Exponential Functions - Ninth Grade Math
Introduction to Exponential Functions
Exponential Function: A function where the variable appears in the exponent
General Form: $f(x) = ab^x$ where $a \neq 0$, $b > 0$, and $b \neq 1$
Base (b): The constant being raised to a power
Coefficient (a): The initial value or starting amount
General Form: $f(x) = ab^x$ where $a \neq 0$, $b > 0$, and $b \neq 1$
Base (b): The constant being raised to a power
Coefficient (a): The initial value or starting amount
Standard Forms of Exponential Functions:
1. Basic Form:
$$f(x) = b^x$$
2. General Form:
$$f(x) = ab^x$$
where $a$ = initial value, $b$ = base
3. Transformation Form:
$$f(x) = ab^{x-h} + k$$
where $h$ = horizontal shift, $k$ = vertical shift
1. Basic Form:
$$f(x) = b^x$$
2. General Form:
$$f(x) = ab^x$$
where $a$ = initial value, $b$ = base
3. Transformation Form:
$$f(x) = ab^{x-h} + k$$
where $h$ = horizontal shift, $k$ = vertical shift
Key Properties:
• When $b > 1$: Exponential GROWTH (increases)
• When $0 < b < 1$: Exponential DECAY (decreases)
• The function NEVER touches the x-axis (has a horizontal asymptote)
• Domain: All real numbers
• Range: $(0, \infty)$ for $f(x) = ab^x$ when $a > 0$
• When $b > 1$: Exponential GROWTH (increases)
• When $0 < b < 1$: Exponential DECAY (decreases)
• The function NEVER touches the x-axis (has a horizontal asymptote)
• Domain: All real numbers
• Range: $(0, \infty)$ for $f(x) = ab^x$ when $a > 0$
1. Evaluate an Exponential Function
Evaluating: Finding the output value for a given input
Steps to Evaluate:
Step 1: Identify the function and the input value
Step 2: Substitute the input value for x
Step 3: Calculate the exponent first
Step 4: Multiply by coefficient if present
Step 5: Simplify to get final answer
Step 1: Identify the function and the input value
Step 2: Substitute the input value for x
Step 3: Calculate the exponent first
Step 4: Multiply by coefficient if present
Step 5: Simplify to get final answer
Example 1: Evaluate $f(x) = 2^x$ when $x = 3$
$f(3) = 2^3 = 2 \times 2 \times 2 = 8$
Answer: $f(3) = 8$
$f(3) = 2^3 = 2 \times 2 \times 2 = 8$
Answer: $f(3) = 8$
Example 2: Evaluate $g(x) = 3 \cdot 2^x$ when $x = 4$
$g(4) = 3 \cdot 2^4 = 3 \cdot 16 = 48$
Answer: $g(4) = 48$
$g(4) = 3 \cdot 2^4 = 3 \cdot 16 = 48$
Answer: $g(4) = 48$
Example 3: Evaluate $h(x) = 5 \cdot (0.5)^x$ when $x = 3$
$h(3) = 5 \cdot (0.5)^3 = 5 \cdot 0.125 = 0.625$
Answer: $h(3) = 0.625$
$h(3) = 5 \cdot (0.5)^3 = 5 \cdot 0.125 = 0.625$
Answer: $h(3) = 0.625$
Example 4: Evaluate $f(x) = 100 \cdot 3^x$ when $x = 0$
$f(0) = 100 \cdot 3^0 = 100 \cdot 1 = 100$
Answer: $f(0) = 100$
$f(0) = 100 \cdot 3^0 = 100 \cdot 1 = 100$
Answer: $f(0) = 100$
Example 5: Evaluate $f(x) = 2^{x+1}$ when $x = 2$
$f(2) = 2^{2+1} = 2^3 = 8$
Answer: $f(2) = 8$
$f(2) = 2^{2+1} = 2^3 = 8$
Answer: $f(2) = 8$
Remember:
• Any number to the power of 0 equals 1: $b^0 = 1$
• Negative exponents mean reciprocal: $b^{-x} = \frac{1}{b^x}$
• Use calculator for non-integer exponents
• Any number to the power of 0 equals 1: $b^0 = 1$
• Negative exponents mean reciprocal: $b^{-x} = \frac{1}{b^x}$
• Use calculator for non-integer exponents
2. Graph Exponential Functions
Key Graph Features:
• Y-intercept: $(0, a)$ where $a$ is the coefficient
• Horizontal Asymptote: Usually $y = 0$ (x-axis)
• End Behavior: Depends on whether it's growth or decay
• Y-intercept: $(0, a)$ where $a$ is the coefficient
• Horizontal Asymptote: Usually $y = 0$ (x-axis)
• End Behavior: Depends on whether it's growth or decay
Characteristics of Exponential Graphs:
Growth ($b > 1$):
• Increases rapidly as $x$ increases
• Approaches 0 as $x$ decreases (never touches)
• Curves upward
• Example: $f(x) = 2^x$
Decay ($0 < b < 1$):
• Decreases rapidly as $x$ increases
• Approaches 0 as $x$ increases (never touches)
• Curves downward
• Example: $f(x) = (0.5)^x$
Growth ($b > 1$):
• Increases rapidly as $x$ increases
• Approaches 0 as $x$ decreases (never touches)
• Curves upward
• Example: $f(x) = 2^x$
Decay ($0 < b < 1$):
• Decreases rapidly as $x$ increases
• Approaches 0 as $x$ increases (never touches)
• Curves downward
• Example: $f(x) = (0.5)^x$
Steps to Graph Exponential Function:
Step 1: Create a table of values (choose x-values like -2, -1, 0, 1, 2)
Step 2: Calculate corresponding y-values
Step 3: Plot the points on coordinate plane
Step 4: Draw smooth curve through points
Step 5: Draw horizontal asymptote (usually $y = 0$)
Step 6: Add arrows to show end behavior
Step 1: Create a table of values (choose x-values like -2, -1, 0, 1, 2)
Step 2: Calculate corresponding y-values
Step 3: Plot the points on coordinate plane
Step 4: Draw smooth curve through points
Step 5: Draw horizontal asymptote (usually $y = 0$)
Step 6: Add arrows to show end behavior
Example 1: Graph $f(x) = 2^x$
Table of values:
Key features:
• Y-intercept: $(0, 1)$
• Asymptote: $y = 0$
• Growth function (increases rapidly)
• Passes through $(1, 2)$, $(2, 4)$, $(3, 8)$
Table of values:
| x | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| y | 0.25 | 0.5 | 1 | 2 | 4 | 8 |
Key features:
• Y-intercept: $(0, 1)$
• Asymptote: $y = 0$
• Growth function (increases rapidly)
• Passes through $(1, 2)$, $(2, 4)$, $(3, 8)$
Example 2: Graph $g(x) = 3 \cdot 2^x$
Table of values:
Key features:
• Y-intercept: $(0, 3)$
• Asymptote: $y = 0$
• Steeper than $f(x) = 2^x$ due to coefficient 3
Table of values:
| x | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|
| y | 1.5 | 3 | 6 | 12 | 24 |
Key features:
• Y-intercept: $(0, 3)$
• Asymptote: $y = 0$
• Steeper than $f(x) = 2^x$ due to coefficient 3
Example 3: Graph $h(x) = (0.5)^x$
Table of values:
Key features:
• Y-intercept: $(0, 1)$
• Decay function (decreases)
• Mirror image of $f(x) = 2^x$ across y-axis
Table of values:
| x | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| y | 4 | 2 | 1 | 0.5 | 0.25 | 0.125 |
Key features:
• Y-intercept: $(0, 1)$
• Decay function (decreases)
• Mirror image of $f(x) = 2^x$ across y-axis
3-4. Match Exponential Functions and Graphs
Matching Strategy: Identify key features to connect function with graph
Key Features to Check:
1. Growth vs Decay:
• If $b > 1$: Graph increases (growth)
• If $0 < b < 1$: Graph decreases (decay)
2. Y-intercept:
• Point where graph crosses y-axis
• For $f(x) = ab^x$, y-intercept is $(0, a)$
3. Steepness:
• Larger coefficient $a$ makes graph steeper
• Larger base $b$ makes growth faster
4. Asymptote:
• Usually $y = 0$, but can shift with transformations
• Graph approaches but never touches asymptote
1. Growth vs Decay:
• If $b > 1$: Graph increases (growth)
• If $0 < b < 1$: Graph decreases (decay)
2. Y-intercept:
• Point where graph crosses y-axis
• For $f(x) = ab^x$, y-intercept is $(0, a)$
3. Steepness:
• Larger coefficient $a$ makes graph steeper
• Larger base $b$ makes growth faster
4. Asymptote:
• Usually $y = 0$, but can shift with transformations
• Graph approaches but never touches asymptote
Example 1: Match the function to characteristics
Function: $f(x) = 2 \cdot 3^x$
Characteristics:
• Y-intercept: $(0, 2)$ because $f(0) = 2 \cdot 3^0 = 2$
• Growth function (base 3 > 1)
• Increases rapidly
• Asymptote at $y = 0$
Graph features to look for:
• Starts at $(0, 2)$
• Curves upward to the right
• Approaches x-axis on the left
Function: $f(x) = 2 \cdot 3^x$
Characteristics:
• Y-intercept: $(0, 2)$ because $f(0) = 2 \cdot 3^0 = 2$
• Growth function (base 3 > 1)
• Increases rapidly
• Asymptote at $y = 0$
Graph features to look for:
• Starts at $(0, 2)$
• Curves upward to the right
• Approaches x-axis on the left
Example 2: Identify function from graph
Graph shows:
• Y-intercept at $(0, 5)$
• Decreasing curve
• Passes through $(1, 2.5)$
Analysis:
• $a = 5$ (y-intercept)
• Decay function, so $0 < b < 1$
• From $(0, 5)$ to $(1, 2.5)$: halved, so $b = 0.5$
Function: $f(x) = 5 \cdot (0.5)^x$
Graph shows:
• Y-intercept at $(0, 5)$
• Decreasing curve
• Passes through $(1, 2.5)$
Analysis:
• $a = 5$ (y-intercept)
• Decay function, so $0 < b < 1$
• From $(0, 5)$ to $(1, 2.5)$: halved, so $b = 0.5$
Function: $f(x) = 5 \cdot (0.5)^x$
Quick Identification Tips:
• Upward curve to right = Growth ($b > 1$)
• Downward curve to right = Decay ($0 < b < 1$)
• Y-intercept = value of $a$
• Steeper = larger $a$ or larger $b$
• Upward curve to right = Growth ($b > 1$)
• Downward curve to right = Decay ($0 < b < 1$)
• Y-intercept = value of $a$
• Steeper = larger $a$ or larger $b$
5-6. Domain and Range of Exponential Functions
Domain: Set of all possible input values (x-values)
Range: Set of all possible output values (y-values)
Range: Set of all possible output values (y-values)
For Basic Exponential Function $f(x) = ab^x$ where $a > 0$:
Domain: All real numbers
$$\text{Domain: } (-\infty, \infty)$$
or in set notation: $\{x | x \in \mathbb{R}\}$
Range: All positive real numbers
$$\text{Range: } (0, \infty)$$
or in set notation: $\{y | y > 0\}$
With Vertical Shift (k):
For $f(x) = ab^x + k$:
• Domain: $(-\infty, \infty)$
• Range: $(k, \infty)$ if $a > 0$
• Asymptote: $y = k$
Domain: All real numbers
$$\text{Domain: } (-\infty, \infty)$$
or in set notation: $\{x | x \in \mathbb{R}\}$
Range: All positive real numbers
$$\text{Range: } (0, \infty)$$
or in set notation: $\{y | y > 0\}$
With Vertical Shift (k):
For $f(x) = ab^x + k$:
• Domain: $(-\infty, \infty)$
• Range: $(k, \infty)$ if $a > 0$
• Asymptote: $y = k$
Example 1: Find domain and range of $f(x) = 2^x$
Domain: Can substitute any real number for x
Domain: $(-\infty, \infty)$ or all real numbers
Range: Output is always positive, never 0 or negative
Range: $(0, \infty)$ or $y > 0$
Asymptote: $y = 0$
Domain: Can substitute any real number for x
Domain: $(-\infty, \infty)$ or all real numbers
Range: Output is always positive, never 0 or negative
Range: $(0, \infty)$ or $y > 0$
Asymptote: $y = 0$
Example 2: Find domain and range of $g(x) = 3 \cdot (0.5)^x$
Domain: $(-\infty, \infty)$
Range: $(0, \infty)$ (always positive)
Asymptote: $y = 0$
Domain: $(-\infty, \infty)$
Range: $(0, \infty)$ (always positive)
Asymptote: $y = 0$
Example 3: Find domain and range of $h(x) = 2^x + 3$
Domain: $(-\infty, \infty)$
Range: Graph is shifted up 3 units
Minimum value approaches 3 (never reaches it)
Range: $(3, \infty)$ or $y > 3$
Asymptote: $y = 3$
Domain: $(-\infty, \infty)$
Range: Graph is shifted up 3 units
Minimum value approaches 3 (never reaches it)
Range: $(3, \infty)$ or $y > 3$
Asymptote: $y = 3$
Example 4: From graph, identify domain and range
Graph shows:
• Curve extends infinitely left and right (horizontally)
• Curve extends upward from asymptote at $y = -2$
Domain: $(-\infty, \infty)$
Range: $(-2, \infty)$ or $y > -2$
Graph shows:
• Curve extends infinitely left and right (horizontally)
• Curve extends upward from asymptote at $y = -2$
Domain: $(-\infty, \infty)$
Range: $(-2, \infty)$ or $y > -2$
7. Write Exponential Functions from Tables and Graphs
Goal: Determine the equation $f(x) = ab^x$ from given data
Steps to Find Exponential Function from Table:
Step 1: Identify the initial value $a$ (when $x = 0$)
Step 2: Find the common ratio $b$ by dividing consecutive y-values
$$b = \frac{y_2}{y_1} = \frac{y_3}{y_2}$$
Step 3: Write the function: $f(x) = ab^x$
Step 4: Verify with other points
Step 1: Identify the initial value $a$ (when $x = 0$)
Step 2: Find the common ratio $b$ by dividing consecutive y-values
$$b = \frac{y_2}{y_1} = \frac{y_3}{y_2}$$
Step 3: Write the function: $f(x) = ab^x$
Step 4: Verify with other points
Example 1: Write function from table
Step 1: Initial value: $a = 5$ (when $x = 0$)
Step 2: Common ratio:
$b = \frac{10}{5} = 2$
$\frac{20}{10} = 2$ ✓
$\frac{40}{20} = 2$ ✓
Answer: $f(x) = 5 \cdot 2^x$
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y | 5 | 10 | 20 | 40 |
Step 1: Initial value: $a = 5$ (when $x = 0$)
Step 2: Common ratio:
$b = \frac{10}{5} = 2$
$\frac{20}{10} = 2$ ✓
$\frac{40}{20} = 2$ ✓
Answer: $f(x) = 5 \cdot 2^x$
Example 2: Write function from table
Initial value: $a = 100$
Common ratio: $b = \frac{50}{100} = 0.5$
Answer: $f(x) = 100 \cdot (0.5)^x$
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y | 100 | 50 | 25 | 12.5 |
Initial value: $a = 100$
Common ratio: $b = \frac{50}{100} = 0.5$
Answer: $f(x) = 100 \cdot (0.5)^x$
Example 3: Write function from two points: $(0, 3)$ and $(2, 12)$
From $(0, 3)$: $a = 3$
Use $(2, 12)$:
$12 = 3 \cdot b^2$
$4 = b^2$
$b = 2$
Answer: $f(x) = 3 \cdot 2^x$
From $(0, 3)$: $a = 3$
Use $(2, 12)$:
$12 = 3 \cdot b^2$
$4 = b^2$
$b = 2$
Answer: $f(x) = 3 \cdot 2^x$
From Graph:
Step 1: Find y-intercept to get $a$
Step 2: Find another clear point
Step 3: Use point to solve for $b$
Step 4: Write function and verify
Step 1: Find y-intercept to get $a$
Step 2: Find another clear point
Step 3: Use point to solve for $b$
Step 4: Write function and verify
8. Write Exponential Functions: Word Problems
Key Words:
• Growth: increases, doubles, triples, appreciates
• Decay: decreases, halves, depreciates, decays
• Initial value: starting amount, original, begins with
• Rate: percent increase/decrease per time period
• Growth: increases, doubles, triples, appreciates
• Decay: decreases, halves, depreciates, decays
• Initial value: starting amount, original, begins with
• Rate: percent increase/decrease per time period
Example 1: A population of bacteria starts at 100 and doubles every hour. Write a function.
Initial value: $a = 100$
Doubles each hour: $b = 2$
Variable: $x$ = number of hours
Answer: $P(x) = 100 \cdot 2^x$
Initial value: $a = 100$
Doubles each hour: $b = 2$
Variable: $x$ = number of hours
Answer: $P(x) = 100 \cdot 2^x$
Example 2: A car worth $25,000 depreciates by 15% each year. Write a function.
Initial value: $a = 25000$
Decreases 15%: Keeps 85% = 0.85
So $b = 0.85$
Answer: $V(t) = 25000 \cdot (0.85)^t$
where $t$ = years
Initial value: $a = 25000$
Decreases 15%: Keeps 85% = 0.85
So $b = 0.85$
Answer: $V(t) = 25000 \cdot (0.85)^t$
where $t$ = years
Example 3: An investment of $1000 triples every 5 years. Write function for amount after x years.
Initial: $a = 1000$
Triples every 5 years:
After 5 years: $1000 \cdot 3$
So base for 1 year: $b = 3^{1/5}$
Answer: $A(x) = 1000 \cdot 3^{x/5}$
Initial: $a = 1000$
Triples every 5 years:
After 5 years: $1000 \cdot 3$
So base for 1 year: $b = 3^{1/5}$
Answer: $A(x) = 1000 \cdot 3^{x/5}$
9. Exponential Growth and Decay: Word Problems
Exponential Growth Formula:
$$A = A_0(1 + r)^t$$
Exponential Decay Formula:
$$A = A_0(1 - r)^t$$
where:
• $A$ = final amount
• $A_0$ = initial amount
• $r$ = rate of growth/decay (as decimal)
• $t$ = time
$$A = A_0(1 + r)^t$$
Exponential Decay Formula:
$$A = A_0(1 - r)^t$$
where:
• $A$ = final amount
• $A_0$ = initial amount
• $r$ = rate of growth/decay (as decimal)
• $t$ = time
Converting Percentage to Decimal:
• 5% = 0.05
• 12% = 0.12
• 100% = 1.00
For Growth: Multiply by $(1 + r)$
• 20% growth → $1 + 0.20 = 1.20$
For Decay: Multiply by $(1 - r)$
• 30% decay → $1 - 0.30 = 0.70$
• 5% = 0.05
• 12% = 0.12
• 100% = 1.00
For Growth: Multiply by $(1 + r)$
• 20% growth → $1 + 0.20 = 1.20$
For Decay: Multiply by $(1 - r)$
• 30% decay → $1 - 0.30 = 0.70$
Example 1: A city's population is 50,000 and grows 3% per year. What will the population be in 10 years?
Given:
$A_0 = 50000$
$r = 0.03$ (3% growth)
$t = 10$ years
Formula:
$A = 50000(1 + 0.03)^{10}$
$A = 50000(1.03)^{10}$
$A = 50000(1.3439...)$
$A \approx 67,195$
Answer: About 67,195 people
Given:
$A_0 = 50000$
$r = 0.03$ (3% growth)
$t = 10$ years
Formula:
$A = 50000(1 + 0.03)^{10}$
$A = 50000(1.03)^{10}$
$A = 50000(1.3439...)$
$A \approx 67,195$
Answer: About 67,195 people
Example 2: A radioactive substance has 200 grams and decays at 5% per hour. How much remains after 8 hours?
Given:
$A_0 = 200$ grams
$r = 0.05$ (5% decay)
$t = 8$ hours
Formula:
$A = 200(1 - 0.05)^8$
$A = 200(0.95)^8$
$A = 200(0.6634...)$
$A \approx 132.7$ grams
Answer: About 132.7 grams
Given:
$A_0 = 200$ grams
$r = 0.05$ (5% decay)
$t = 8$ hours
Formula:
$A = 200(1 - 0.05)^8$
$A = 200(0.95)^8$
$A = 200(0.6634...)$
$A \approx 132.7$ grams
Answer: About 132.7 grams
Example 3: A car bought for $30,000 depreciates 12% each year. What is its value after 5 years?
$A = 30000(1 - 0.12)^5$
$A = 30000(0.88)^5$
$A = 30000(0.5277...)$
$A \approx 15,832$
Answer: About $15,832
$A = 30000(1 - 0.12)^5$
$A = 30000(0.88)^5$
$A = 30000(0.5277...)$
$A \approx 15,832$
Answer: About $15,832
Example 4: An investment of $5000 grows 8% annually. How much after 15 years?
$A = 5000(1 + 0.08)^{15}$
$A = 5000(1.08)^{15}$
$A = 5000(3.1722...)$
$A \approx 15,861$
Answer: About $15,861
$A = 5000(1 + 0.08)^{15}$
$A = 5000(1.08)^{15}$
$A = 5000(3.1722...)$
$A \approx 15,861$
Answer: About $15,861
10. Compound Interest: Word Problems
Compound Interest: Interest calculated on initial principal AND accumulated interest
Principal (P): Initial amount invested or borrowed
Rate (r): Annual interest rate (as decimal)
Time (t): Number of years
Compounding Frequency (n): Number of times interest compounds per year
Principal (P): Initial amount invested or borrowed
Rate (r): Annual interest rate (as decimal)
Time (t): Number of years
Compounding Frequency (n): Number of times interest compounds per year
Compound Interest Formula:
$$A = P\left(1 + \frac{r}{n}\right)^{nt}$$
where:
• $A$ = final amount
• $P$ = principal (initial amount)
• $r$ = annual interest rate (as decimal)
• $n$ = number of times compounded per year
• $t$ = time in years
Interest Earned:
$$I = A - P$$
$$A = P\left(1 + \frac{r}{n}\right)^{nt}$$
where:
• $A$ = final amount
• $P$ = principal (initial amount)
• $r$ = annual interest rate (as decimal)
• $n$ = number of times compounded per year
• $t$ = time in years
Interest Earned:
$$I = A - P$$
Common Compounding Frequencies:
• Annually: $n = 1$
• Semi-annually (twice per year): $n = 2$
• Quarterly (4 times per year): $n = 4$
• Monthly: $n = 12$
• Weekly: $n = 52$
• Daily: $n = 365$
• Annually: $n = 1$
• Semi-annually (twice per year): $n = 2$
• Quarterly (4 times per year): $n = 4$
• Monthly: $n = 12$
• Weekly: $n = 52$
• Daily: $n = 365$
Example 1: $2000 is invested at 6% annual interest, compounded annually. Find amount after 10 years.
Given:
$P = 2000$
$r = 0.06$
$n = 1$ (annually)
$t = 10$
Solution:
$A = 2000\left(1 + \frac{0.06}{1}\right)^{1 \cdot 10}$
$A = 2000(1.06)^{10}$
$A = 2000(1.7908...)$
$A \approx 3581.70$
Answer: $3,581.70
Given:
$P = 2000$
$r = 0.06$
$n = 1$ (annually)
$t = 10$
Solution:
$A = 2000\left(1 + \frac{0.06}{1}\right)^{1 \cdot 10}$
$A = 2000(1.06)^{10}$
$A = 2000(1.7908...)$
$A \approx 3581.70$
Answer: $3,581.70
Example 2: $5000 is invested at 4% annual interest, compounded quarterly. Find amount after 5 years.
Given:
$P = 5000$
$r = 0.04$
$n = 4$ (quarterly)
$t = 5$
Solution:
$A = 5000\left(1 + \frac{0.04}{4}\right)^{4 \cdot 5}$
$A = 5000(1 + 0.01)^{20}$
$A = 5000(1.01)^{20}$
$A = 5000(1.2202...)$
$A \approx 6,101$
Answer: $6,101
Given:
$P = 5000$
$r = 0.04$
$n = 4$ (quarterly)
$t = 5$
Solution:
$A = 5000\left(1 + \frac{0.04}{4}\right)^{4 \cdot 5}$
$A = 5000(1 + 0.01)^{20}$
$A = 5000(1.01)^{20}$
$A = 5000(1.2202...)$
$A \approx 6,101$
Answer: $6,101
Example 3: $10,000 invested at 3% compounded monthly for 8 years. Find total and interest earned.
Given: $P = 10000$, $r = 0.03$, $n = 12$, $t = 8$
$A = 10000\left(1 + \frac{0.03}{12}\right)^{12 \cdot 8}$
$A = 10000(1.0025)^{96}$
$A = 10000(1.2702...)$
$A \approx 12,702$
Interest earned:
$I = 12702 - 10000 = 2,702$
Answer: Total = $12,702, Interest = $2,702
Given: $P = 10000$, $r = 0.03$, $n = 12$, $t = 8$
$A = 10000\left(1 + \frac{0.03}{12}\right)^{12 \cdot 8}$
$A = 10000(1.0025)^{96}$
$A = 10000(1.2702...)$
$A \approx 12,702$
Interest earned:
$I = 12702 - 10000 = 2,702$
Answer: Total = $12,702, Interest = $2,702
Example 4: Compare: $1000 at 5% compounded annually vs. monthly for 10 years.
Annually ($n = 1$):
$A = 1000(1.05)^{10} = 1000(1.6289) \approx 1,628.89$
Monthly ($n = 12$):
$A = 1000(1 + \frac{0.05}{12})^{120} = 1000(1.0042)^{120} \approx 1,647.01$
Difference: $1647.01 - 1628.89 = $18.12$
Answer: Monthly compounding earns $18.12 more
Annually ($n = 1$):
$A = 1000(1.05)^{10} = 1000(1.6289) \approx 1,628.89$
Monthly ($n = 12$):
$A = 1000(1 + \frac{0.05}{12})^{120} = 1000(1.0042)^{120} \approx 1,647.01$
Difference: $1647.01 - 1628.89 = $18.12$
Answer: Monthly compounding earns $18.12 more
Summary: Key Formulas
| Type | Formula | When to Use |
|---|---|---|
| Basic Exponential | $f(x) = ab^x$ | General exponential function |
| Exponential Growth | $A = A_0(1 + r)^t$ | Population, appreciation, increase |
| Exponential Decay | $A = A_0(1 - r)^t$ | Depreciation, radioactive decay, decrease |
| Compound Interest | $A = P\left(1 + \frac{r}{n}\right)^{nt}$ | Investments, savings accounts, loans |
Growth vs Decay Comparison
| Feature | Exponential Growth | Exponential Decay |
|---|---|---|
| Base (b) | $b > 1$ | $0 < b < 1$ |
| Direction | Increases | Decreases |
| Graph Shape | Curves upward | Curves downward |
| Example Base | $b = 2, 3, 1.5$ | $b = 0.5, 0.8, 0.25$ |
| Real-World | Population growth, investments | Radioactive decay, depreciation |
| With Rate (r) | $b = 1 + r$ (where $r > 0$) | $b = 1 - r$ (where $0 < r < 1$) |
Quick Reference Guide
Evaluating Exponential Functions:
• Substitute x-value into function
• Calculate exponent first
• Multiply by coefficient
• Remember: $b^0 = 1$ always
• Substitute x-value into function
• Calculate exponent first
• Multiply by coefficient
• Remember: $b^0 = 1$ always
Graphing Key Points:
• Y-intercept: $(0, a)$
• Asymptote: usually $y = 0$
• Growth: increases to right
• Decay: decreases to right
• Never crosses asymptote
• Y-intercept: $(0, a)$
• Asymptote: usually $y = 0$
• Growth: increases to right
• Decay: decreases to right
• Never crosses asymptote
Domain and Range (Standard):
• Domain: Always all real numbers $(-\infty, \infty)$
• Range: $(0, \infty)$ for $f(x) = ab^x$ with $a > 0$
• Range shifts with vertical translations
• Domain: Always all real numbers $(-\infty, \infty)$
• Range: $(0, \infty)$ for $f(x) = ab^x$ with $a > 0$
• Range shifts with vertical translations
Word Problem Strategy:
1. Identify initial value ($a$ or $P$)
2. Determine if growth or decay
3. Find rate and convert to decimal
4. Identify time variable
5. Choose appropriate formula
6. Substitute and solve
1. Identify initial value ($a$ or $P$)
2. Determine if growth or decay
3. Find rate and convert to decimal
4. Identify time variable
5. Choose appropriate formula
6. Substitute and solve
Success Tips for Exponential Functions:
✓ Remember: variable is in the EXPONENT
✓ Growth when $b > 1$, decay when $0 < b < 1$
✓ Domain is always all real numbers
✓ Range is always positive (unless shifted)
✓ Y-intercept is the coefficient $a$
✓ For growth: use $(1 + r)$; for decay: use $(1 - r)$
✓ Convert percentages to decimals (divide by 100)
✓ Compound interest: remember to divide rate by $n$
✓ Graph never touches the asymptote
✓ Use calculator for non-integer exponents
✓ Remember: variable is in the EXPONENT
✓ Growth when $b > 1$, decay when $0 < b < 1$
✓ Domain is always all real numbers
✓ Range is always positive (unless shifted)
✓ Y-intercept is the coefficient $a$
✓ For growth: use $(1 + r)$; for decay: use $(1 - r)$
✓ Convert percentages to decimals (divide by 100)
✓ Compound interest: remember to divide rate by $n$
✓ Graph never touches the asymptote
✓ Use calculator for non-integer exponents