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CanadaON
Grade 12

Transformations of Exponential Graphs

Lesson

 

A general exponential curve

The exponential curve given by $$y=A×bmx+c+k represents a transformation of the basic curve $$y=bx. Introducing constants enables the model to become a powerful tool in the investigation of certain types of growth and decay phenomena. Modelling with theoretical functions in this way provides a great example of why the study of mathematics is so crucial to our understanding of nature. 

Examples 

The functions $$y=25x+3 and $$y=120×2x are examples of the general exponential function given by $$y=A×bmx+c+k, with both $$A and $$m non-zero. The number $$b is known as the base of the function, and it is strictly defined as a positive number not equal to 1. 

For $$y=25x+3, we would say that $$A=1, $$b=2, $$m=5, $$c=3 and $$k=0. For $$y=120×2x, we would say that $$A=120, $$b=2, $$m=1, $$c=0 and $$k=0.

The function $$y=42×(0.5)x has $$A=2, $$b=0.5, $$m=1, $$c=0 and $$k=4. Each constant has a particular effect on the overall graph.   

The constant $$m is often called the growth constant (or decay constant if $$m is negative). It can take on a range of non-zero values designed to suit particular real life growth or decay rates. However, for our immediate purposes, we will restrict $$m to non-zero integer values only.

We introduce all of these constants in order to accurately model real world phenomena. This is the power of a generalised model. We can adjust the constants to fit reality and in so doing learn more about the way nature works.

A few key points

Whilst the general form is a comprehensive tool for sketching exponential curves, there are a few simpler observations to keep in mind. We can summarise them using examples as shown in this table:

Specific Example Observation
$$y=3x Reflect $$y=3x across the $$x-axis
$$y=3x5 Translate $$y=3x horizontally  to the right by $$5 units
$$y=3x5 Translate $$y=3x vertically downward by $$5 units
$$y=2×3x Double every $$y value of  $$y=3x
$$y=83x Reflect $$y=3x across the $$x axis then translate $$8 units upward

More complex forms of the exponential require more thought. For example, the function $$y=32x5 is quite interesting to think about. The applet below can produce the graph as a plot of points, but we can think about what the curve might look like without it.

For example, we can rewrite the function as follows:

 $$y=32(x52)=(32)(x52)=9(x52).

Hence, the function could be thought of as the function $$y=9x translated to the right by $$212 units.

 

The applet

The applet below is extremely versatile, but we need to keep in mind that it is a learning tool exploring the effects of the different constants involved. As a guide, it might be helpful to use the applet to create the four graphs shown in this table.

Verify the $$y-intercepts of each graph, the limiting value of $$y (this is the value that the function gets close to without actually ever reaching) and whether or not the graph is rising or falling:

Function $$y-intercept limiting value rising/falling
$$y=2x $$y=1 $$y=0 rising
$$y=3x+1 $$y=1 $$y=0 falling
$$y=3×4x2 $$y=1 $$y=2 rising
$$y=(0.5)x $$y=1 $$y=0 falling

After experimenting with these, try other combinations of constants. What can you learn? 

 

BASES BETWEEN 0 AND 1

One final point that should be noted is that a curve like $$y=(0.5)x is none other than $$y=2x in disguise. Thus: 

$$y=(0.5)x=(12)x=12x=2x 

 

In a similar way we can say that $$y=(1b)x=bx, and so every exponential curve of the form $$y=bx, with a base $$b in the interval $$0<b<1, can be re-expressed as $$y=(1b)x. Since $$b is a positive number, this means that exponential functions of the form  $$y=bx where $$0<b<1 are in fact decreasing curves.  

Worked Examples

Question 1

The function $$y=5x has been transformed into the function $$y=5x+4+2

Identify the:

  1. Horizontal translation

  2. Vertical translation

  3. Growth constant

Question 2

Consider the function $$y=10x and its inverse function:

  1. Plot the graph of $$y=10x.

    Loading Graph...

  2. Find the equation of the inverse function of $$y=10x.

  3. Hence determine the correct graph for $$y=logx.

    Loading Graph...

    A

    Loading Graph...

    B

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    C

    Loading Graph...

    D

Question 3

Consider the function $$y=3x :

  1. Find the $$y-intercept of the curve $$y=3x.

  2. Fill in the table of values for $$y=3x.

    $$x $$3 $$2 $$1 $$0 $$1 $$2 $$3
    $$y $$ $$ $$ $$ $$ $$ $$
  3. Find the horizontal asymptote of the curve $$y=3x.

  4. Hence plot the curve $$y=3x.

    Loading Graph...

  5. Is the function $$y=3x, an increasing or decreasing function?

    Increasing function

    A

    Decreasing function

    B

 

 

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