#### Graphs of the exponential function : answers.

1.1
y = 3x + 1  − 1
1.2
y = 3x − 1   + 2
Horizontal asymptote: y = − 1
Horizontal asymptote: y = 2
Base > 0, function is increasing.
Base > 0, function is increasing
Y-intercept:   y = 30 + 1  − 1
Y-intercept:   y = 30 − 1   + 2
= 3 − 1
= 3 − 1  + 2
= 2
= 2,333
Y-intercept is (0 ; 2)
Y-intercept is (0 ; 2,33)
X-intercept:   3x + 1  − 1 = 0
No X-intercept because q > 0
3x + 1 = 1 = 30
x + 1 = 0
x = − 1
X-intercept is (− 1 ; 0)
1.3
y = 21 − x   + 3
1.4
y = 32 − x  − 3
Horizontal asymptote: y = 3
Horizontal asymptote: y = − 2
Base < 0, function decreases.
Base > 0, function increases.
Y-intercept:   y = 21 − 0   + 3
Y-intercept:   y = 32 − 0  − 3
= 2 + 3
= 9 − 3
= 5
= 6
Y-intercept is (0 ; 5)
Y-intercept is (0 ; 6)
No X-intercept because q > 0
X-intercept:   32 − x  − 3 = 0
32 − x = 3
2 − x = 1
x = 1
X-intercept is (1 ; 0)
1.5
y = 2.3x − 1   + 2
1.6
y = 3.21 − x   − 3
Horizontal asymptote: y = 2
Horizontal asymptote: y = − 3
Base > 0, function increases.
Base < 0, function decreases.
Y-intercept:   y = 2.30 − 1   + 2
Y-intercept:   y = 3.21 − 0   − 3
= 2.3 − 1   + 2
= 3.21   − 3
= 2,667
= 3
Y-intercept is (0 ; 2,67)
Y-intercept is (0 ; 3)
No X-intercept because q > 0
X-intercept:   3.21 − x   − 3 = 0
3.21 − x   = 3
21 − x  = 1 = 20
1 − x = 0
x = 1
X-intercept is (1 ; 0)
1
2.1
y = 2x + p   + q
2.2
y = (——)x + p   + q
3
Horizontal asymptote: y = 1 and thus q = 1
Horizontal asymptote: y = − 3 and thus q = −3
1
y = 2x + p   + 1
y = (——)x + p    −3
3
1
At (0 ; 1,5):  20 + p   + 1 = 1,5
At (2;−2):  (——)2 + p    −3 = −2
3
2p   = 0,5   = 2−1
3− 2 − p    = 1 = 30
p = −1
− 2 − p = 0
p = − 2
1
y = 2x − 1   + 1
y = (——)x − 2    −3
3
1
2.3
y = 2x + p   + q
2.4
y = (——)x + p   + q
2
Horizontal asymptote: y = −2 and thus q = −2
Horizontal asymptote: y = − 2 and thus q = 2
1
y = 2x + p   − 2
y = (——)x + p   + 2
2
1
At (2 ; −1):  22 + p   − 2 = −1
At (1;3):  (——)1 + p   + 2 = 3
2
1
1
22 + p   = 1 = 20
(——)1 + p   = 1 = (——)0
2
2
2 + p = 0
1 + p = 0
p = − 2
p = − 1
1
Equation : y = 2x − 2   − 2
Equation : y = (——)x − 1   + 2
2
2.5
y = a.2   + q
2.6
y = a.b   + q
Horizontal asymptote: y = −1 and thus q = 1
Horizontal asymptote: y = − 3 and thus q = − 3
y = a.b   + 1
y = a.b   − 3
At (0 ; 3):  a.b0   + 1 = 3
At (0;0):  ab0   − 3 = 0
a.1 = 2
a − 3 = 0
a = 2
a = 3
y = 2.b   + 1
y = 3.b   − 3
At (1 ; 7):  2.b1   + 1 = 7
At (−2;9):  3.b−2   − 3 = 9
2b = 6
3−2 = 12
1
b = 3
b = 2−1  = (—)
2
1
Equation : y = 2x − 2   − 2
Equation : y = (——)x − 1   + 2
2
3.1
y = 4x − 1   − 2
4.1
y = 2−x + 1   + 2
Y-intercept:  y = 40 − 1   − 2
Y-intercept:  y = 20 + 1   + 2
= 4− 1   − 2
= 21   + 2
= − 1,75
= 4
Y-intercept is (0 ; − 1,75)
Y-intercept is (0 ; 4)
X-intercept:  4x − 1   − 2 = 0
X-intercept: The horizontal asymptote is greater
4x − 1  = 2 =  40,5
than 0 and therefore there is
x − 1 = 0,5
no X-intercept.
x = 1,5
X-intercept is (1,5 ; 0)
3.2
At P(2 ; p):  p = 42 − 1   − 2
4.2
At P(−1 ; p)  p = 2−(−1) + 1   + 2
= 41   − 2
= 22   + 2
= 2
= 6
3.3
At R(r ; −1):  4r − 1   − 2 = −1
4.3
At R(r ; 2,25)  2−r + 1   + 2 = 2,25
4r − 1 = 1  = 40
2−r + 1  = 0,25  = 2− 2
r − 1 = 0
−r + 1 = − 2
r = 1
r = 3
3.4
Domain : {x | x ∈ R}
4.4
Domain : {x | x ∈ R}
3.5
Range : {y | y ≠ − 2;  y ∈ R}
4.5
Range : {y | y > 2;  y ∈ R}
4.6
h(x) = 2−x + 1   + 2 − 4
= 2−x + 1   − 2
1
1
5.1
y = (——) x − 1   − 4
5.2
At C(c;4):  (——) c − 1   − 4 = 4
2
2
1
Y-intercept:  y = (——) 0 − 1   − 4
(2 − 1) c − 1   = 8
2
1
= (——) 0 − 1   − 4
2 − c + 1   = 23
2
= 4 − 4 = 0
− c + 1 = 3
1
X-intercept:  (——) x − 1   − 4 = 0
c = − 2
2
(2 − 1) x − 1   = 4
5.3
Domain : { x | x ∈ R }
2 − x + 1   = 22
5.4
Range:  { y | y > −4; y ∈ R }
1
− x + 1 = 2
5.5
h(x) = (——) x − 1   − 4 + 2
2
x = − 1
1
= (——) x − 1   − 2
2
1
1
6.1
y = (——)x + p   + q
6.2
y = (——)x − 2   − 1
3
3
1
Horizontal asymptote: y = − 1 and thus q = − 1
At P(p;2):  (——) p − 2   − 1 = 2
3
1
1
At (0;8):  (——) 0 + p   − 1 = 8
(——) p − 2   = 3
3
3
(3 −1) p  = 9 = 3 2
(3 −1) p − 2  = 3
−p = 2
−p + 2 = 1
p = − 2
p = 1
p = − 2  and  q = − 1
6.3
Domain: { x | x ∈ R }
1
6.5
h(x) = (——)x − 2   − 1 + 3
6.4
Range: { y | y > −1; y  ∈ R }
3
6.6
The graph is translated 3 units to the top and thus
1
= (——)x − 2   + 2
3
all points are translated 3 units to the top.
The Y-intercept of h(x) will be 8 + 3 = 11