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Exponential and Logarithmic Functions

Basic Exponential Function & Its Characteristics

The following are graphs of the basic logarithmic function, \(f(x)=b^x, \\ b>0, \\ b\ne 1\):

\(f(x)=b^x\) where \(0<b<1\)


\(f(x)=b^x\) where \(b>1\)


Characteristics of the Exponential Function

Using the graphs, we can identify some characteristics of the basic exponential function \(f(x)=b^x, \ b>0, b\ne 1\), including:

  • Domain: \(D:\{ x \in \mathbb{R}\}\) 
  • Range: \(R:\{ y \in \mathbb{R} \ | \ y > 0\}\) 
  • Horizontal asymptote: \(y=0\)
  • \(y\)-intercept: \((0,1)\)

Tip: If you recall the characteristics of the basic exponential function (\(f(x)=log_b(x)\)) graph (which can be found here), you'll see that the the basic exponential and logarithmic functions are very similar, and are, in fact, related. This is because both functions are inverses of each other, so their characteristics are also the inverse of each other.

So, if you can recall the characteristics of one function, you can then use the fact that they're inverse functions to recall the other functions' characteristics. Specifically, characteristics associated with the \(x\) and \(y\) variables are switched.

The related characteristics are summarized in the following tables:


Graphing Exponential Functions with Transformations

Given the graph of the parent function \(f(x)=b^x\), we are able to graph any logarithmic function of the form:

\(f(x)=ab^{k(x-d)}+c\), for any \(a,k,c,d\in \mathbb{R}\) 

by applying transformations to the parent graph. Each of the four parameters, \(a,b,c,d\) correspond to certain transformations, as summarized in the following table:

Function Notation/Parameter Corresponding Transformation(s) Coordinate Point Transformation
\(f(x)=\)\(\ a\)\(b^{k(x-d)}+c\)

Vertical stretch/compression & reflection

  • If \(a<0\), vertical reflection (i.e. over the x-axis)
  • If \(|a|>1\), vertical stretch by a factor of \(|a|\)
  • If \(0<|a|<1\), vertical compression by a factor of \(|a|\)
\((x,y) \rightarrow\) \((x,\)\(\ a\)\(y)\)

Horizontal stretch/compression & reflection

  • If \(k<0\), horizontal reflection (i.e. over the y-axis)
  • If \(|k|>1\), horizontal compression by a factor of \(|\frac{1}{k}|\)
  • If \(0<|k|<1\), horizontal stretch by a factor of \(|\frac{1}{k}|\)
\((x,y) \rightarrow(\)\(\frac{1}{k}\)\(x,y)\)

Horizontal shift/translation left or right by \(d\) units

  • If \(d>0\), shift right by \(d\) units
  • If \(d<0\), shift left by \(d\) units
\((x,y) \rightarrow\)\(\ (x\)\(+ d\)\(,y)\)
\(f(x)=ab^{k(x-d)}+\)\(\ c\)

Vertical shift/translation up or down by \(c\) units

  • If \(c>0\), shift up by \(c\) units
  • If \(c<0\), shift down by \(c\) units
\((x,y) \rightarrow\) \((x,y\)\(+ c\)\()\)

Tip: To help remember what parameters correspond to which transformations, note that parameters "outside" of the function \(f(x)=b^x\) (i.e. \(a, c\)) transform the \(y\) or vertical value, while the ones "inside" the function (i.e. \(k, d\)) transform the \(x\) or horizontal value.


  • After transforming, the horizontal asymptote is located at \(y=c\) (Why?)
  • The domain of the transformed function is dependent on whether it lies above or below the asymptote:
    • If the function is on the below of the asymptote, the domain is \(y<c\)
    • If the function is on the above of the asymptote, the domain is \(y>c\)
  • Regardless of transformations, the domain of an exponential function will always be \(\{x\in\mathbb{R}\}\) (Why?)


Try this interactive tool. 

Adjust the sliders for \(a\), \(b\), \(c\), \(d\), and \(k\). Observe what happens to the exponential function.


Example: Sketch the function \(f(x)=2(3^{x-1})-2\) by first:
a) Graphing the parent function, then applying transformations. b) Plotting any key points and drawing the asymptote.


a) Graphing the parent function, then applying transformations.

1. In this case, the parent function is \(f(x)=3^x\), which has a horizontal asymptote of \(y=0\). So we first sketch this function:


*Sketch this graph by creating a table values then plotting the points, or by finding the y-intercept, horizontal asymptote and another point on the graph, then sketching the function.

2. Identifying the transformation parameters, we see that we have \(a=2\), so we have a vertical stretch by a factor of 2:


3. Next, we have \(d=1\), so we have a horizontal shift to the right by 1 unit:


4. Finally, we have \(c=-2\), so we have a vertical shift down by 2 units. This also shifts our original horizontal asymptote down to \(y=-2\):


*Note: It is also possible to graph this function by transforming each individual point from the parent function or by applying all the transformations graphically at the same time (instead of one-by-one like we did above).

We compare the graphs of the two functions, \(f(x)=3^x\) and \(f(x)=2(3^{x-1}-2\), and summarize their characteristics in the following table:

\(f(x)=3^x\) \(f(x)=2(3^{x-1})-2\)
undefined undefined

Domain: \(D:\{x\in\mathbb{R}\}\)

Range: \(R:\{y\in\mathbb{R} \ | \ y>0\}\)

Horizontal asymptote: \(y=0\)

\(x\)-intercept: None

\(y\)-intercept: \((0,1)\)

Domain: \(D:\{x\in\mathbb{R}\}\)

Range: \(R:\{y\in\mathbb{R} \ | \ y>-2\}\)

Horizontal asymptote: \(y=-2\)

\(x\)-intercept: \((1,0)\)

\(y\)-intercept: \((0,-\frac{4}{3})\)

b) Plotting any key points and drawing the asymptote:

1. The function \(f(x)=2(3^{x-1})-2\) has:

  • \(x\)-intercept at \((1,0)\),
  • \(y\)-intercept at \((0,-\frac{4}{3})\),
  • a point at \((2,4)\), and
  • horizontal asymptote \(y=-2\).

We can plot and draw these characteristics on the Cartesian plane first:




2. After plotting/drawing these characteristics, we can draw a smooth curve through the points and approaching the asymptote, in order to finish sketching the function:


Example: Determine the exponential equation in the form \(f(x)=a2^{kx}+c\) that is represented by the following graph:


Watch this video for the solution! 

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