Part I - Graphing Exponential FunctionsDirections: Graph Each Exponential Function By Creating A Table, Then Identify Its Key Characteristics.1. Y = ( 1 3 ) X Y=\left(\frac{1}{3}\right)^{x} Y = ( 3 1 ​ ) X - Growth / Decay: - Domain: - Range: - Y Y Y -intercept:

Directions: Graph each exponential function by creating a table, then identify its key characteristics.

1. $y=\left(\frac{1}{3}\right)^{x}$

Growth / Decay: This function represents a decay function because the base $\frac{1}{3}$ is less than 1.

Domain: The domain of an exponential function is all real numbers. Therefore, the domain of this function is $(-\infty, \infty)$.

Range: The range of an exponential function is all positive real numbers. Therefore, the range of this function is $(0, \infty)$.

$y$-intercept: To find the $y$-intercept, we substitute $x=0$ into the function:

$$y=\left(\frac{1}{3}\right)^{0}=1$$

Therefore, the $y$-intercept is $(0, 1)$.

Graphing the Function:

To graph the function, we can create a table of values:

$x$	$y=\left(\frac{1}{3}\right)^{x}$
-2	$\left(\frac{1}{3}\right)^{-2}=9$
-1	$\left(\frac{1}{3}\right)^{-1}=3$
0	$\left(\frac{1}{3}\right)^{0}=1$
1	$\left(\frac{1}{3}\right)^{1}=\frac{1}{3}$
2	$\left(\frac{1}{3}\right)^{2}=\frac{1}{9}$

Using this table, we can plot the points on a coordinate plane and draw a smooth curve through them.

Key Characteristics:

• The function has a horizontal asymptote at $y=0$.
• The function has a $y$-intercept at $(0, 1)$.
• The function is continuous and smooth.
• The function has a domain of $(-\infty, \infty)$ and a range of $(0, \infty)$.

2. $y=2^{x}$

Growth / Decay: This function represents a growth function because the base 2 is greater than 1.

Domain: The domain of an exponential function is all real numbers. Therefore, the domain of this function is $(-\infty, \infty)$.

Range: The range of an exponential function is all positive real numbers. Therefore, the range of this function is $(0, \infty)$.

$y$-intercept: To find the $y$-intercept, we substitute $x=0$ into the function:

$$y=2^{0}=1$$

Therefore, the $y$-intercept is $(0, 1)$.

Graphing the Function:

To graph the function, we can create a table of values:

$x$	$y=2^{x}$
-2	$2^{-2}=\frac{1}{4}$
-1	$2^{-1}=\frac{1}{2}$
0	$2^{0}=1$
1	$2^{1}=2$
2	$2^{2}=4$

Using this table, we can plot the points on a coordinate plane and draw a smooth curve through them.

Key Characteristics:

• The function has a horizontal asymptote at $y=0$.
• The function has a $y$-intercept at $(0, 1)$.
• The function is continuous and smooth.
• The function has a domain of $(-\infty, \infty)$ and a range of $(0, \infty)$.

3. $y=3^{x}$

Growth / Decay: This function represents a growth function because the base 3 is greater than 1.

Domain: The domain of an exponential function is all real numbers. Therefore, the domain of this function is $(-\infty, \infty)$.

Range: The range of an exponential function is all positive real numbers. Therefore, the range of this function is $(0, \infty)$.

$y$-intercept: To find the $y$-intercept, we substitute $x=0$ into the function:

$$y=3^{0}=1$$

Therefore, the $y$-intercept is $(0, 1)$.

Graphing the Function:

To graph the function, we can create a table of values:

$x$	$y=3^{x}$
-2	$3^{-2}=\frac{1}{9}$
-1	$3^{-1}=\frac{1}{3}$
0	$3^{0}=1$
1	$3^{1}=3$
2	$3^{2}=9$

Using this table, we can plot the points on a coordinate plane and draw a smooth curve through them.

Key Characteristics:

• The function has a horizontal asymptote at $y=0$.
• The function has a $y$-intercept at $(0, 1)$.
• The function is continuous and smooth.
• The function has a domain of $(-\infty, \infty)$ and a range of $(0, \infty)$.

4. $y=\left(\frac{1}{2}\right)^{x}$

Growth / Decay: This function represents a decay function because the base $\frac{1}{2}$ is less than 1.

Domain: The domain of an exponential function is all real numbers. Therefore, the domain of this function is $(-\infty, \infty)$.

Range: The range of an exponential function is all positive real numbers. Therefore, the range of this function is $(0, \infty)$.

$y$-intercept: To find the $y$-intercept, we substitute $x=0$ into the function:

$$y=\left(\frac{1}{2}\right)^{0}=1$$

Therefore, the $y$-intercept is $(0, 1)$.

Graphing the Function:

To graph the function, we can create a table of values:

$x$	$y=\left(\frac{1}{2}\right)^{x}$
-2	$\left(\frac{1}{2}\right)^{-2}=4$
-1	$\left(\frac{1}{2}\right)^{-1}=2$
0	$\left(\frac{1}{2}\right)^{0}=1$
1	$\left(\frac{1}{2}\right)^{1}=\frac{1}{2}$
2	$\left(\frac{1}{2}\right)^{2}=\frac{1}{4}$

Using this table, we can plot the points on a coordinate plane and draw a smooth curve through them.

Key Characteristics:

• The function has a horizontal asymptote at $y=0$.
• The function has a $y$-intercept at $(0, 1)$.
• The function is continuous and smooth.
• The function has a domain of $(-\infty, \infty)$ and a range of $(0, \infty)$.

5. $y=\left(\frac{1}{4}\right)^{x}$

Growth / Decay: This function represents a decay function because the base $\frac{1}{4}$ is less than 1.

Domain: The domain of an exponential function is all real numbers. Therefore, the domain of this function is $(-\infty, \infty)$.

Range: The range of an exponential function is all positive real numbers. Therefore, the range of this function is $(0, \infty)$.

$y$-intercept: To find the $y$-intercept, we substitute $x=0$ into the function:

$$y=\left(\frac{1}{4}\right)^{0}=1$$

Therefore, the $y$-intercept is $(0, 1)$.

Graphing the Function:

To graph the function, we can create a table of values:

$x$	$y=\left(\frac{1}{4}\right)^{x}$
-2	$\left(\frac{1}{4}\right)^{-2}=16$
-1	$\left(\frac{1}{4}\right)^{-1}=4$
0	$\left(\frac{1}{4}\right)^{0}=1$
1	$\left(\frac{1}{4}\right)^{1}=\frac{1}{4}$
2	$\left(\frac{1}{4}\right)^{2}=\frac{1}{16}$

Part II - Graphing Exponential Functions: Q&A

Directions: Review the key characteristics of exponential functions and answer the following questions.

Q1: What is the domain of an exponential function?

A1: The domain of an exponential function is all real numbers, denoted as $(-\infty, \infty)$.

Q2: What is the range of an exponential function?

A2: The range of an exponential function is all positive real numbers, denoted as $(0, \infty)$.

Q3: What is the $y$-intercept of an exponential function?

A3: The $y$-intercept of an exponential function is the point where the function intersects the $y$-axis, which is $(0, 1)$ for all exponential functions.

Q4: What is the horizontal asymptote of an exponential function?

A4: The horizontal asymptote of an exponential function is the horizontal line that the function approaches as $x$ approaches infinity or negative infinity, which is $y=0$ for all exponential functions.

Q5: What is the key characteristic of a growth function?

A5: A growth function is an exponential function with a base greater than 1, which means that the function increases as $x$ increases.

Q6: What is the key characteristic of a decay function?

A6: A decay function is an exponential function with a base less than 1, which means that the function decreases as $x$ increases.

Q7: How do you graph an exponential function?

A7: To graph an exponential function, you can create a table of values and plot the points on a coordinate plane. You can also use a graphing calculator or software to graph the function.

Q8: What is the significance of the $y$-intercept in an exponential function?

A8: The $y$-intercept in an exponential function represents the initial value of the function, which is the value of the function when $x=0$.

Q9: What is the significance of the horizontal asymptote in an exponential function?

A9: The horizontal asymptote in an exponential function represents the value that the function approaches as $x$ approaches infinity or negative infinity.

Q10: How do you determine whether an exponential function is a growth or decay function?

A10: To determine whether an exponential function is a growth or decay function, you can look at the base of the function. If the base is greater than 1, the function is a growth function. If the base is less than 1, the function is a decay function.

Q11: What is the relationship between the base and the exponent in an exponential function?

A11: The base and the exponent in an exponential function are related in that the base raised to the power of the exponent gives the value of the function.

Q12: How do you evaluate an exponential function at a given value of $x$?

A12: To evaluate an exponential function at a given value of $x$, you can substitute the value of $x$ into the function and simplify the expression.

Q13: What is the significance of the domain and range of an exponential function?

A13: The domain and range of an exponential function represent the set of all possible input values and output values of the function, respectively.

Q14: How do you graph an exponential function with a negative exponent?

A14: To graph an exponential function with a negative exponent, you can use the fact that $a^{-x}=\frac{1}{a^x}$, where $a$ is the base of the function.

Q15: What is the relationship between exponential functions and logarithmic functions?

A15: Exponential functions and logarithmic functions are inverse functions, which means that they are related in that the logarithm of an exponential function is the exponent, and the exponential of a logarithmic function is the base.

Q16: How do you use exponential functions to model real-world phenomena?

A16: Exponential functions can be used to model a wide range of real-world phenomena, including population growth, chemical reactions, and financial investments.

Q17: What are some common applications of exponential functions?

A17: Exponential functions have a wide range of applications, including population growth, chemical reactions, financial investments, and electrical circuits.

Q18: How do you use exponential functions to solve problems?

A18: Exponential functions can be used to solve a wide range of problems, including problems involving population growth, chemical reactions, and financial investments.

Q19: What are some common mistakes to avoid when working with exponential functions?

A19: Some common mistakes to avoid when working with exponential functions include forgetting to include the base, forgetting to include the exponent, and forgetting to simplify the expression.

Q20: How do you check your work when working with exponential functions?

A20: To check your work when working with exponential functions, you can use a graphing calculator or software to graph the function and verify that it matches the expected graph.