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Introduction

In Grade 10, you studied graphs of many different forms. In this chapter, you will learn a little more about the graphs of exponential functions.

Functions of the form y = a b ( x + p ) + q For b > 0

This form of the exponential function is slightly more complex than the form studied in Grade 10.

General shape and position of the graph of a function of the form f ( x ) = a b ( x + p ) + q .

Investigation : functions of the form y = a b ( x + p ) + q

  1. On the same set of axes, with 5 x 3 and 35 y 35 , plot the following graphs:
    1. f ( x ) = - 2 · 2 ( x + 1 ) + 1
    2. g ( x ) = - 1 · 2 ( x + 1 ) + 1
    3. h ( x ) = 0 · 2 ( x + 1 ) + 1
    4. j ( x ) = 1 · 2 ( x + 1 ) + 1
    5. k ( x ) = 2 · 2 ( x + 1 ) + 1
    Use your results to understand what happens when you change the value of a . You should find that the value of a affects whether the graph curves upwards ( a > 0 ) or curves downwards ( a < 0 ). You should also find that a larger value of a (when a is positive) stretches the graph upwards. However, when a is negative, a lower value of a (such as -2 instead of -1) stretches the graph downwards. Finally, note that when a = 0 the graph is simply a horizontal line. This is why we set a 0 in the original definition of these functions.
  2. On the same set of axes, with 3 x 3 and 5 y 20 , plot the following graphs:
    1. f ( x ) = 1 · 2 ( x + 1 ) - 2
    2. g ( x ) = 1 · 2 ( x + 1 ) - 1
    3. h ( x ) = 1 · 2 ( x + 1 ) + 0
    4. j ( x ) = 1 · 2 ( x + 1 ) + 1
    5. k ( x ) = 1 · 2 ( x + 1 ) + 2
    Use your results to understand what happens when you change the value of q . You should find that when q is increased, the whole graph is translated (moved) upwards. When q is decreased (poosibly even made negative), the graph is translated downwards.
  3. On the same set of axes, with 5 x 3 and 35 y 35 , plot the following graphs:
    1. f ( x ) = - 2 · 2 ( x + 1 ) + 1
    2. g ( x ) = - 1 · 2 ( x + 1 ) + 1
    3. h ( x ) = 0 · 2 ( x + 1 ) + 1
    4. j ( x ) = 1 · 2 ( x + 1 ) + 1
    5. k ( x ) = 2 · 2 ( x + 1 ) + 1
    Use your results to understand what happens when you change the value of a . You should find that the value of a affects whether the graph curves upwards ( a > 0 ) or curves downwards ( a < 0 ). You should also find that a larger value of a (when a is positive) stretches the graph upwards. However, when a is negative, a lower value of a (such as -2 instead of -1) stretches the graph downwards. Finally, note that when a = 0 the graph is simply a horizontal line. This is why we set a 0 in the original definition of these functions.
  4. Following the general method of the above activities, choose your own values of a and q to plot 5 graphs of y = a b ( x + p ) + q on the same set of axes (choose your own limits for x and y carefully). Make sure that you use the same values of a , b and q for each graph, and different values of p . Use your results to understand the effect of changing the value of p .

These different properties are summarised in [link] .

Table summarising general shapes and positions of functions of the form y = a b ( x + p ) + q .
p < 0 p > 0
a > 0 a < 0 a > 0 a < 0
q > 0
q < 0

Domain and range

For y = a b ( x + p ) + q , the function is defined for all real values of x . Therefore, the domain is { x : x R } .

The range of y = a b ( x + p ) + q is dependent on the sign of a .

If a > 0 then:

b ( x + p ) > 0 a · b ( x + p ) > 0 a · b ( x + p ) + q > q f ( x ) > q

Therefore, if a > 0 , then the range is { f ( x ) : f ( x ) [ q , ) } . In other words f ( x ) can be any real number greater than q .

If a < 0 then:

b ( x + p ) > 0 a · b ( x + p ) < 0 a · b ( x + p ) + q < q f ( x ) < q

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Source:  OpenStax, Siyavula textbooks: grade 11 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11243/1.3
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