MAC 1105
Final Exam Review
And Practice Final Exam Solutions
• You will need to have your own graphing
calculator for the test.
• You may not share calculators or use any type
of communication device in place of a
calculator.
• The final exam is comprehensive over the
entire course (not just the second half).
• Students who do not take the final exam will
receive a zero for the final exam unless prior
arrangements have been made for a rare
incomplete.
• If you miss the final and do not email me by the
day of the exam, I will assume you wish to take
a zero on your final exam. Communication
with me via email is key!
How to Study for the Final Exam
• Complete all assigned homework. Remember that an overall
(for the entire semester) homework score of at least 80% gives
you 10 bonus points on the test.
• Review your notes and applicable worksheet questions for each
of the 25 objectives in the chart below.
• Set up a study system (note cards for example) for each of the 25
objectives in the chart below. See the column below labeled
“important ideas”.
• Complete the practice test and check your solutions in class.
• Work the suggested text problems for each of the 25 objectives
in the chart below and check your solutions. The problems are
odd numbered so answers should be readily available.
Exam Topics 1 – 5
Objective
Sections
1) Solve a quadratic
equation using the
quadratic formula.
1.4
2) Use the discriminant to
determine the number and
type of solutions of a
quadratic equation.
1.4
3) Solve equations
graphically.
1.5
Suggested
Text
Problems
Important Ideas
p. 165: 31, 33,
35
𝑥=
−𝑏 ±
𝑏 2 − 4𝑎𝑐
2𝑎
𝐷 = 𝑏 2 − 4𝑎𝑐
p. 165: 47, 49
D = 0 → one real root
D > 0 → two real roots
D < 0 → two imaginary roots
See Notes
Enter each equation.
(Y = )
Begin with the standard window.
(ZOOM 6)
Adjust the window to see the intersection
(WINDOW)
Find the intersection
(2nd TRACE (CALC) 5)
The x coordinate is the solution.
4) Solve an absolute value
inequality.
1.7
5) Given two points, find
the distance between the
two points and the
midpoint.
2.1
p. 166:
99, 101
p. 293:
9a, 11a, 13a
|u| < a means
|u| ≤ a means
|u| > a means
|u| ≥ a means
−a < u < a
−a ≤ u ≤ a
u> a or u < −a
u≥ a or u ≤ −a
𝑑=
(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
𝑥1 + 𝑥2 𝑦1 + 𝑦2
𝑀=(
,
)
2
2
Exam Topics 6 – 10
6) Test an equation for
symmetry.
2.2
p. 293:
25, 27, 29, 31,
33
(symmetry
only)
1) Substitute:
x−axis symmetry: (−y) for y
y−axis symmetry: (−x) for x
origin symmetry: (−x) for x and (−y) for y
2) Determine if the result is the original equation.
7) Find the center and
radius of a circle.
2.2
p. 189:
57, 59, 71
8) Write the equation of a
circle.
2.2
p. 293:
35, 37
9) Find the slope of a line.
2.3
p. 293:
9c, 11c, 13c
10) Write a linear equation
given two ordered pairs of
data.
2.3
p. 295:
105a
Complete the square (if necessary) to transform the equation
to
𝑥 − ℎ 2 + 𝑦 − 𝑘 2 = 𝑟2
𝑐𝑒𝑛𝑡𝑒𝑟 ℎ, 𝑘 , 𝑟𝑎𝑑𝑖𝑢𝑠 𝑟
𝑐𝑒𝑛𝑡𝑒𝑟 ℎ, 𝑘 , 𝑟𝑎𝑑𝑖𝑢𝑠 𝑟
𝑥 − ℎ 2 + 𝑦 − 𝑘 2 = 𝑟2
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦
m =𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥 =
𝑦2 − 𝑦1
𝑥2 − 𝑥1
1) Find the slope.
2) Use the point slope formula:
y – y1 = m(x– x1)
Exam Topics 11 – 15
11) Solve problems involving
linear regression.
2.3
p. 295:
111a, 111c
Enter the data
(STAT EDIT 1)
Find the regression coefficients.
(STAT CALC 4)
Write the equation of the line.
(𝑦 = 𝑎𝑥 + 𝑏)
12) Solve a system of two
equations by substitution or
elimination
5.1
p. 487:
13,15,17,19
13) Work an application of
systems of linear equations.
5.4
p. 488:
87, 91,95, 99
Set up a system of two equations in two unknowns. Solve the
system.
14) Use your calculator to
find intervals of increasing
and decreasing.
2.5
p. 236:
11
Graph the function.
Adjust the window to show all of the turning points.
The intervals of increasing/decreasing are intervals of the domain
(x).
15) Classify a function as
even, odd, or neither.
2.5
p. 237:
33, 35, 37, 41,
43, 45
1)
2)
3)
Substitution:
Solve one of the equations for one the variables (usually y).
Substitute that result into the other equation.
Solve.
Elimination:
1) Get the coefficients of one of the variables to be additive
inverses if needed (by multiplication).
2) Add the equations to eliminate that variable.
3) Solve.
4) Back substitute to find the value of the other variable.
Find −𝑥 .
𝑓 −𝑥 = 𝑓 𝑥 → 𝑒𝑣𝑒𝑛
𝑓 −𝑥 = −𝑓 𝑥 → 𝑜𝑑𝑑
Exam Topics 16 – 20
16) Evaluate a difference
quotient.
17) Find function values of a
piece-wise function.
18) Write the equation of a
function whose graph has been
transformed.
2.5
2.6
2.7
p. 239:
79
p. 248:
21a, 47
p. 265: 67, 69,
71, 73
𝑓 𝑥 + ℎ − 𝑓(𝑥)
ℎ
Make sure to pay attention to which “piece” of the function you should
use.
Be sure to know the ten different transformations from section 2.7:
𝑓 𝑥 ± 𝑎 → 𝑢𝑝 𝑑𝑜𝑤𝑛 𝑎 𝑢𝑛𝑖𝑡𝑠 (𝑢𝑝 𝑖𝑠+)
𝑓 𝑥 ± 𝑎 → 𝑙𝑒𝑓𝑡 𝑟𝑖𝑔ℎ𝑡 𝑎 𝑢𝑛𝑖𝑡𝑠 (𝑙𝑒𝑓𝑡 𝑖𝑠+)
𝑘𝑓(𝑥) → 𝑠𝑡𝑟𝑒𝑡𝑐ℎ𝑒𝑑 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑏𝑦 𝑎 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑘
𝑖𝑓 𝑘 > 1,
1
𝑠ℎ𝑟𝑢𝑛𝑘 𝑏𝑦 𝑖𝑓 0 < 𝑘 < 1
𝑘
𝑓 𝑘𝑥 → 𝑠ℎ𝑟𝑢𝑛𝑘 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙𝑙𝑦 𝑓𝑜𝑟 𝑘 > 1,
𝑠𝑡𝑟𝑒𝑡𝑐ℎ𝑒𝑑 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙𝑙𝑦 𝑓𝑜𝑟 0 < 𝑘 < 1
−𝑓 𝑥 → 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑 𝑎𝑏𝑜𝑢𝑡 𝑥
𝑓 −𝑥 → 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑 𝑎𝑏𝑜𝑢𝑡 𝑦
19) Find the vertex of a
quadratic function.
3.1
p. 307: 53, 55
(vertex only)
Use the formula:
𝑏
𝑏
The vertex of 𝑥 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 is − 2𝑎 , 𝑓 − 2𝑎
OR
Graph the function and find the max/min.
20) Find any vertical or
horizontal asymptotes for a
given rational function.
3.6
p. 373: 25, 27,
29, 31, 35, 37,
39, 41
Vertical: Simplify the function completely. The vertical line 𝑥 = 𝑎 is a
vertical asymptote of the function if 𝑎 causes the denominator to be
zero (after simplification).
Horizontal: n = degree of numerator, m = degree of denominator
𝑛 < 𝑚 → 𝑥 − 𝑎𝑥𝑖𝑠
𝑛 = 𝑚 → 𝑦 = 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓.
𝑛 > 𝑚 → 𝑛𝑜 𝐻. 𝐴.
Exam Topics 21 – 25
21) Work a variation
application.
22) Solve an exponential
equation by writing both sides
to the same base and/or by
taking the (natural/common)
logarithm of both sides.
3.8
p. 393:
29, 33, 35, 37
1.
2.
3.
4.
4.1, 4.5
p. 473:
55, 57, 59, 63
Convert both sides to the same base and set the exponents equal to
one another (if possible).
Use the given information to set up an equation.
Solve for k → the constant of variation.
Write a new equation using all variables and the constant k.
Input the new information and compute the answer.
OR
Take the (natural/common) logarithm of both sides. Use the third
property of logarithms to “move” the variable to the outside.
Solve for the variable.
23) Work with compound
interest formulas (including
compounding continuously).
4.2
24) Write a logarithmic
expression in expanded form
or condensed form.
4.4
p. 425:
19, 21
𝑟
𝐴 = 𝑃(1 + )𝑛𝑡
𝑛
𝐴 = 𝑃𝑒 𝑟𝑡
p. 453:
19, 21, 23, 25,
39, 43
𝑙𝑜𝑔𝑎 𝑀𝑁 = 𝑙𝑜𝑔𝑎 𝑀 + 𝑙𝑜𝑔𝑎 𝑁
𝑀
𝑙𝑜𝑔𝑎
= 𝑙𝑜𝑔𝑎 𝑀 − 𝑙𝑜𝑔𝑎 𝑁
𝑁
𝑙𝑜𝑔𝑎 𝑀𝑟 = 𝑟𝑙𝑜𝑔𝑎 𝑀
25) Solve a logarithmic
equation by consolidating the
logarithms to a single
logarithmic expression.
4.3, 4.5
p. 467:
53, 55, 57, 59
1) Isolate all logarithmic expressions on one side of the equation
and use logarithmic properties to rewrite the logarithmic
expressions as a single logarithmic expression .
2) Rewrite the logarithmic expression as an exponential
expression.
3) Solve.
4) Reject values that produce the logarithm of a negative number
or the logarithm of zero.
Special Note
• It is entirely possible that the questions on
the test may have additional parts that differ
slightly from those in the practice test.
• To prepare for these additional parts, be
sure you are familiar with your notes, your
worksheets, and your homework.
• Objectives will remain unchanged.
Practice Test Solutions
1a) Solve: 𝑥 2 = 5 − 8𝑥
𝑥 2 + 8𝑥 − 5 = 0 ⇨ 𝑎 = 1, 𝑏 = 8, 𝑐 = −5
−𝑏 ± 𝑏2 − 4𝑎𝑐
𝑥=
2𝑎
−8 ± 82 − 4(1)(−5)
=
2(1)
−8 ± 64 + 20
=
2
−8 ± 84
=
2
−8 ± 4 ∙ 21
=
2
−8 ± 2 21
=
2
= −4 ± 21
{−4 − 21, −4 + 21 }
1b) Solve: 𝑥 2 + 4𝑥 + 8 = 0
𝑥 2 + 4𝑥 + 8 = 0 ⇨ 𝑎 = 1, 𝑏 = 4, 𝑐 = 8
−𝑏 ± 𝑏 2 − 4𝑎𝑐
𝑥=
2𝑎
−4 ± 42 − 4(1)(8)
=
2(1)
−4 ± 16 − 32
=
2
−4 ± −16
=
2
−4 ± 4𝑖
=
2
= −2 ± 2𝑖
{−2 + 2𝑖, −2 − 2𝑖 }
1c) Solve: 𝑥 2 = 15 + 9𝑥
𝑥 2 − 9𝑥 − 15 = 0 ⇨ 𝑎 = 1, 𝑏 = −9, 𝑐 = −15
−𝑏 ± 𝑏2 − 4𝑎𝑐
𝑥=
2𝑎
−(−9) ± (−9)2 − 4(1)(−15)
=
2(1)
9 ± 81 + 60
=
2
9 ± 141
=
2
9 − 141 9 + 141
{
,
}
2
2
2) Find the discriminant and the number and type of roots:
8𝑦 2 = −5𝑦 − 3
8𝑦 2 + 5𝑦 + 3 = 0 ⇨ 𝑎 = 8, 𝑏 = 5, 𝑐 = 3
𝐷 = 𝑏2 − 4𝑎𝑐
= 5 2− 4 8 3
= 25 − 96
= −71
𝐷 𝑖𝑠 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 ⇨ 2 𝑖𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦 𝑟𝑜𝑜𝑡𝑠
𝐷 = −71, two imaginary roots
3) Use the intersection – of – graphs method to approximate the
solution to nearest hundredth: −3.5 20 + 7 𝑥 + 6.4(14𝜋𝑥 +
𝑌1 = −3.5 20 + 7 𝑥 + 6.4 14𝜋𝑥 + 0.6
𝑌2 = −5
𝐴𝑑𝑗𝑢𝑠𝑡 𝑡ℎ𝑒 𝑥 𝑡𝑜 −2,2 𝑡ℎ𝑒𝑛 𝑐ℎ𝑜𝑜𝑠𝑒 2𝑛𝑑 𝑇𝑟𝑎𝑐𝑒 𝐶𝑎𝑙𝑐 5
𝑇ℎ𝑒 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑖𝑠 𝑡ℎ𝑒 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑜𝑓 𝑡ℎ𝑒
𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛.
{0.22}
4a) Solve: 8𝑥 − 8 ≥ 9
8𝑥 − 8 ≤ −9 𝑜𝑟 8𝑥 − 8 ≥ 9
8𝑥 ≤ −1 𝑜𝑟 8𝑥 ≥ 17
1
17
𝑥 ≤ − 𝑜𝑟 𝑥 ≥
8
8
1
17
(−∞, − ]𝑈 [ , ∞)
8
8
4b) Solve: 9𝑥 + 3 < 19
−19 < 9𝑥 + 3 < 19
−22 < 9𝑥 < 16
22
16
−
<𝑥<
9
9
22 16
(− , )
9 9
5a) Find the distance between P and Q.
𝑃 4, −4 , 𝑄 −4, 2
𝑑=
(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
=
((−4) − 4)2 + (2 − (−4))2
=
−8
2
+ 6
2
10
=
100 = 10
5b) Find the midpoint of P and Q.
𝑃 4, −4 , 𝑄 −4, 2
𝑥1 + 𝑥2 𝑦1 + 𝑦2
𝑀=(
,
)
2
2
4 + −4 −4 + 2
=(
,
)
2
2
0 2
=
,−
2 2
(0, −1)
6a) Test the equation for symmetry:
𝑦 = −2𝑥 3 + 5𝑥
𝑥: −𝑦 = −2𝑥 3 + 5𝑥 ⇨ 𝑦 = 2𝑥 3 − 5𝑥 𝑛𝑜
𝑦: 𝑦 = −2 −𝑥
3
+ 5 −𝑥 ⇨ 𝑦 = 2𝑥 3 − 5𝑥 𝑛𝑜
𝑜𝑟𝑖𝑔𝑖𝑛: −𝑦 = −2 −𝑥 3 + 5 −𝑥 ⇨
−𝑦 = 2𝑥 3 − 5𝑥
⇨ 𝑦 = −2𝑥 3 + 5𝑥 𝑦𝑒𝑠
𝑜𝑟𝑖𝑔𝑖𝑛
6b) Test the equation for symmetry:
𝑦 = 2𝑥 2 − 2
𝑥: −𝑦 = 2𝑥 2 − 2 ⇨ 𝑦 = −2𝑥 2 − 2 𝑛𝑜
𝑦: 𝑦 = 2 −𝑥
2
− 2 ⇨ 𝑦 = 2𝑥 2 − 2 𝑦𝑒𝑠
𝑜𝑟𝑖𝑔𝑖𝑛: −𝑦 = 2 −𝑥 2 − 2 ⇨
−𝑦 = 2𝑥 2 − 2 𝑛𝑜
𝑦 − 𝑎𝑥𝑖𝑠
7) Find the center and radius of the circle.
𝑥 2 + 𝑦 2 + 8𝑥 + 10𝑦 = 23
𝑥 2 + 8𝑥 +
+ 𝑦 2 + 10𝑦 +
1
8 = 4, 42 = 16
2
1
10 = 5, 52 = 25
2
= 23
𝑥 2 + 8𝑥 + 16 + 𝑦 2 + 10𝑦 + 25 = 23 + 16 + 25
𝑥+4
𝑥 −ℎ
2
+ 𝑦 + 5 2 = 64
2 + 𝑦 − 𝑘 2 = 𝑟2
−4, −5 , 𝑟 = 8
8) Find the equation of the circle that satisfies the given
conditions: C𝑒𝑛𝑡𝑒𝑟 𝑎𝑡 0, −7 , 𝑟𝑎𝑑𝑖𝑢𝑠 9
𝑥 −ℎ
𝑥−0
2
2
+ 𝑦 −𝑘
2
= 𝑟2
+ 𝑦— −7
2
= 92
𝑥2 + 𝑦 + 7
2
= 81
9a) Find the slope of the line through the given pair of points.
−9, −1 , (4, 7)
𝑦2 −
𝑚=
𝑥2 −
7 − (−1)
=
4 − (−9)
8
=
13
8
13
𝑦1
𝑥1
9b) Find the slope of the line through the given pair of points.
−7, −1 , 8, −1
𝑦2 −
𝑚=
𝑥2 −
−1 − −1
=
8 − −7
0
=
15
=0
0
𝑦1
𝑥1
10) After three seconds, a bottle has 15 ounces of water. After 10
seconds, the bottle has 43 ounces of water. Write a linear equation
that models the amount of water in the bottle in terms of x.
3 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 ⇨ 15 𝑜𝑢𝑛𝑐𝑒𝑠: 3, 15
10 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 ⇨ 43 𝑜𝑢𝑛𝑐𝑒𝑠: 10, 43
43 − 15 28
𝑚=
=
=4
10 − 3
7
𝑦 − 15 = 4 𝑥 − 3
𝑦 − 15 = 4𝑥 − 12
𝑦 = 4𝑥 + 3
CHECK:
𝑦 = 4 3 + 3 = 12 𝑦𝑒𝑠 , 𝑦 = 4 10 + 3 = 43 (𝑦𝑒𝑠)
𝑦 = 4𝑥 + 3
11) The paired data consists of the costs of advertising (in thousands
of dollars) and the number of products sold (in thousands). Find the
equation of the regression line that models the data.
Cost
9
2
3
4
2
5
9
10
Number
85
52
55
68
67
86
83
73
𝑃𝑙𝑎𝑐𝑒 𝑡ℎ𝑒 "Costs" 𝑖𝑛 𝐿1 𝑎𝑛𝑑 𝑡ℎ𝑒 "𝑁𝑢𝑚𝑏𝑒𝑟𝑠" 𝑖𝑛 𝐿2
𝑆𝑇𝐴𝑇 ⇨ 𝐶𝐴𝐿𝐶 ⇨ 4
𝑎 ≈ 2.79, 𝑏 ≈ 55.8
𝑦 = 2.79𝑥 + 55.79
12) Solve the system of equations.
6𝑥 + 9𝑦 = −69
3𝑥 − 2𝑦 = −2
6𝑥 + 9𝑦 = −69
−2 3𝑥 − 2𝑦 = −2
6𝑥 + 9𝑦 = −69
−6𝑥 + 4𝑦 = 4
13𝑦 = −65
𝑦 = −5
6𝑥 + 9 −5 = −69
6𝑥 − 45 = −69
6𝑥 = −24
𝑥 = −4
{ −4, −5 }
13) Jim wants to plan a meal with 127 grams of carbohydrates and 1150 calories. If green
beans have 7 grams of carbohydrates and 30 calories per half – cup serving and if fried
shrimp have 9 grams of carbohydrates and 190 calories per three – ounce serving, how
many servings of green beans and shrimp should he use?
1
𝑥 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑢𝑝 𝑠𝑒𝑟𝑣𝑖𝑛𝑔𝑠 𝑜𝑓 𝑔𝑟𝑒𝑒𝑛 𝑏𝑒𝑎𝑛𝑠,𝑦 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡ℎ𝑟𝑒𝑒 𝑜𝑢𝑛𝑐𝑒 ℎ𝑒𝑙𝑝𝑖𝑛𝑔𝑠 𝑜𝑓 𝑠ℎ𝑟𝑖𝑚𝑝
2
7𝑥 + 9𝑦 = 127
30𝑥 + 190𝑦 = 1150
𝑒𝑙𝑖𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛
−30 7𝑥 + 9𝑦 = 127
7 30𝑥 + 190𝑦 = 1150
−210𝑥 − 270𝑦 = −3810
210𝑥 + 1330𝑦 = 8050
1060𝑦 = 4240
𝑦=4
7𝑥 + 9 4 = 127
7𝑥 + 36 = 127
7𝑥 = 91
𝑥 = 13
13 ℎ𝑎𝑙𝑓 𝑐𝑢𝑝𝑠 𝑜𝑓 𝑏𝑒𝑎𝑛𝑠 𝑎𝑛𝑑 4 𝑡ℎ𝑟𝑒𝑒 𝑜𝑢𝑛𝑐𝑒 ℎ𝑒𝑙𝑝𝑖𝑛𝑔𝑠 𝑜𝑓 𝑠ℎ𝑟𝑖𝑚𝑝
14) By using a graphing calculator, find the increasing interval(s)
for the function f x = 2𝑥 3 − 3𝑥 2 − 36𝑥 + 30
Choose windows large enough to allow
you to see the turning points:
𝑥: −10, 10
𝑦: −200, 200
−∞, −2 , (3, ∞)
15a) Determine if the given function is even, odd, or neither.
𝑓 𝑥 = 4𝑥 2 − 2
𝑓 −𝑥 = 4 −𝑥 2 − 2
= 4𝑥 2 − 2
𝐸𝑣𝑒𝑛
15b) Determine if the given function is even, odd, or neither.
𝑓 𝑥 = −5𝑥 5 + 9𝑥 3
𝑓 −𝑥 = −5 −𝑥 5 + 9 −𝑥
= −5 −𝑥 5 + 9 −𝑥 3
= 5𝑥 5 − 9𝑥 3
= −1 −5𝑥 5 + 9𝑥 3
𝑂𝑑𝑑
3
15c) Determine if the given function is even, odd, or neither.
𝑓 𝑥 = 9𝑥 4 − 4𝑥 − 6
𝑓 −𝑥 = 9 −𝑥
4
− 4 −𝑥 − 6
= 9𝑥 4 + 4𝑥 − 6
𝑁𝑒𝑖𝑡ℎ𝑒𝑟
15d) Determine if the given function is even, odd, or neither.
𝑥
𝑓 𝑥 = 3
𝑥 − 7𝑥
−𝑥
𝑓 −𝑥 =
−𝑥 3 − 7(−𝑥)
−𝑥
=
−𝑥 3 + 7𝑥
−1 𝑥
=
−1 𝑥 3 − 7𝑥
𝑥
= 3
𝑥 − 7𝑥
Even
𝑓 𝑥+ℎ −𝑓 𝑥
16) Determine the difference quotient,
ℎ
2
𝑓 𝑥 = 10𝑥 + 13𝑥 − 11
, for the function
𝑓 𝑥+ℎ −𝑓 𝑥
ℎ
10 𝑥 + ℎ 2 + 13 𝑥 + ℎ − 11 − (10𝑥 2 + 13𝑥 − 11)
=
ℎ
10 𝑥 2 + 2𝑥ℎ + ℎ2 + 13𝑥 + 13ℎ − 11 − 10𝑥 2 − 13𝑥 + 11
=
ℎ
10𝑥 2 + 20𝑥ℎ + 10ℎ2 + 13𝑥 + 13ℎ − 11 − 10𝑥 2 − 13𝑥 + 11
=
ℎ
20𝑥ℎ + 10ℎ2 + 13ℎ
=
ℎ
= 20𝑥 + 10ℎ + 13
20𝑥 + 13 + 10ℎ
17) In Country X, the average hourly wage in dollars from
1945 to 1995 can be modeled by
𝑓 𝑥 =
0.076 𝑥 − 1945 + 0.36 𝑖𝑓 1945 ≤ 𝑥 < 1970
0.184 𝑥 − 1970 + 3.07 𝑖𝑓 1970 ≤ 𝑥 ≤ 1995
Use 𝑓 to estimate the average hourly wages
in 1950, 1970, and 1990.
𝑓 1950 = 0.076 1950 − 1945 + 0.36 = 0.74
𝑓 1970 = 0.184 𝑥 − 1970 + 3.07 = 3.07
𝑓 1990 = 0.184 1990 − 1970 + 3.07 = 6.75
$0.74, $3.07, $6.75
18) Write an equation for the function whose
graph fits the following description.
The graph of 𝑓 𝑥 = |𝑥| is shifted three units left,
reflected in the y-axis and shifted two units
down.
𝑠ℎ𝑖𝑓𝑡𝑒𝑑 𝑡ℎ𝑟𝑒𝑒 𝑢𝑛𝑖𝑡𝑠 𝑙𝑒𝑓𝑡: 𝑦 = 𝑥 + 3
𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑦 − 𝑎𝑥𝑖𝑠: 𝑦 = −𝑥 + 3
𝑠ℎ𝑖𝑓𝑡𝑒𝑑 2 𝑢𝑛𝑖𝑡𝑠 𝑑𝑜𝑤𝑛: 𝑦 = −𝑥 + 3 − 2
𝑦 = −𝑥 + 3 − 2
19) Find the vertex of the parabola 𝑦 = 𝑥 2 + 6𝑥 + 10.
𝑎 = 1, 𝑏 = 6, 𝑐 = 10
𝑏
6
6
−
=−
= − = −3
2𝑎
2∙1
2
𝑓 −3 = −3
2
+ 6 −3 + 10
= 9 − 18 + 10
=1
OR
graph the function and find the minimum.
(−3,1)
20a) Find the vertical asymptotes, if any, of
𝑥+5
𝑔 𝑥 =
𝑥−4
𝑥−4=0
⇨𝑥=4
𝑥=4
20b) Find the horizontal asymptotes, if any, of
−4𝑥 2 + 7
ℎ 𝑥 =
4𝑥 2 − 1
𝑑𝑒𝑔𝑟𝑒𝑒𝑠 𝑎𝑟𝑒 𝑒𝑞𝑢𝑎𝑙
4
⇨𝑦= −
4
⇨ 𝑦 = −1
𝑦 = −1
21) The volume V of a gas varies inversely as the pressure P on it. The volume of a
𝑘𝑔
gas is 190 𝑐𝑚3 under a pressure of 21 𝑐𝑚2 . What will its volume be under a
𝑘𝑔
pressure of 35 2?
𝑐𝑚
Assume the temperature is constant.
𝑘
𝑉=
𝑃
𝑘
190 =
21
𝑘 = 3990
3990
3990
𝑉=
⇨𝑉=
= 114
𝑃
35
114 𝑐𝑚3
22a) Solve the equation 8𝑥−2 = 644𝑥
8
𝑥−2
= 82
4𝑥
𝑥 − 2 = 2 4𝑥
𝑥 − 2 = 8𝑥
−7𝑥 = 2
2
𝑥= −
7
2
{− }
7
22b) Solve. Please round your answer in part b to
three decimal places. 2𝑥 = 52𝑥+1
ln 2𝑥 = ln 52𝑥+1
𝑥 ln 2 = 2𝑥 + 1 ln 5
𝑥 ln 2 = 2𝑥 ln 5 + ln 5
𝑥 ln 2 − 2𝑥 ln 5 = ln 5
𝑥 (ln 2 − 2 ln 5) = ln 5
ln 5
𝑥=
≈ −0.637
ln 2 − 2 ln 5
{−0.637}
23a) Determine the final value of the given amount:
$820 at 8% compounded continuously for 2 years
𝐴 = 𝑃𝑒 𝑟𝑡
𝐴 = 820𝑒 0.08
= 820𝑒 .16
≈ 962.28
$962.28
2
23b) Randy invested his inheritance in an account that paid 6.2%
interest, compounded continuously. After ten years, he found that
he now had $62,978.62. What was the original amount of his
inheritance?
62978.62 = 𝑃𝑒 .062∙10
62978.62
𝑃=
𝑒 .62
≈ 33879
$33,879
23c) Find the time required for an initial investment of $10,000 to grow to
$18,000 at an annual rate of 6% if the interest is compounded quarterly.
𝑟
𝐴=𝑃 1+
𝑛
𝑛𝑡
0.06
18000 = 10000 1 +
4
1.8 = 1.0154𝑡
ln 1.8 = ln 1.0154𝑡
ln 1.8 = 4𝑡 ln 1.015
ln 1.8
𝑡=
≈ 9.87
4 ln 1.015
9.87 𝑦𝑒𝑎𝑟𝑠
4𝑡
24a) Write the expression in expanded form.
𝑥4𝑧
log 8
𝑦
𝑥4𝑧
log 8
𝑦
= log 𝑥 4 𝑧 − log 𝑦 8
= log 𝑥 4 + log 𝑧 − 8 log 𝑦
= 4 log 𝑥 + log 𝑧 − 8 log 𝑦
4 log 𝑥 + log 𝑧 − 8 log 𝑦
24b) Write the expression in condensed form.
1
𝑙𝑜𝑔𝑎 𝑥 + 5𝑙𝑜𝑔𝑎 𝑦 − 2𝑙𝑜𝑔𝑎 𝑥
2
1
𝑙𝑜𝑔𝑎 𝑥 2
+ 𝑙𝑜𝑔𝑎 𝑦 5 − 𝑙𝑜𝑔𝑎 𝑥 2
= 𝑙𝑜𝑔𝑎 𝑥 1/2 𝑦 5 − 𝑙𝑜𝑔𝑎 𝑥 2
= 𝑙𝑜𝑔𝑎
= 𝑙𝑜𝑔𝑎
𝑙𝑜𝑔𝑎
1
𝑥 2𝑦5
𝑥2
𝑦5
𝑥 3/2
𝑦5
3
𝑥2
𝑂𝑅
𝑙𝑜𝑔𝑎
𝑦5
𝑥3
3
25a) Solve. log 27 1 − 𝑥 =
1
273
3
1
3
3
= 1−𝑥
3
27 = 1 − 𝑥
27 = 1 − 𝑥
1 − 𝑥 = 27
−𝑥 = 26
𝑥 = −26
Note the lack of domain issues. When you substitute −26
in for 𝑥, the result is the cube root of 27 which is 3 and
certainly greater than zero.
{−26}
25b) Solve.
2
log(𝑥 +5𝑥
+ 16) = 1
101 = 𝑥 2 + 5𝑥 + 16
10 = 𝑥 2 + 5𝑥 + 16
𝑥 2 + 5𝑥 + 16 = 10
𝑥 2 + 5𝑥 + 6 = 0
𝑥+3 𝑥+2 =0
𝑥 = −3, −2
Note there are no domain issues. The first step indicates
that the solutions will make the expression inside the
logarithm equal to 10, which is certainly greater than zero.
{−3, −2}
25c) Solve.
log 𝑥 − 9 = 1 − log 𝑥
log 𝑥 − 9 + log 𝑥 = 1
log 𝑥 − 9 𝑥 = 1
101 = 𝑥 − 9 𝑥
10 = 𝑥 2 − 9𝑥
𝑥 2 − 9𝑥 − 10 = 0
𝑥 − 10 𝑥 + 1 = 0
𝑥 = 10, 𝑥 = −1
There are domain issues.
𝑥 ≠ −1 𝑎𝑠 log −1 − 9 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡
{10}
Extra Practice
1) Solve.
5𝑥 2 − 6𝑥 = 4𝑥 2 + 6𝑥 − 3
1) Solve.
5𝑥 2 − 6𝑥 = 4𝑥 2 + 6𝑥 − 3
𝑥 2 − 12𝑥 + 3 = 0
⇨ 𝑎 = 1, 𝑏 = −12, 𝑐 = 3
𝑥=
=
=
−𝑏 ±
−(−12) ±
12 ±
𝑏2 − 4𝑎𝑐
2𝑎
(−12)2 − 4(1)(3)
2(1)
144 − 12 12 ± 132
=
2
2
12 ± 4 ∙ 33 12 ± 2 33
=
=
2
2
= 6 ± 33
{6 − 33, 6 + 33 }
2) Find the discriminant and the number and type of roots:
16x 2 − 88x + 121 = 0
2) Find the discriminant and the number and type of roots:
16x 2 − 88x + 121 = 0
𝑏 2 − 4𝑎𝑐 = −88
2
− 4 16 121
= 7744 − 7744
=0
1 𝑟𝑒𝑎𝑙 𝑟𝑜𝑜𝑡
3) Use the intersection – of – graphs method to approximate the
solution to nearest hundredth:
4
−π𝑥 + 5 = 9 + 0.4𝑥
3) Use the intersection – of – graphs method to approximate the
solution to nearest hundredth:
4
−π𝑥 + 5 = 9 + 0.4𝑥
𝑌1 = −π𝑥 + 5
4
𝑌2 = 9 + 0.4𝑥
{0.92}
4a) Solve.
3𝑥 − 3 ≥ 15
4a) Solve.
3𝑥 − 3 ≥ 15
3𝑥 − 3 ≤ −15 𝑜𝑟 3𝑥 − 3 ≥ 15
3𝑥 ≤ −12 𝑜𝑟 3𝑥 ≥ 18
𝑥 ≤ −4 𝑜𝑟 𝑥 ≥ 6
(−∞, −4]𝑈 [6, ∞)
4b)Solve.
3𝑥 + 4 ≤ 19
4b) Solve.
3𝑥 + 4 ≤ 19
−19 ≤ 3𝑥 + 4 ≤ 19
−23 ≤ 3𝑥 ≤ 15
23
15
−
≤𝑥≤
3
3
23
[− , 5]
3
5a) Find the distance between P and Q.
𝑃 −4, 1 , 𝑄 −7, −9
5a) Find the distance between P and Q.
𝑃 −4, 1 , 𝑄 −7, −9
𝑑=
=
(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
( −7 − −4 )2 + ((−9) − (1))2
=
−3
=
2
+ −10
109
2
5b) Find the midpoint of P and Q.
𝑃 −4, 1 , 𝑄 −7, −9
5b) Find the midpoint of P and Q.
𝑃 −4, 1 , 𝑄 −7, −9
𝑥1 + 𝑥2 𝑦1 + 𝑦2
𝑀=(
,
)
2
2
−4 + −7 1 + −9
=(
,
)
2
2
−11 −8
=
,
2
2
11
(− , −4)
2
6) Test the equation for symmetry:
𝑦 = 2𝑥 2 − |𝑥|
6) Test the equation for symmetry:
𝑦 = 2𝑥 2 − |𝑥|
𝑥: −𝑦 = 2𝑥 2 − |𝑥| 𝑛𝑜
𝑦: 𝑦 = 2 −𝑥
2
− |𝑥| ⇨ 𝑦 = 2𝑥 2 − |𝑥| 𝑦𝑒𝑠
𝑜𝑟𝑖𝑔𝑖𝑛: −𝑦 = 2 −𝑥
2
− 𝑥 ⇨ −𝑦 = 2𝑥 2 − |𝑥|
no
𝑦 − 𝑎𝑥𝑖𝑠
7) Find the center and radius of the circle.
𝑥 2 + 𝑦 2 − 2𝑥 − 2𝑦 − 4 = 0
7) Find the center and radius of the circle.
𝑥 2 + 𝑦 2 − 2𝑥 − 2𝑦 − 4 = 0
𝑥 2 − 2𝑥 +
+ 𝑦 2 − 2𝑦 +
=4
1
−2 = −1, (−1)2 = 1
2
𝑥 2 − 2𝑥 + 1 + 𝑦 2 − 2𝑦 + 1 = 4 + 1 + 1
𝑥−1 2+ 𝑦−1 2 =6
𝑥 − ℎ 2 + 𝑦 − 𝑘 2 = 𝑟2
1, 1 , 𝑟 = 6
8) Find the equation of the circle that satisfies the given
conditions: C𝑒𝑛𝑡𝑒𝑟 𝑎𝑡 −2, −3 , 𝑟𝑎𝑑𝑖𝑢𝑠 7
8) Find the equation of the circle that satisfies the given
conditions: C𝑒𝑛𝑡𝑒𝑟 𝑎𝑡 −2, −3 , 𝑟𝑎𝑑𝑖𝑢𝑠 7
𝑥 −ℎ
𝑥 − (−2
2
2
+ 𝑦 −𝑘
2
+ 𝑦— (−3)
(𝑥 + 2)2 + 𝑦 + 3
2
= 𝑟2
2
= 7
=7
2
9) Find the slope of the line through the given pair of points.
3, −2 , (2, −3)
9) Find the slope of the line through the given pair of points.
3, −2 , (2, −3)
𝑦2 −
𝑚=
𝑥2 −
𝑦1
𝑥1
−3 − (−2)
=
2−3
−1
=
−1
=1
10) A manufacturer produces 50 TVs at a cost of $17,500 and 75 TVs at a cost of
$21,250. Write a linear equation that models the cost in terms of the number of TVs
10) A manufacturer produces 50 TVs at a cost of $17,500 and 75 TVs at a cost of
$21,250. Write a linear equation that models the cost in terms of the number of TVs
50 𝑇𝑉𝑠 ⇨ $17,500: 50, 17500
75 𝑇𝑉𝑠 ⇨ $21,250: 75, 21250
21250 − 17500 3750
𝑚=
=
= 150
75 − 50
25
𝑦 − 17500 = 150 𝑥 − 50
𝑦 − 17500 = 150𝑥 − 7500
𝑦 = 150𝑥 + 10000
CHECK:
𝑦 = 150 50 + 10000 = 17500 𝑦𝑒𝑠 ,
𝑦 = 150 75 + 10000 = 21250 (𝑦𝑒𝑠)
𝑦 = 150𝑥 + 10000
11) The paired data consists of the number of newspapers (in thousands)
that sell at a given price (in nickels) per copy for a college newspaper.
Find the equation of the regression line that models the data.
Newspapers
1
2
3
4
5
Price
11
8
6
4
3
11) The paired data consists of the number of newspapers (in thousands)
that sell at a given price (in nickels) per copy for a college newspaper.
Find the equation of the regression line that models the data.
Newspapers
1
2
3
4
5
Price
11
8
6
4
3
𝑃𝑙𝑎𝑐𝑒 𝑡ℎ𝑒 "Newspapers" 𝑖𝑛 𝐿1
𝑎𝑛𝑑 𝑡ℎ𝑒 "𝑃𝑟𝑖𝑐𝑒" 𝑖𝑛 𝐿2
𝑆𝑇𝐴𝑇 ⇨ 𝐶𝐴𝐿𝐶 ⇨ 4
𝑎 = −2, 𝑏 = 12.4
𝑦 = −2𝑥 + 12.4
12) Solve.
5𝑥 − 𝑦 = 5
3𝑥 + 2𝑦 = −10
12) Solve.
5𝑥 − 𝑦 = 5
3𝑥 + 2𝑦 = −10
2(5𝑥 − 𝑦 = 5)
3𝑥 + 2𝑦 = −10
10𝑥 − 2𝑦 = 10
3𝑥 + 2𝑦 = −10
13𝑥 = 0
𝑥=0
5 0 −𝑦 =5
−𝑦 = 5
𝑦 = −5
{ 0, −5 }
13) Last year Mrs. Garcia invested $50,000. She put part of the money in a real estate
venture that paid 7.5% and the rest in a small – business venture that paid 12%. The
combined income from the two investments was $5190. How much did she invest in each?
13) Last year Mrs. Garcia invested $50,000. She put part of the money in a real estate
venture that paid 7.5% and the rest in a small – business venture that paid 12%. The
combined income from the two investments was $5190. How much did she invest in each?
𝑥 = 𝑚𝑜𝑛𝑒𝑦 𝑖𝑛 𝑟𝑒𝑎𝑙 𝑒𝑠𝑡𝑎𝑡𝑒, 𝑦 = 𝑚𝑜𝑛𝑒𝑦 𝑖𝑛 𝑠𝑚𝑎𝑙𝑙 𝑏𝑢𝑠𝑖𝑛𝑒𝑠𝑠
𝑥 + 𝑦 = 50000
0.075𝑥 + 0.12𝑥 = 5190 → 75𝑥 + 120𝑦 = 5190000
𝑒𝑙𝑖𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛
−75 𝑥 + 𝑦 = 50000
75𝑥 + 120𝑦 = 5190000
−75𝑥 − 75𝑦 = −3750000
75𝑥 + 120𝑦 = 5190000
45𝑦 = 1440000
𝑦 = 32000
𝑥 + 32000 = 50000 → 𝑥 = 18000
$18,000 in real estate, $32,000 in small business
14) By using a graphing calculator, find the increasing interval(s)
1
for the function f x = 𝑥 3 − 4𝑥
3
14) By using a graphing calculator, find the increasing interval(s)
1
for the function f x = 𝑥 3 − 4𝑥
3
Choose windows large enough to allow
you to see the turning points:
𝑥: −10, 10
𝑦: −10, 10
−∞, −2 , (2, ∞)
15) Determine if the given function is even, odd, or neither.
𝑥2 + 2
𝑓 𝑥 = 4
𝑥 +1
15) Determine if the given function is even, odd, or neither.
𝑥2 + 2
𝑓 𝑥 = 4
𝑥 +1
(−𝑥)2 +2
𝑓 −𝑥 =
(−𝑥)4 +1
𝑥2 + 2
= 4
𝑥 +1
𝐸𝑣𝑒𝑛
𝑓 𝑥+ℎ −𝑓 𝑥
16) Determine the difference quotient,
ℎ
2
𝑓 𝑥 = 3𝑥 − 2𝑥 + 5
, for the function
𝑓 𝑥+ℎ −𝑓 𝑥
16) Determine the difference quotient,
ℎ
2
𝑓 𝑥 = 3𝑥 − 2𝑥 + 5
, for the function
𝑓 𝑥+ℎ −𝑓 𝑥
ℎ
2
3 𝑥 + ℎ − 2 𝑥 + ℎ + 5 − (3𝑥 2 − 2𝑥 + 5)
=
ℎ
3 𝑥 2 + 2𝑥ℎ + ℎ2 − 2𝑥 − 2ℎ + 5 − 3𝑥 2 + 2𝑥 − 5
=
ℎ
3𝑥 2 + 6𝑥ℎ + 3ℎ2 − 2𝑥 − 2ℎ + 5 − 3𝑥 2 + 2𝑥 − 5
=
ℎ
6𝑥ℎ + 3ℎ2 − 2ℎ
=
ℎ
= 6𝑥 + 3ℎ − 2
17) Suppose a state’s income tax code states that the tax
liability T on x dollars of taxable income as follows
T 𝑥 =
0.04𝑥
800 + 0.06𝑥
𝑖𝑓 0 ≤ 𝑥 < 20,000
𝑖𝑓 𝑥 ≥ 20,000
Use 𝑇 to find the tax liability on each taxable income:
$12,000, $20,000, $50,000
17) Suppose a state’s income tax code states that the tax
liability T on x dollars of taxable income as follows
T 𝑥 =
0.04𝑥
800 + 0.06𝑥
𝑖𝑓 0 ≤ 𝑥 < 20,000
𝑖𝑓 𝑥 ≥ 20,000
Use 𝑇 to find the tax liability on each taxable income:
$12,000, $20,000, $50,000
𝑓 12000 = 0.04(12000) = $480
𝑓 20000 = 800 + 0.06 20000 = $2000
𝑓 50000 = 800 + 0.06 50000 = $3800
$480, $2000, $3800
18) Write an equation for the function whose
graph fits the following description.
The graph of 𝑓 𝑥 = 𝑥 2 is shifted five units right,
stretched vertically by a factor of three, reflected in
the y-axis and shifted four units up.
18) Write an equation for the function whose
graph fits the following description.
The graph of 𝑓 𝑥 = 𝑥 2 is shifted five units right, stretched vertically by a
factor of three, reflected in the y-axis and shifted four units up.
𝑠ℎ𝑖𝑓𝑡𝑒𝑑 𝑓𝑖𝑣𝑒 𝑢𝑛𝑖𝑡𝑠 𝑟𝑖𝑔ℎ𝑡: 𝑦 = 𝑥 − 5
2
𝑠𝑡𝑟𝑒𝑡𝑐ℎ𝑒𝑑 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑏𝑦 𝑎 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑡ℎ𝑟𝑒𝑒: 𝑦 = 3 𝑥 − 5
𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑦 − 𝑎𝑥𝑖𝑠: 𝑦 = 3 −𝑥 − 5
𝑠ℎ𝑖𝑓𝑡𝑒𝑑 4 𝑢𝑛𝑖𝑡𝑠 𝑢𝑝: 𝑦 = 3 −𝑥 − 5
𝑦 = 3 −𝑥 − 5
2
+4
2
+4
2
2
19) Find the vertex of the parabola 𝑦 = −2𝑥 2 + 10𝑥 − 3.
19) Find the vertex of the parabola 𝑦 = −2𝑥 2 + 10𝑥 − 3.
𝑎 = −2, 𝑏 = 10, 𝑐 = −3
𝑏
10
10 5
−
=−
=−
= (𝑜𝑟 2.5)
2𝑎
2 ∙ −2
−4 2
5
5
𝑓
= −2
2
2
=−
2
+ 10
5
−3
2
25
+ 25 − 3
2
19
=
(𝑜𝑟 9.5)
2
OR
graph the function and find the maximum.
5 19
,
𝑜𝑟 (2.5, 9.5)
2 2
20a) Find the vertical asymptotes, if any, of
(𝑥 − 3)(2𝑥 − 2)
𝑔 𝑥 =
(𝑥 − 3)(𝑥 + 4)
20a) Find the vertical asymptotes, if any, of
(𝑥 − 3)(2𝑥 − 2)
𝑔 𝑥 =
(𝑥 − 3)(𝑥 + 4)
𝑂𝑛𝑙𝑦 𝑤ℎ𝑒𝑟𝑒 𝑥 + 4 = 0
𝑥 = −4
(𝑛𝑜𝑡 " − 4" → 𝑦𝑜𝑢 𝑚𝑢𝑠𝑡 𝑤𝑟𝑖𝑡𝑒 𝑥 = −4
𝑠𝑖𝑛𝑐𝑒 𝑖𝑡 𝑖𝑠 𝑎 𝑙𝑖𝑛𝑒)
20b) Find the horizontal asymptotes, if any, of
2𝑥 − 3
ℎ 𝑥 =
3𝑥 + 5
20b) Find the horizontal asymptotes, if any, of
2𝑥 − 3
ℎ 𝑥 =
3𝑥 + 5
𝑑𝑒𝑔𝑟𝑒𝑒𝑠 𝑎𝑟𝑒 𝑒𝑞𝑢𝑎𝑙
2
⇨𝑦=
3
2
2
(𝑛𝑜𝑡 → 𝑦𝑜𝑢 𝑚𝑢𝑠𝑡 𝑤𝑟𝑖𝑡𝑒 𝑦 =
3
3
𝑠𝑖𝑛𝑐𝑒 𝑖𝑡 𝑖𝑠 𝑎 𝑙𝑖𝑛𝑒)
21) The safe load that a rectangular beam can support varies
jointly as the width and square of the depth of the beam and
inversely as its length. A beam 4 inches wide, 6 inches deep,
and 25 feet long can support a safe load of 576 pounds. Find
the safe load for a beam that is of the same material but is 6
inches wide, 10 inches deep, and 20 feet long.
21) The safe load that a rectangular beam can support varies
jointly as the width and square of the depth of the beam and
inversely as its length. A beam 4 inches wide, 6 inches deep,
and 25 feet long can support a safe load of 576 pounds. Find
the safe load for a beam that is of the same material but is 6
inches wide, 10 inches deep, and 20 feet long.
𝑘𝑤𝑑 2
𝑆=
𝑙
𝑘(4)(62 )
576 =
25
14400 = 144𝑘
𝑘 = 100
100𝑤𝑑 2
100(6)(102 )
𝑆=
⇨𝑆=
= 3000
𝑙
20
3000 𝑝𝑜𝑢𝑛𝑑𝑠
22a) Solve.
3𝑥 = 9 𝑥−1 ⋅ 27
1−3𝑥
22a) Solve.
3𝑥 = 9 𝑥−1 ⋅ 27
𝑥−1
1−3𝑥
1−3𝑥
3𝑥 = 3 2
⋅ 3 3
3𝑥 = 32𝑥−2 · 33−9𝑥
3𝑥 = 31−7𝑥
𝑥 = 1 − 7𝑥
8𝑥 = 1
1
𝑥=
8
1
{ }
8
22b) Solve. Please round your answer to three decimal places.
3𝑥 −2 = 43−5𝑥
22b) Solve. Please round your answer to three decimal places.
3𝑥 −2 = 43−5𝑥
ln 3𝑥−2 = ln 43−5𝑥
𝑥 − 2 ln 3 = 3 − 5𝑥 ln 4
𝑥 ln 3 − 2 ln 3 = 3 ln 4 − 5𝑥 ln 4
𝑥 ln 3 + 5𝑥 ln 4 = 2 ln 3 + 3 ln 4
𝑥(ln 3 + 5 ln 4) = 2 ln 3 + 3 ln 4
2 ln 3 + 3 ln 4
𝑥=
≈ 0.792
ln 3 + 5 ln 4
{0.792}
23a) How much does Amy have to invest at 8% compounded
continuously to have a balance of $19584.76 after ten years?
23a) How much does Amy have to invest at 8% compounded
continuously to have a balance of $19584.76 after ten years?
𝐴 = 𝑃𝑒 𝑟𝑡
19584.76 = 𝑃𝑒 0.08∗10
19584.76 = 𝑃𝑒 0.8
19584.76
𝑃=
≈ 8800
0.8
𝑒
$8,800
23b) How long will it take an investment of $500 to double
if the money is invested at 6% compounded monthly ?
23b) How long will it take an investment of $500 to double
if the money is invested at 6% compounded monthly ?
𝑟
𝐴=𝑃 1+
𝑛
𝑛𝑡
0.06
1000 = 500 1 +
12
12𝑡
2 = 1.00512𝑡
ln 2 = ln 1.00512𝑡
ln 2 = 12𝑡 ln 1.005
ln 2
𝑡=
≈ 11.58
12 ln 1.005
11.58 𝑦𝑒𝑎𝑟𝑠
24a) Write the expression in expanded form.
𝑦+3 2
l𝑛[
]
3
𝑥 −2
24a) Write the expression in expanded form.
𝑦+3 2
l𝑛[
]
3
𝑥 −2
= ln 𝑦 + 3
2
− ln 𝑥 − 2
3
= 2 ln(𝑦 + 3) − 3 ln(𝑥 − 2)
24b) Write the expression in condensed form.
1
2 ln 𝑥 − ln(𝑥 2 + 1)
2
24b) Write the expression in condensed form.
1
2 ln 𝑥 − ln(𝑥 2 + 1)
2
1
2 ln 𝑥 − ln(𝑥 2 + 1)
2
= ln 𝑥 2 − ln 𝑥 2 + 1
1
2
= ln 𝑥 2 − ln 𝑥 2 + 1
= ln
𝑥2
𝑥2 + 1
25a) Solve.
log 2 5𝑥 + 1 = −2
25a) Solve.
log 2 5𝑥 + 1 = −2
2−2 = 5𝑥 + 1
1
= 5𝑥 + 1
22
1
= 5𝑥 + 1
4
1
5𝑥 + 1 =
4
5𝑥 =
1
−1
4
5𝑥 = −
𝑥=−
3
4
3
20
1
𝑅𝑒𝑚𝑒𝑚𝑏𝑒𝑟 − 𝑡ℎ𝑒𝑟𝑒 𝑎𝑟𝑒 𝑛𝑜 𝑑𝑜𝑚𝑎𝑖𝑛 𝑖𝑠𝑠𝑢𝑒𝑠 𝑠𝑖𝑛𝑐𝑒 5𝑥 + 1 𝑤𝑎𝑠 𝑠𝑒𝑡 𝑡𝑜 .
4
1
𝑥 𝑤𝑖𝑙𝑙 𝑐𝑎𝑢𝑠𝑒 5𝑥 + 1 𝑡𝑜 𝑏𝑒 𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 𝑧𝑒𝑟𝑜.
4
{−
3
}
20
25b) Solve.
log 𝑥 + 8 + log 𝑥 − 1 = 1
25b) Solve.
log 𝑥 + 8 + log 𝑥 − 1 = 1
log 𝑥 + 8 + log(𝑥 − 1) = 1
log 𝑥 + 8 𝑥 − 1 = 1
101 = 𝑥 + 8 𝑥 − 1
10 = 𝑥 2 + 7𝑥 − 8
𝑥 2 + 7𝑥 − 18 = 0
𝑥+9 𝑥−2 = 0
𝑥 = −9 𝑥 = 2
𝑥 ≠ −9
{2}