C2,C3

Exponential Functions

Definition and Identification

An exponential function is a function of the form $f(x) = a imes b^x$, where $a$ and $b$ are constants, $a \neq 0$, $b > 0$, and $b \neq 1$. Exponential functions are characterized by a constant ratio between successive outputs for equally spaced inputs.

• Key Properties:

  • The base $b$ determines growth ($b > 1$) or decay ($0 < b < 1$).

  • The initial value is $a$.

  • Exponential functions differ from linear functions, which have a constant difference between outputs.

• Example: $f(x) = 3 \times 2^x$ is exponential with $a = 3$, $b = 2$.

Recognizing Exponential Functions

Given a function, determine if it is exponential by checking if it can be written in the form $a \times b^x$.

• Examples:

  • $f(x) = 2^x$ → Yes, $a = 1$, $b = 2$

  • $f(x) = x^2$ → No, not exponential (polynomial)

  • $f(x) = 5 \times (1.2)^x$ → Yes, $a = 5$, $b = 1.2$

  • $f(x) = 3^x + 2$ → No, not in standard exponential form

Graphs of Exponential Functions

Basic Shapes: Growth and Decay

The graph of $y = a b^x$ has two basic shapes depending on the value of $b$:

• Exponential Growth: $b > 1$

• Exponential Decay: $0 < b < 1$

Type	Shape	Domain	Range	Horizontal Asymptote
Growth ($b > 1$)	Increasing curve	$(-\infty, \infty)$	$(0, \infty)$	$y = 0$
Decay ($0 < b < 1$)	Decreasing curve	$(-\infty, \infty)$	$(0, \infty)$	$y = 0$

• Example: $y = 2^x$ (growth), $y = (1/2)^x$ (decay)

Tables and Graphs

Exponential functions show constant multiplicative changes in $y$ for equal changes in $x$.

x	f(x) = 3 \times 2^x
0	3
1	6
2	12
3	24

• Each increase in $x$ multiplies $f(x)$ by 2.

• Contrast: Linear functions add a constant amount for each increase in $x$.

Modeling with Exponential Functions

Population Growth and Decay

Exponential functions are used to model population growth, radioactive decay, and other processes with constant percentage change.

• General Formula: $P(t) = P_0 (1 + r)^t$ where $P_0$ is the initial value, $r$ is the growth/decay rate (as a decimal), and $t$ is time.

• Example: A town has 1000 people and grows by 5% per year:

  • First year: $P(1) = 1000 \times 1.05$

  • Second year: $P(2) = 1000 \times (1.05)^2$

  • General: $P(t) = 1000 \times (1.05)^t$

• Decay Example: If population decreases by 12% per year: $P(t) = 1000 \times (0.88)^t$

Finding an Exponential Equation Through Two Points

Solving for Parameters

Given two points $(x_1, y_1)$ and $(x_2, y_2)$, find $a$ and $b$ in $y = a b^x$:

• Set up two equations using the points.

• Solve for $b$ by dividing the equations.

• Substitute $b$ back to find $a$.

• Example: Through $(2, 12)$ and $(4, 27)$:

  • $12 = a b^2$

  • $27 = a b^4$

  • Divide: $\frac{27}{12} = \frac{a b^4}{a b^2} = b^2 \Rightarrow b = \sqrt{\frac{27}{12}}$

  • Find $a$ using one equation.

Transformations of Exponential Functions

Translations and Reflections

Transformations of $f(x) = a b^x$ include shifts, stretches, and reflections:

• Horizontal shift: $f(x) = a b^{x-h}$ shifts right by $h$ units.

• Vertical shift: $f(x) = a b^x + k$ shifts up by $k$ units.

• Reflection: $f(x) = -a b^x$ reflects over the $x$-axis.

• Stretch/Compression: Changing $a$ stretches or compresses vertically.

• Example: $f(x) = 2^{x-3} + 4$ is $2^x$ shifted right 3 units and up 4 units.

Practice Problems and Applications

Sample Problems

• Determine which functions are exponential.

• Model population growth with exponential functions.

• Find exponential equations through given points.

• Apply transformations to exponential functions and describe the effects.

Summary Table: Linear vs. Exponential Functions

Function Type	Change in y for equal x	General Form
Linear	Constant addition	$f(x) = mx + b$
Exponential	Constant multiplication	$f(x) = a b^x$

Key Formulas

• Exponential Function: $f(x) = a b^x$

• Population Growth/Decay: $P(t) = P_0 (1 + r)^t$

• Average Rate of Change: $\text{AROC} = \frac{f(x+h) - f(x)}{h}$

Additional info:

• These notes cover the core concepts of exponential functions, including identification, graphing, modeling, and transformations, as required in College Algebra.

• Practice problems and tables reinforce the distinction between linear and exponential growth.