Using Transformations to Investigate Functions

Transformations are often used to describe the relationship between functions.
 

Let's investigate how transformations may be applied to functions:

Example 1:
 Remember to utilize your graphing calculator to get a feel for the function and the changes.

Transform the function f(x) = ex with a vertical stretch by a factor of 3, followed by a translation 5 units to the right.

  • Write an equation for the transformed function.

  • Graph the transformed function.
Answer:
Remember:
• a vertical stretch will change the y values.
• a translation to the right will affect the x values.

The original function, f(x), (the parent function) is graphed in blue.
f(x) with a vertical stretch by a factor of 3 is graphed in red.
f(x) with the vertical stretch AND the translation of 5 units to the right is graphed in green.

The final transformed function is represented by the equation
 f(x) = 3 e(x-5)

 

Example 2:
This problem appears in the Core Curriculum Resource for Math B.

 Given f(x) = x2 - 2x

     A.  Determine an expression for h(x), if h(x) = f(-x).
     B.  Determine an expression for g(x), if g(x) is represented by
           the rotation of 180º of f(x) about the origin.
     C.  Rotate f(x) 90º about the origin.  Find the coordinates of
           the point(s) for which x = -1, under the rotation.
Answer:
 Things to remember:
• Rotation of 180º
r180º(x,y) = (-x,-y)

• Rotation of 90º
r90º(x,y) = (-y,x)

• Examine points that are easily readable from the original graph.

• Again, your graphing calculator could assist you in finding your answers.

• While graphs are NOT required in this problem, they certainly help in analyzing the problem.

The original function, f(x), is graphed in blue.

A.  the expression for h(x) is
h(x) = (-x)2 -2(-x) = x2 +2x

B.  the expression for g(x) is
g(x) = -x2 -2x

C.  the 90º rotation is indicated by the dotted line.  The coordinates for which
 x = -1 are (-1, -0.414) and (-1, 2.414) *
 
 * x = -1 under the rotation is equivalent to y = 1 under the original graph.
Therefore, we are interested in  x2 - 2x = 1   which gives    x2 -2x -1 = 0
Use your graphing calculator to solve.  One possible calculator solution method is shown below:
Y1=x2 -2x - 1
Y2 = 0
Use 2nd - Calc - #5 Intersect to find the points of intersection

 

Example 3:
The Math B exam may present you with questions where you will utilize your graphing calculator indirectly.

Such questions often give you the "answer" and ask for the original function.


Write an equation of the graph shown at the right.  Assume that the original (parent) function is one of the following:

      

Answer:
 Use your graphing calculator to help you decide if your answer is correct and to help you make adjustments to your equation.


 

 


• When the graph y=f(x) is stretched vertically by a positive factor of b, the equation of the new graph is  y=bf(x).

 Use your graphing calculator to see which function most closely resembles the graph.  The answer will be the absolute value function.  Graph it to see what has changed.

 
The vertex of the absolute value function has been translated 7 units to the right and 2 units up.  Start by making these adjustments.

Unfortunately, the new graph also has vertical stretch.  The vertical stretch factor is 3/2.

    Ans.

 


Roberts