Transformation of Exponential Functions

Key Concepts

• Move the graph of an exponential function vertically.
• Explain about the horizontal translation of an exponential graph.
• Compare two different transformations of f(x)=2^x 

Transformation of Exponential Functions

1. Quadratic function 

A function  f defined by  f(x)=ax²+bx+c, where  a, b, and  c are real numbers and  a≠0 is called a quadratic function. 

• The graph of a quadratic function is a curve called a parabola. 

• The quadratic parent function is f(x)=x²
• For 0<|a|<1, the shape of the parabola is wider than the parent function. 
• For |a|>1, the shape of the parabola is narrower than the parent function. 

• f(x)=ax² is the reflection of f(x)=−ax² over the x-axis. 

2. Graph of g(x) = x²+h

• The value of kk in g(x)=x²+k=translates the graph of parent function f, vertically k units. 
• The value of k does not affect the axis of symmetry. 

3. Graph of (x−h)²

• The value of h in g(x)=(x−h)² translates the graph of parent function f, horizontally h units. 
• The vertex of the graph g is (0, h). 
• The value of h translates the axis of symmetry. 

4. Exponential function 

• The product of an initial amount and a constant ratio raised to a power is an exponential function.  
• Exponential functions are modeled using f(x)=a.b^x where a is a non-zero constant, b>0 and b≠1.

1. Vertical translations of graphs of exponential functions 

The graph of f(x)=2^x+k is a vertical translation of the graph of f(x)=2^x 

If k is positive, the graph is moved up.  

If k is negative, the graph is moved down. 

Example: Compare the graph of f(x)=a^x−k with the parent function.  

The graph moves downwards. 

Example: Compare the graph of f(x)=a^x+k

The graph moves upwards. 

2. Horizontal translations of graphs of exponential functions 

The graph of f(x)=2^x−h is a horizontal translation of the graph of f(x)=2^x

• If hh is positive, the graph is translated to the right.  
• If hh is negative, the graph is translated to the left. 

Example: Compare the graph of f(x)=a^x−h with the parent function when h>0. 

The graph moves to the right by h units. 

Example: Compare the graph of f(x)=a^x−h with the parent function when h<0.  

The graph moves to the left by h units. 

3. Compare two different transformations of f(x)=2^x

• We can compare two different transformations of f(x)=2^x

1. Compare the asymptote, y-intercept of each transformation function (from the function or the graph of the function) with respect to the f(x) 
2. Identify the difference between the asymptotes of the two transformation functions. 

Exercise

• How does the graph of g(x)=2^x+1 compare to the graph of f(x)=2^x?
• Compare the graph of f(x)=2^(x+2) with the graph of f(x)=2^x.
• How does the graph of m(x)=3^x-4 compare to the graph of p(x)=3^x+7?
• Compare the function represented by the graph of g(x)=2^x-3 to the function represented by the table.

• Find the value of k from the graph.

Concept Map 

What we have learned

• The graph of f(x)=2^x+k is a vertical translation of the graph of f(x)=2^x.
• The graph of f(x)=2^(x-h) is a horizontal translation of the graph of f(x)=2^x.