THE EXPONENTIAL FUNCTION

The exponential function takes the form , where the independent variable is the exponent.

Graphs with a > 1

An example of an exponential function is . So, how does the graph of compare to that of ?

We know that the graph of represents a parabola with its vertex at the origin, and is symmetrical about the y–axis. To determine the properties of the exponential function we set up a table of values and use these values to sketch a graph of .

x –3 –2 –1 0 1 2 3 4 5

We can now plot both graphs on the same set of axes and compare their properties:

Notice how different the graphs of the two functions are, even though their rules appear similar.

The difference being that for the quadratic function, the variable x is the base, whereas for the exponential, the variable x is the power.

We can now investigate the exponential function for different bases. Consider the exponential

functions and :

From the graphs we can see that

increases much faster than for x > 0.

For example, at x = 1, and

then, at x = 2, .

However, for x < 0 we have the opposite, decreases faster than . Notice then that at x = 0, both graphs pass through the point (0, 1).

From the graphs we can see that for values of x less than zero, the graph of lies below that of . Whereas for values of x greater than zero, then the graph of lies

above that of .

Exponential functions that display these properties are referred to as exponential growth functions.

(4,16)

(2,4) 1

0

y

x 4

16

2 4

Properties of

1. The function increases for all values of x (i.e., as x increases so too do the values of y).

2. The function is always positive (i.e., it lies above the x-axis).

3. As then

then .

i.e., the x-axis is an asymptote.

4. When

i. then ,

ii. then

iii. then .

f x( ) = 2^x

x→∞ y→∞

x→–∞ y→0

x>0 y>1

x = 0 y = 1

x<0 0< <y 1 f x( ) = 2^x

f x( ) = x²

f x( ) = 3^x g x( ) = 4^x y

x g x( ) = 3^x f x( ) = 4^x

(0, 1)

For x < 0, f(x) < g(x).

For x > 0, f(x) > g(x).

f x( ) = 4^x g x( ) = 3^x f 1( ) = 4 g 1, ( ) = 3 f 2( ) = 16 g 2, ( ) = 9

f x( ) = 4^x g x( ) = 3^x

f x( ) = 4^x

f x( ) = 3^x f x( ) = 4^x

f x( ) = 3^x

What happens when 0 < a < 1?

We make use of the TI–83 to investigate such cases. Consider the case where . Rather than using a table of values we

provide a sketch of the curve. The graph shows that the function is decreasing – such exponential functions are referred to as exponential decay. In fact, from the second screen we can see that the graph of

is a reflection of about the y–axis.

We note that the function can also be written as . Meaning that there are two ways to represent an exponential decay function, either as or

. For example, the functions and are identical.

We can summarise the exponential function as follows:

There also exists an important exponential function known as the natural exponential function.

This function is such that the base has the special number ‘e’. The number ‘e’, which we will consider in more detail in Chapter 7 has a value that is given by the expression

a = ¹₂ Intercepts : Cuts y–axis at (0,1) Other : Increases [growth]

Continuous Intercepts : Cuts y–axis at (0,1) Other : Decreases [decay]

Continuous

However, at this stage it suffices to realise that the number ‘ e’ is greater than one. This means that a function of the form will have the same properties as that of for a > 1.

That is, it will depict an exponential growth. Whereas the function will depict an exponential decay.

In Chapter 6 we will look at transformations of functions and consider terms such as ‘stretching’,

‘translations’, ‘dilations’ and so on – terms that are applicable to the examples we are about to examine. However, at this stage, we will consider sketching exponential curves from first principles only (and make general observations).

(a) Making use of the TI–83 we have:

(b)

From our observations we can make the following generalisation about the graph of .

Because of the extremities between the large and small values encountered with exponential functions, sketching these graphs to scale is often difficult. When sketching exponential functions, it is important to include the main features of the graph – for example, make sure that the intercepts and asymptotes are clearly labelled and then provide the general shape of the curve.

f x( ) = e^x f x( ) = a^x

f x( ) = e^–x

On the same set of axes sketch the following.

(a) f x( ) = 2^x,g x( ) = 2^x–1 (b) f x( ) = 3^x,g x( ) = 3^x+2

E

^XAMPLE5.21

S

o l u t i o

n _(0,1) ^(1,2)

(2,4)

(1,1) (2,2) (0,1)

(1,2)

(2,4) (3,4)

g x( ) = 2^x–1 f x( ) = 2^x

Observations:

The graph of g(x) is the graph of f(x) moved (i.e., translated) one unit to the right.

Observations:

The graph of g(x) is the graph of f(x) moved (i.e., translated) two units to the left.

Notice that we had to change the window settings to see both graphs on the same screen.

g x( ) = 3^x+2

f x( ) = 3^x

(0,9) (2,9)

(–2,1) (0,1)

y = a^x±k 1. The graph of is identical to but moved ‘ k’ units to the right.

2. The graph of is identical to but moved ‘ k’ units to the left.

y = a^x–k,k>0 y = a^x y = a^x+k,k>0 y = a^x

(a)

(b)

From our observations we can make the following generalisation about the graph of .

(a) (b)

On the same set of axes sketch the following.

(a) f x( ) = 2^x,g x( ) = 2^x–1 (b) f x( ) = 3^x,g x( ) = 3^x+2 (i.e., translated) one unit down. Notice too that the asymptote has also moved down one unit, from y = 0 to y = –1. (i.e., translated) two units up. Notice too that the asymptote has also moved up two units, from y = 0 to y = 2.

On the same set of axes sketch the following.

(a) f x( ) = 2^x,g x( ) = 3×2^x (b) f x( ) 3^x,g x( ) 1 stretched by a factor of 3 along the y–axis.

Notice too that the asymptote has not changed.

Observations:

The graph of g(x) is the graph of f(x) shrunk by a factor of 2 along the y–axis and reflected about the x–axis. Notice too that the asymptote has not changed.

g x( ) = 3×2^x

From our observations we can make the following generalisation about the graph of .

Obviously we can use a combination of these ‘effects’ on the basic exponential function:

Notice that in both cases the general shape of the exponential growth remains unaltered. Only the main features of the graph are of interest when sketching is involved.

(a) (b)

1. On separate sets of axes sketch the graphs of the following functions and determine the range of each function.

(a) (b) (c)

i. Stretched along the y–axis if k > 1.

ii. Shrunk along the y–axis if 0 < k < 1.

2. The graph of is identical to but

i. Reflected about the x–axis and stretched along the y–axis if k < –1.

ii. Reflected about the x–axis and shrunk along the y–axis if –1 < k < 0.

y = k×a^x,k>0 y = a^x

2. Sketch the following on the same set of axes, clearly labelling the y–intercept.

(a) where i. c = 1 ii. c = –2

(b) where i. c = 0.5 ii. c = –0.5

3. Sketch the following on the same set of axes, clearly labelling the y–intercept.

(a) where i. b = 2 ii. b = –2

(b) where i. b = 3 ii. b = – 2

4. On the same set of axes, sketch the following graphs

(a) and (b) and

(c) and (d) and

5. Find the range of the following functions:

(a) (b)

(c) (d)

(e) (f)

6. Sketch the graphs of the following functions, stating their range.

(a) (b)

8. Find the range of the following functions

(a) .

(b) .

(c) .

9. (a) Sketch the graph of , clearly labelling all intercepts with the axes and the equation of the asymptote.

(b) Solve for x, where .

10. Sketch the graphs of the following functions

(a) (b) (c)

11. Sketch the graphs of the following functions

(a) (b) (c)

12. Sketch the graphs of the following functions and find their range.

(a) (b)

(c) (d)

13. Sketch the graphs of the following, and hence state the range in each case.

(a) (b)

(c) (d)

14. Sketch the graph of the functions

(a) (b)

(c) (d)

(e) (f)

15. (a) On the same set of axes, sketch and where a > 1.

Hence, sketch the graph of the function , where a > 1.

(b) On the same set of axes, sketch and , where a > 1.

Hence, deduce the graph of , where a > 1.

16. Sketch the graph of the following functions and determine their range.

(a) (b)

(c) (d)

(e) (f)

In document IB Maths SL Textbook (Page 147-154)