Exponential Functions and Graphs

28.1 Introduction

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

28.2 Functions of the Form

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

Figure 28.1: General shape and position of the graph of a function of the form .

Activity :: Investigation : Functions of the Form

1. On the same set of axes, plot the fol lowing graphs :

Use your results to deduce the effect of a.

2. On the same set of axes, plot the following graphs:

Use your results to deduce the effect of q.

3. Following the general method of the above activities, choose your own values
of a and q to plot 5 different graphs of to deduce the effect of
p.

You should have found that the value of a affects whether the graph curves upwards (a > 0) or
curves downwards (a < 0).

You should have also found that the value of p affects the position of the x-intercept.

You should have also found that the value of q affects the position of the y-intercept.

These different properties are summarised in Table 28.1. The axes of symmetry for each graph
is shown as a dashed line .

Table 28.1: Table summarising general shapes and positions of functions of the form .

28.2.1 Domain and Range

For , the function is defined for all real values of x. Therefore, the domain is
{x : x ∈ R}.

The range of is dependent on the sign of a .

If a > 0 then:

Therefore, if a > 0, then the range is {f(x) : f(x) ∈ [q,∞)}.

If a < 0 then:

Therefore, if a < 0, then the range is {f(x) : f(x) ∈ (−∞,q]}.

For example, the domain of g(x) = 3 · 2^x+1 + 2 is {x : x ∈ R}. For the range,

Therefore the range is {g(x) : g(x) ∈ [2,∞)}.

Exercise: Domain and Range 1. Give the domain of y = 3x. 2. What is the domain and range of f(x) = 2x ? 3. De termine the domain and range of y = (1,5)x+3.

28.2.2 Intercepts

For functions of the form, , the intercepts with the x and y axis is calulated by
setting x = 0 for the y-intercept and by setting y = 0 for the x-intercept.

The y-intercept is calculated as follows :

For example, the y-intercept of g(x) = 3 · 2^x+1 + 2 is given by setting x = 0 to get:

The x-intercepts are calculated by setting y = 0 as follows:

Which only has a real solution if either a < 0 or Q < 0. Otherwise, the graph of the function of
form does not have any x-intercepts.

For example, the x-intercept of g(x) = 3 · 2^x+1 + 2 is given by setting x = 0 to get:

which has no real solution. Therefore, the graph of g(x) = 3 · 2^x+1 + 2 does not have any
x-intercepts.

Exercise: Intercepts 1. Give the y-intercept of the graph of y = bx + 2. 2. Give the x- and y-intercepts of the graph of y = 1/2 (1,5)x+3 − 0,75.

28.2.3 Asymptotes

There are two asymptotes for functions of the form . They are determined by
examining the domain and range.

We saw that the function was undefined at x = −p and for y = q. Therefore the asymptotes
are x = −p and y = q.

For example, the domain of g(x) = 3 · 2^x+1 + 2 is {x : x ∈ R, x ≠ −1} because g(x) is
undefined at x = −1. We also see that g(x) is undefined at y = 2. Therefore the range is
{g(x) : g(x) ∈ (−∞,2) ∪ (2,∞)}.

From this we deduce that the asymptotes are at x = −1 and y = 2.

Exercise: Asymptotes 1. Give the equation of the asymptote of the graph of y = 3x − 2. 2. What is the equation of the horizontal asymptote of the graph of y = 3(0,8)x-1 − 3 ?

28.2.4 Sketching Graphs of the Form

In order to sketch graphs of functions of the form, , we need to calculate
determine four characteristics:

1. domain and range

2. y-intercept

3. x-intercept

For example, sketch the graph of g(x) = 3 · 2^x+1 + 2. Mark the intercepts.

We have determined the domain to be {x : x ∈ R} and the range to be {g(x) : g(x) ∈ [5,∞)}.

The y-intercept is = 8 and there are no x-intercepts.

Figure 28.2: Graph of g(x) = 3 · 2^x+1 + 2.

Exercise: Sketching Graphs 1. Draw the graphs of the following on the same set of axes. Label the horizontal aymptotes and y-intercepts clearly. Draw the graph of f(x) = 3x. B Explain where a solution of 3x = 5 can be read off the graph.

28.3 End of Chapter Exercises

1. The following table of values has columns giving the y-values for the graph y = a^x,
y = a^x+1 and y = a^x + 1. Match a graph to a column.

2. The graph of f(x) = 1 + a.2^x (a is a constant) passes through the origin.

A Determine the value of a.
B Determine the value of f(−15) correct to FIVE decimal places .
C Determine the value of x, if P(x; 0,5) lies on the graph of f.
D If the graph of f is shifted 2 units to the right to give the function h, write down the
equation of h.

3. The graph of f(x) = a.bx (a ≠ 0) has the point P(2;144) on f.

A If b = 0,75, calculate the value of a.
B Hence write down the equation of f.
C Determine, correct to TWO decimal places, the value of f(13).
D Describe the transformation of the curve of f to h if h(x) = f(−x).