Chapter 3 Test
Exponential functions
Part A - Multiple Choice - (5 pts)
1. The values of a,b, q in the exponential function in the form of y = ab x + q , with its graph shown
below, would be described as:
a) a = 1,b > 1,q > 0
b) a > 1,b > 1,q > 0
†
c) a ≥ 1,0 < b < 1,q > 0
d) a = 1,0 < b < 1,q < 0
†
x
†
2. The domain and range of the exponential function, f ( x ) = 40(1.05) - 20 , would be described
†
by:
†
a) X | x ΠR,Y | y > -20, y ΠR
b) X | x ΠR,Y | y > 20, y ΠR
c) X | x > -20, x ΠR,Y | y ΠR
d) X | x Œ R,Y | y ≥ 40, y Œ R
†
†
3.† The expression
†
125
a)
27
†
Ê 5 ˆ-3
Á- ˜ , is equivalent to the fraction:
Ë 3¯
125
27
27
b) c)
d) 27
125
125
†
4. Which exponential
equation best represents the following graph?
(a, b, h, and k are positve real numbers)
†
†
†
†
a) y = -ab( x- h) + k
b) y = -ab-( x-h ) + k
c) y = ab( x- h ) - k
d) y = -ab( x +h) + k
†
†
t
†
†
Ê 1 ˆ1620
5. The formula describing the decay of the radioactive isotope radium radium-226 is: A = AoÁ ˜ ,
Ë 2¯
where A is the amount of radium present, Ao is the original quantity of
radium and t is the time in years. The statement that best describes this functions is:
a) Radium has a half-life of 1620 years.
b) The amount is Radium doubles every 1620 years.
†
c) The decay rate is at 25%
†
d) Ao will decay to 50% of its original amount after 1620 seconds.
Extended questions
1.† State whether the following sequences are arithmetic, power, or exponential. Give a reason for
your choice.
(6 pts)
a) 3, 5, 7, 9, 11, ...
1 1 1 1
, , , ,...
3 6 18 54
c) 2,2, 8,...
b)
linear - first level common difference
other - no common ratio or level of difference
exponential - common ratio of 2
†
2. Ellen’s response to the question that follows is shown below. Ellen’s response is incorrect.
What error(s) has she made? Explain to Ellen what she should do to correct her error(s).
From the table, determine the equation that represents the relationship between the xvalues and the y-values.
r1 40 1
=
=
r2 80 2
r4 320 1
=
=
r5 640 2
Ê 1 ˆx
y = 40Á ˜
Ë 2¯
(5 pts)
† the formula tn . In this case the common
To find the common ratio, you must use
t n-1
†
ratio is 2. You must also take note of the horizontal stretch of 10. It is shown in the
formula as:
y = 40(2)
x
10
†
3. Find the equation of the image y = (2) x under each of the following mapping rule,
( x, y ) Æ (3x - 1, y + 5)
(3 pts)
†
y = (2)
1
( x +1)
3
+5
x -1
†
4. Given that f (x) = (1.2 ) - 3 ,
a) What are the domain and the range of the function?
(2 pts)
X | x ΠR,Y | y > -3, y ΠR
b) Do the values of f(x) increase or decrease? Explain
(1 pts)
The values of f(x) increase for x ΠR because the base of the power is greater than 1.
† on the y-axis. Therefore it is a growth curve.
There is also no reflection
c) What is the approximate zero of f(x)? Show how you approximate your answer. (1 pts)
†
x=7
(1,-2)
†
d) Describe the function as a combination of transformations of f (x) = (1.2 )x .
(1 pts)
x,
y
Æ
x
+
1,
y
3
( ) (
)
e) Sketch the graph of this function. Label the focal point and the asymptote.
(3 pts)
see above
5. Kate bought a computer for $2 000, to use in a business she is setting up. If it depreciates at a
rate of 30% per year;
a) What will the depreciated value be after one year, two years, ... five years?
1400, 980, 686, 480.20, 336.14
b) Find an expression for its value after n years and show this on a graph.
t n = 2000(0.7) n
c) What would the value of the computer be after 18 months?
t n = 2000(0.7)1.5 = $1171.32
6. Evaluate the following without the use of a calculator (show your work):
†
(6 pts)
1
Ê 27 ˆ 3 3
a) Á
˜ =
Ë125 ¯
5
†
†
1
Ê 4
ˆ
0
3
b) Á27 - 2 ˜ ÷16 2 = (81-1) ÷ 4 = 20
Ë
¯
(4 pts)
7. Solve for x
x
32x - 27 ( 3) = 0
Ê1ˆ
Ê
3Á ˜ ˆ
x
Á
3 Á 3 - 3 Ë 2 ¯ ˜˜ = 0
Ë
¯
x
x
x
Ê1ˆ
3Á ˜
Ë 2¯
x
x
Ê1 ˆ
3Á ˜
Ë2¯
3 = 0,3 - 3
3 = 0,3 = 3
no solution, x =
†
=0
3
2
(4pts)