Exponential Functions

In this guide, we will focus on graphing exponential functions, specifically functions with bases such as 2 and e. We will explore key concepts such as horizontal asymptotes, domain, and range for these functions. We will begin with basic exponential functions and gradually introduce more complex variations.

Introduction to Exponential Functions

Example Function: y = 2^x

To graph the function $y = 2^x$, we will start by creating a table of values. For illustrative purposes, we will choose input values of 0 and 1.

1. Calculate Function Values:

   • For $x = 0$:
     

     $2^0 = 1$

   • For $x = 1$:
     

     $2^1 = 2$

2. Plot Points:

   • The points obtained from the calculations are $(0, 1)$ and $(1, 2)$.

3. Graphing Process:

   • Begin at the horizontal asymptote, which for this function is the x-axis ($y = 0$). Connect the points with a smooth curve, showing that the function rises as $x$ increases.

Determining Horizontal Asymptotes

The horizontal asymptote for the function $y = 2^x$ is $y = 0$.

• If the function had been modified to include a constant, for example $y = 2^x + 3$, this would shift the horizontal asymptote 3 units upward to $y = 3$.

• Similarly, for $y = 2^{(x - 4)}$, the horizontal asymptote would shift to $y = 4$.

• If no vertical shift is present, the horizontal asymptote remains at $y = 0$.

Domain and Range of Exponential Functions

Domain:

The domain of the function $y = 2^x$ is all real numbers. In interval notation, this is:

$(- \text{∞}, \text{∞})$

This indicates that $x$ can take on any value without restriction.

Range:

The range represents the possible $y$-values of the function. For this specific function:

• The lowest $y$-value approaches the horizontal asymptote, which is 0, but it does not actually include 0.

• The highest possible value tends towards infinity. Thus, the range is given in interval notation as:
  

  $(0, \text{∞})$

Example Function: y = 3^(x + 1) - 2

Calculating the Graph

1. Identification of Transformations:- The term $-2$ indicates that the graph will be shifted downward 2 units, creating a new horizontal asymptote at $y = -2$.

   • Additionally, the term $(x + 1)$ indicates a horizontal shift of 1 unit to the left.

Change of Input Values:

Instead of evaluating at $x = 0$ and $x = 1$, we will evaluate:

• For $x + 1 = 0 \Rightarrow x = -1$

• For $x + 1 = 1 \Rightarrow x = 0$
  

  Thus, we will plug in values $-1$ and $0$.

1. Calculate Function Values:

   • For $x = -1$:
     

     $3^{(0)} - 2 = 1 - 2 = -1$

   • For $x = 0$:
     

     $3^{(1)} - 2 = 3 - 2 = 1$

2. Plot Points:

   • The points obtained are $(-1, -1)$ and $(0, 1)$.

Graphing Process:

• Start with the horizontal asymptote at $y = -2$, and plot the points connecting them, indicating the curve's behavior as it approaches the horizontal asymptote.

Domain and Range

Domain:

The domain remains all real numbers:

$(- \text{∞}, \text{∞})$

Range:

For the range of this function:

• The lowest point begins at the horizontal asymptote, $y = -2$, extending to infinity.
  

  Thus, the range is:
  

  $(-2, \text{∞})$

Example Function: y = e^(x + 1)

Characteristics of Base e

In this scenario, we have the base $e$ ($\text{approximately } 2.718 \text{ or often simplified as } 2.7$).

1. Calculating Function Values:

   • For $x = 0$:
     

     $e^{(0 + 1)} = e^1 = e \text{ (approximately } 2.7 \text{)}</p></li><li><p>For$x = 1$: <br></p><p>$e^{(1 + 1)} = e^2 \text{ (approximately } 7.389 \text{)}

2. Determine Horizontal Asymptote:

   • The function has a horizontal asymptote at $y = 1$.

3. Points to Graph:

   • After evaluating the function, using the input $x=0$ gives us $y=1$ and $x=1$ gives us approximately $1.7$. Thus, we plot the points $(0, 0)$ and $(1, 1.7)$.

Domain and Range

Domain:

The domain is the same as previous functions:

$(- \text{∞}, \text{∞})$

Range:

The function's range extends from the horizontal asymptote up to infinity:

$(1, \text{∞})$

Example Function: y = 3 - e^(x - 2)

Identifying Changes in the Function

1. Input Selection:

   • Set $x - 2$ equal to 0 and 1 to determine new input values, resulting in $x = 2$ and $x = 3$.

   • The horizontal shift to the right indicates we will select inputs at $2$ and $3$.

2. Determine Horizontal Asymptote:

   • The horizontal asymptote is at $y = 3$, designated by the constant component of the function. The negative sign in front of e indicates a reflection over the horizontal asymptote.

Calculating Function Values:

• For $x = 2$:
  

  $3 - e^{(2 - 2)} = 3 - e^{0} = 3 - 1 = 2$

• For $x = 3$:
  

  $3 - e^{(3 - 2)} = 3 - e^{1} = 3 - 2.7 = 0.3$

Plotting Points:

The calculated points are $(2, 2)$ and $(3, 0.3)$.

Domain and Range

Domain:

The domain continues to be:

$(- \text{∞}, \text{∞})$

Range:

The range moves from negative infinity up to the horizontal asymptote:

$(- \text{∞}, 3)$ (with 3 not included).

This concludes our comprehensive exploration of graphing exponential functions, their characteristics, domain, and range. We have illustrated how transformations affect their behavior and provided step-by-step examples to solidify understanding.