Download 1. With linear functions as x increases by

Name _____________________________
Period _____
Date ________________
Big Ideas:
1. With linear functions as x increases by one, you add the same number to (or subtract the
same number from) the previous y term; whereas with exponential functions, as x increases
by one, you multiply the previous y term by the same number.
2. If the multiplier is a number greater than 1, it is an exponential growth function. If the
multiplier is between 0 and 1 it is an exponential decay function.
3. An exponential growth function will eventually exceed a linear or quadratic function.
4. Know which part of the equation is the asymptote and how to find the y-intercept of an
exponential function.
5. Know how to write the recursive and explicit formulas for arithmetic and geometric
sequences.
6. Know key characteristics of functions (i.e. intercepts, domain and range, intervals of
increase and decrease, and end behavior
7. Find Rate of Change given a graph, table of equation.
Part I. Exponential Functions
y = abx
The multiplier is the base, b.
y = a (1 + r )t
The multiplier is the growth
factor, 1 + r.
The multiplier determines if the function is
growth or decay
y = a (1 - r )t
The multiplier is the decay
factor, 1 – r.
a is the initial output (when no time has
passed); it is the y-intercept as long as
there is nothing being added or
subtracted from the exponent
1. EOCT Question: Andrew invested $1,000 in his savings account. The interest rate, r, is
compounded annually. Which equation shows the amount, A, in his account after x years.
A. A = 1000(1- r )x
C. A = 1000(r - 1 )x
B. A = 1000(1+ r )x
D. A = 1000(r )x
2. EOCT Question: Which function BEST describes a function with exponential decay?
3. State if the following represents exponential growth, decay, or neither. Then state the
asymptote.
f ( x)  3(.25) x  4
4. A certain population of bacteria has a growth rate of 0.04 bacteria/hour. The
formula for the growth of the bacteria’s population is A = P0 (2.71828)0.04t, where
P0 is the original population and t is the time in hours. If you begin with 400
bacteria, approximately how many of the bacteria can you expect after 100
hours?
A. 40,000
C. 888,601,488
B. 271,828
D. 21,839
5. A population of squirrels doubles every
year. Initially there were 5 squirrels. A
biologist studying the squirrels created a
function to model their population
growth, P(t) = 5(2)t where t is time.
The graph of the function is shown.
What is the valid range of the function if
it is to correctly model the population?
A. any real number
B. any whole number greater than 0
C. any whole number greater than 5
D. any whole number greater than or
equal to 5
6. The function graphed on this coordinate
grid models the maximum height, y, of a
dropped ball in feet after its xth bounce.
On which bounce was the height of the
ball approximately 4 feet?
A.
B.
C.
D.
bounce 1
bounce 2
bounce 3
bounce 4
7. The points (0, 1), (1, 4), (2, 16), (3, 64) are on the graph of a function. Which
equation represents that function?
A. f(x) = 2x
C. f(x) = 4x
B. f(x) = 3x
D. f(x) = 5x
Part II. Comparing Linear and Exponential
An exponential function eventually exceeds a linear (or quadratic function)
8. EOCT Question. The coordinate plane to the
right shows two functions.
 f(x) is an increasing linear function
 g(x) is an increasing exponential
Based on the information, which statement is
true for all real values of the domain x ≥ 0?
A.
B.
C.
D.
f(x) = g(x) for only one value in the domain
f(x) = g(x) for many values in the domain
f(x) > g(x) for all values in the domain
f(x) < g(x) for all values in the domain
9. EOCT Question. As the value of x increases, which function has the greatest rate of
growth?
A. f(x) = x2 + 7
C. h(x) = 7 - x2
B. g(x) = 2 + 7 x
D. k(x) = 2 x + 7
10. Observe the table of the following two functions. Based on the information, which
statement is true for all real values of the domain x ≥ 0?
A.
B.
C.
D.
The graph of 1.001x will cross the graph of 2x in one place for the domain given
The graph of 1.001x will cross the graph of 2x in two places for the domain given
2x will be greater than 1.001x for all values of the domain
1.001x will be greater than 2x for all values of the domain
11. Which table represents a linear function?
A.
X
0
1
2
3
4
y
2
4
8
16
32
C.
X
0
1
2
3
4
y
5
6
7
8
9
B.
X
y
0
1
1
3
2
9
3
27
4
81
X
y
0
0
1
1
2
4
3
9
4
16
D.
12. Find the y-intercept for each function and state which is the greatest
A.
f (x )  3(.25) x  4
B. g(x) = 6x -5
C. h(x) = 2x2 – 6x – 4
13. To rent a canoe, the cost is $3 for the oars and life preserver, plus $5 an hour for the canoe. Which
graph models the cost of renting a canoe?
A.
B.
C.
D.
Part III. Arithmetic and Geometric Sequences
GERALD (Geometric Exponential Ratio/ Arithmetic Linear Difference)
common ratio
common difference
Arithmetic Sequence Formulas
Geometric Sequence Formulas
Recursive formula - defined in terms of the
previous term
𝑎1 is the starting point.
𝑎𝑛 = 𝑎𝑛−1 + d
Recursive formula - defined in terms of the
previous term
𝑎1 is the starting point.
𝑎𝑛 = 𝑟(𝑎𝑛−1 )
Explicit Formula – defined in terms of n
Explicit Formula – defined in terms of n
𝑎𝑛 = 𝑑𝑛 + 𝑎0
𝑎𝑛 = 𝑎1 (𝑟)𝑛−1
𝑎0 is the y-intercept
Where r is the multiplier and 𝑎1 is the first term
Where d is the common difference and
14. Which function represents this sequence?
n
an
1
6
2
18
3
54
4
162
5
486
A. f(n) = 3n – 1
C. f(n) = 3(6)n – 1
B. f(n) = 6n-1
D. f(n) = 6(3)n – 1
15. Which function represents this sequence?
n
an
1
3
2
10
A. f(n) = n + 3
C. f(n) = 3n + 7
3
17
4
24
5
31
B. f(n) = 7n - 4
D. f(n) = n + 7
Part IV. Key Characteristics
For Questions 16 – 19 use the graph below. Several copies are given if needed.
domain:
16. What are the domain and range of this function?
Use inequality notation
range:
increase interval(s):
17. What are the intervals of increase and decrease of this function?
Use interval notation: ( ) [ ] 
decrease interval(s):
18. Describe the end behavior of this function. (2 points)
As x  -∞, y  _____
As x  ∞, y  _____
19. Which function is increasing at an increasing rate, increasing at a decreasing rate and increasing at
a constant rate?
A.
B.
C.
20. Which function is increasing the fastest?
A.
B.
C.
D.
x
y
x
y
x
y
x
y
0
4
7
𝟏⁄
𝟐
0
-45
-1
1
1
12
8
1
1
-15
0
6
2
36
9
1𝟐
2
-5
1
36
𝟏
Key
1.
4.
7.
10.
13.
16.
B
D
C
A
C
Domain: -∞ < x < ∞
Range: y < 4
2.
5.
8.
11.
14.
17.
19. A. increasing at a constant rate
B. increasing at a decreasing rate
C. increasing at an increasing rate
A
D
A
C
D
Interval of inc: (-∞, ∞)
Interval of dec: there is no
interval where the function
is decreasing
3.
6.
9.
12.
15.
18.
Decay; asymptote = -4
B
B
B
B
As x  -∞, y  -∞
As x  ∞, y  4
20. A and D are both
increasing at an increasing
rate, but D is increasing
faster than A because the
multiplier is greater.