x and y = 3x.
y = 2x y = 3x
The red line along the x axis is the horizontal asymptote. The graphed equations will never intersect the asymptote.
Graphing Exponential Functions The graph below shows the equations y = 2x and y = 3x.
y = 2x y = 3x
The graphed equations will get exponentially closer to, and appear to touch, the asymptote but they will NEVER intersect the asymptote.
Graphing Exponential Functions The graph below shows the equations y = 0.1x and y = 0.5x.
y = 0.1x y = 0.5x
The horizontal asymptote is still along the x axis. As before, the graphed equations will NEVER intersect the asymptote.
Graphing Exponential Functions Exponential functions with bases > 1 are increasing functions.
y = 0.1x
y = 0.5x
y = 2x
y = 3x
Graphing Exponential Functions With increasing functions, larger bases (> 1) indicate a faster increase.
y = 0.1x
y = 0.5x
y = 2x
y = 3x
Graphing Exponential Functions Exponential functions with bases > 0 but < 1 are decreasing functions.
y = 0.1x
y = 0.5x
y = 2x
y = 3x
Graphing Exponential Functions With decreasing functions, bases (> 0, < 1) closer to 0 indicate a faster decrease.
y = 0.1x
y = 0.5x
y = 2x
y = 3x
Applications of Exponential Functions Exponential functions are used in many applications.
Population
Radioactive Decay Compound Interest
P Po e
kt
N Noe
t
i S a 1 n
nt
Applications of Exponential Functions Exponential functions are used in many applications.
Population
P Po e
kt
e is a constant NOT a variable e is Euler’s number It is an irrational number equal to… 2.718281828… or
1 1 1 2 ... 2! 3! 4!
Evaluating Exponential Functions Exponential functions can be used to calculate compounding interest.
A total of $1250 is deposited with a compounding interest rate of 4.6%. If interest is compounded monthly, what will the total amount accumulate to in 3 years?
Evaluating Exponential Functions Step 1: Locate and input the values for variables a, i, n and t.
A total of $1250 is deposited with a compounding interest rate of 4.6%. If interest is compounded monthly, what will the total amount accumulate to in 3 years?
i S = a 1 n
nt
a = initial amount (principal) i = interest rate (decimal) n = number of compounding periods per year t = time in years
Evaluating Exponential Functions Step 1: Locate and input the values for variables a, i, n and t.
A total of $1250 is deposited with a compounding interest rate of 4.6%. If interest is compounded monthly, what will the total amount accumulate to in 3 years?
i S = a 1 n
nt
a = initial amount (principal) i = interest rate (decimal) n = number of compounding periods per year t = time in years
Evaluating Exponential Functions Step 1: Locate and input the values for variables a, i, n and t.
A total of $1250 is deposited with a compounding interest rate of 4.6%. If interest is compounded monthly, what will the total amount accumulate to in 3 years?
(12)(3)
0.046 S = (1250) 1 12
a = initial amount (principal) i = interest rate (decimal) n = number of compounding periods per year t = time in years
Evaluating Exponential Functions Step 2: Evaluate the function.
A total of $1250 is deposited with a compounding interest rate of 4.6%. If interest is compounded monthly, what will the total amount accumulate to in 3 years?
(12)(3)
0.046 S = (1250) 1 12
a = initial amount (principal) i = interest rate (decimal) n = number of compounding periods per year t = time in years
Evaluating Exponential Functions Step 2: Evaluate the function.
(0.046) S (1250)1 (12)
(12 )( 3)
S (1250)1 0.003833...
36
S (1250)1.003833...
36
S (1250)1.147672...
S 1434.590905... $1434.59 The total amount accumulated in 3 years would be $1434.59.
Evaluating Exponential Functions Continuous exponential growth can be calculated using a formula.
2000 units grow at an exponential rate of 3.5% a day for 19 days. What is the final amount at the end of this period?
Evaluating Exponential Functions Step 1: Locate and input the values for variables a, n and t.
2000 units grow at an exponential rate of 3.5% a day for 19 days. What is the final amount at the end of this period?
a = initial amount (principal)
y = aent
n = rate of growth t = time period
Evaluating Exponential Functions Step 1: Locate and input the values for variables a, n and t.
2000 units grow at an exponential rate of 3.5% a day for 19 days. What is the final amount at the end of this period?
a = initial amount (principal)
y = aent
n = rate of growth t = time period
Evaluating Exponential Functions Step 1: Locate and input the values for variables a, n and t.
2000 units grow at an exponential rate of 3.5% a day for 19 days. What is the final amount at the end of this period?
a = initial amount (principal)
y = (2000)e(0.035)(19)
n = rate of growth t = time period
Evaluating Exponential Functions Step 2: Evaluate the function.
2000 units grow at an exponential rate of 3.5% a day for 19 days. What is the final amount at the end of this period?
a = initial amount (principal)
y = (2000)e(0.035)(19)
n = rate of growth t = time period
Evaluating Exponential Functions Step 2: Evaluate the function.
y = (2000)e(0.035)(19) y = (2000)e(0.665) y = (2000)(2.718281…)(0.665)
y = (2000)(1.944490…)
Remember, e is Euler’s number – a known constant
y = 3888.981042… = 3889 The final amount of units after 19 days is 3889.
Evaluating Exponential Functions Continuous exponential decay proportional to the present amount can be calculated using a formula.
A population of bacteria colonies is exposed to antibiotics and is decreasing at a rate of 8% an hour. What is the final number of colonies after 1 day if there was 271 colonies to begin with?
Evaluating Exponential Functions Step 1: Locate and input the values for variables a, n and t.
A population of bacteria colonies is exposed to antibiotics and is decreasing at a rate of 8% an hour. What is the final number of colonies after 1 day if there was 271 colonies to begin with?
a = initial amount (principal)
y = ae –nt
n = rate of decay t = time period
Evaluating Exponential Functions Step 1: Locate and input the values for variables a, n and t.
A population of bacteria colonies is exposed to antibiotics and is decreasing at a rate of 8% an hour. What is the final number of colonies after 1 day if there was 271 colonies to begin with?
a = initial amount (principal)
y = ae –nt
n = rate of decay t = time period
Evaluating Exponential Functions Step 1: Locate and input the values for variables a, n and t.
A population of bacteria colonies is exposed to antibiotics and is decreasing at a rate of 8% an hour. What is the final number of colonies after 1 day if there was 271 colonies to begin with?
a = initial amount (principal)
y = (271)e –(0.08)(24)
n = rate of decay t = time period
Note: ‘1 day’ becomes ‘24 hours’ because n and t must be in the same unit.
Evaluating Exponential Functions Step 2: Evaluate the function.
A population of bacteria colonies is exposed to antibiotics and is decreasing at a rate of 8% an hour. What is the final number of colonies after 1 day if there was 271 colonies to begin with?
a = initial amount (principal)
y = (271)e –(0.08)(24)
n = rate of decay t = time period
Note: ‘1 day’ becomes ‘24 hours’ because n and t must be in the same unit.
Evaluating Exponential Functions Step 2: Evaluate the function.
y = (271)e –(0.08)(24) y = (271)e –(1.92)
y = (271)(2.718281…) –(1.92) y = (271)(0.146606…)
Remember, e is Euler’s number – a known constant
y = 39.730486… = 40 The final number of colonies after 1 day is 40.
Evaluate the following equation:
𝒚 = 𝟏𝟖𝒙
6 x 4
x 0.7
