MATHEMATICS
MORE EXERCISES
Graphs of the exponential function.

Question  1
Sketch the graphs of the following
functions :
$$\hspace*{2 mm}\mathrm{1.1\kern3mmy = 3^{x+1} − 1\kern2mm\ }$$                      [ A 1.1 ]
$$\hspace*{2 mm}\mathrm{1.2\kern3mm3^{x − 1} + 2\kern2mm\ }$$                           [ A 1.2 ]
$$\hspace*{2 mm}\mathrm{1.3\kern3mmy = 2^{1 − x} + 3\kern2mm\ }$$                     [ A 1.3 ]
$$\hspace*{2 mm}\mathrm{1.4\kern3mmy = 3^{2 − x} − 3\kern2mm\ }$$                       [ A 1.4 ]
$$\hspace*{2 mm}\mathrm{1.5\kern3mmy = 2.3^{x − 1} + 2\kern2mm\ }$$                    [ A 1.5 ]
$$\hspace*{2 mm}\mathrm{1.6\kern3mmy = 3.2^{1 − x} − 3\kern2mm\ }$$                   [ A 1.6 ]

Question  2
The figures show the graphs of the
$$\hspace*{3 mm}\mathrm{\kern3mmfunctions\ with\ equation\ y = a^{x + p} + q\kern2mm\ }$$
Use the information given in the figure
to detemine the values of p and q :

2.1

$$\hspace*{5 mm}\mathrm{\kern3mmy = 2^{x + p} + q\kern2mm\ }$$

B is the point (0 ; 1,5)
[ A 2.1 ]

2.2

$$\hspace*{5 mm}\mathrm{\kern3mmy = \Big(\frac{1}{3}\Big)^{x + p} + q\kern2mm\ }$$

Horizontal asymptote : y = − 3 and A(1 ; 0)
B is the point (0 ; 6) and P(2 ; − 2)
[ A 2.2 ]

2.3

$$\hspace*{5 mm}\mathrm{\kern3mmy = 2^{x + p} + q\kern2mm\ }$$

Given : A(3 ; 0), B(0 ; −1,75) and P(2 ; − 1)
[ A 2.3 ]

2.4

$$\hspace*{5 mm}\mathrm{\kern3mmy = \Big(\frac{1}{2}\Big)^{x + p} + q\kern2mm\ }$$

Given : B(0 ; 2,33) and P(−2 ; 5)
[ A 2.4 ]

2.5

$$\hspace*{5 mm}\mathrm{\kern3mmy = a.b^x + q and y = 1\kern2mm\ }$$

Given : A(1 ; 7) and B(0 ; 3)
[ A 2.5 ]

2.6

$$\hspace*{5 mm}\mathrm{\kern3mmy = a.b^x + q\kern2mm\ }$$

Given : A(0 ; 0) and B(− 2 ; 9)

[ A 2.6 ]

Question  3
The figure shows the graph of
y = 4x − 1   − 2

3.1  Calculate the intercepts on
the axes.                                            [ A 3.1 ]

3.2  Calculate the value of p if P(2;p) is
a point on the graph.                      [ A 3.2 ]

3.3  Calculate the value of r if R(r;−1) is
a point on the graph.                     [ A 3.3 ]

3.4  For which values of x will
4x − 1   − 2 > 0?                               [ A 3.4 ]

3.5  Write down the domain of
the function.                                  [ A 3.5 ]

3.6  Write down the range of
the function.                                  [ A 3.6 ]

Question  4
The figure shows the graph of
y = 2− x + 1   + 2

4.1  Calculate the intercepts on
the axes.                                         [ A 4.1 ]

4.2  Calculate the value of p if P(-1 ; p)
is a point on the graph.              [ A 4.2 ]

4.3  Calculate the value of r if R(r ; 2,25)
is a point on the graph.              [ A 4.3 ]

4.4  Write down the domain of
the function.                                [ A 4.4 ]

4.5  Write down the range of
the function.                                [ A 4.5 ]

4.6  For which values of x will
the function be positive?.         [ A 4.6 ]

4.7  The function h(x) is formed
when the function above is
translated 4 units downwards.
Write down the equation of
the function h(x).                        [ A 4.7 ]

Question  5
The figure shows the graph of
$$\hspace*{5 mm}\mathrm{\kern3mmy = \Big(\frac{1}{2}\Big)^{x − 1} − 4\kern2mm\ }$$

5.1  Calculate the intercepts on
the axes.                                         [ A 5.1 ]

5.2  Calculate the value of c if C(c ; 4)
is a point on the graph.              [ A 5.2 ]

5.3  Write down the domain of
the function.                                [ A 5.3 ]

5.4  Write down the range of
the function.                                [ A 5.4 ]

5.5  For which values of x will
$$\hspace*{12 mm}\mathrm{\kern3mmy = \Big(\frac{1}{2}\Big)^{x − 1} − 4 > 0?\kern2mm\ }$$             [ A 5.5 ]

5.6  The function h(x) is formed
when the function above is
translated 2 units upwards.
Write down the equation of
the function h(x).                        [ A 5.6 ]

Question  6
The figure shows the graph of
$$\hspace*{5 mm}\mathrm{\kern3mmy = \Big(\frac{1}{3}\Big)^{x + p} + q\kern2mm\ }$$

6.1  Calculate the values of p and q.
[ A 6.1 ]

6.2  Calculate the value of p if P(p ; 2)
is a point on the graph.               [ A 6.2 ]

6.3  Write down the domain of
the function.                                [ A 6.3 ]

6.4  Write down the range of
the function.                                [ A 6.4 ]

6.5  The function h(x) is formed
when the function above is
translated 3 units upwards.
Write down the equation of
the function h(x).                        [ A 6.5 ]

6.6  Write down the coordinates of
the Y-intercept of h(x).               [ A 6.6 ]

Question  7
The diagram shows the graph of h(x) = 2x − 2
P (p ; 30) is a point on the graph of h(x).

7.1  Write down the equation of the
horizontal asymptote.                   [ A 7.1 ]

7.2  Calculate the coordinates of points
the intercepts on the axes.             [ A 7.2 ]

7.3  Determine the value of p.               [ A 7.3 ]

7.4  For which values of x will h(x) ≤ 30?
[ A 7.4 ]

Question  8
The diagram shows the graph of
f(x) = p.2−x  + q
D(−4 ;−12) and E(2 ; e) are points on
the graph of f(x).

8.1  Write down the equation of the
horizontal asymptote.                   [ A 8.1 ]

8.2  Determine the values of p and q
and write down the equation
of f(x).                                               [ A 8.2 ]

8.3  Determine the value of e.             [ A 8.3 ]

8.4  For which value(s) of x is
f(x) < 3,75?                                        [ A 8.4 ]

8.5  For which value(s) of x is
f(x) > 0?                                             [ A 8.5 ]

Question  9
The diagram shows the graph of
g(x) = 2x + p  + q
A(2 ;11) is a point on the graph of g(x) and
the y-intercept is (0 ; −1).
9.1  Determine the values of p and q and
thus the equation of g(x).                 [ A 9.1 ]

9.2  Write down the equation of the
horizontal asymptote.                       [ A 9.2 ]

9.3  The graph of h is obtained
by moving the graph of g 4 units
downwards. Write down the
equation of h(x).;                               [ A 9.3 ]

9.4  The graph of k is obtained
by moving the graph of g 3 units
to the right. Write down the
equation of k(x).;                               [ A 9.3 ]

Question  10
It is given that f(x) = 2x.

10.1  The graph of g is obtained
by moving the graph of f 3 units
downwards. Write down the
equation of g(x).                          [ A 10.1 ]

10.2  The graph of h is obtained
by moving the graph of f 2 units
to the left. Write down the
equation of h(x).                           [ A 10.2 ]

Question  11
Given that f(x) = 3−x.

11.1  The graph of g is obtained
by moving the graph of f 2 units
upward.. Write down the
equation of g(x).                           [ A 11.1 ]

11.2  The graph of h is obtained
by moving the graph of f 4 units
to the right. Write down the
equation of g(x).                           [ A 11.2 ]

Question  12
Given that f(x) = 5x + 3.

12.1  The graph of g is obtained
by moving the graph of f 4 units
downwards. Write down the
equation of g(x).                           [ A 12.1 ]

12.2  The graph of h is obtained
by moving the graph of f 4 units
to the left. Write down the
equation of h(x).                           [ A 12.2 ]

Question  13
Given that f(x) = − 3−x + 2.

13.1  The graph of g is obtained
by moving the graph of f 2 units
upward. Write down the
equation of g(x).                           [ A 13.1 ]

13.2  The graph of h is obtained
by moving the graph of f 3 units
to the right. Write down the
equation of h(x).                           [ A 12.2 ]