Day 12 Weekend HW Assignment #1: ~45min.
Assignment #2: ~45min. shall follow.
1. <10min. Basic graphing Review
Some graphing tips – study if you CARE enough…
NOTES A. You must master the graphs of parent functions:

y = x2,

y = x3,

y = √x = x½,

y = 3√x = x1/3,

y = |x|

y = 1/x

y = 1/x2

y = log x

y = ln x

y = ax [growth and decay]

y = ex

y = sin x

y = cos x

y = x2/3 [the graph with a cusp]
NOTES B. You must master the effects of simply transformations to f(x): Understand, dont
memorize!

y = af(x) → makes the graph rise faster [a > 1] or slower [0 < a < 1] than the parent
function

y = f(x ± a) → shifts the graph a units left / right

y = f(x) ± a → shifts the graph a units up / down

y = -f(x) → reflects the graph about the x-axis [flips it upside down]

y = f(-x) → reflects the graph about the y-axis [left- and right- potions swap]

y = f(|x|) → left potion of the graph is the same as the right, since the “heights” on the
negative axis are the same as that on the positive ~ whatever’s on the right is what
we have on the left

y = |f(x)| → all negative heights, if any, turn positive...and get flipped upwards
Plot all graphs on separate axes: copy the Q so I can do a quick assessment.
a) -3x + 4y = 8
a’
y = -2/x
b) y = ½ (x + 1)2 – 2
b’
y = 4/x
c) y = 5(x – 1)3 + 3
c’
y = 7/x2
d) y = ½|x + 3| – 1
d’
y = -1/x2
e) y = 4(x + 1)(x – 6)
e’
y = -√x + 2
f) y = -2·4x – 1 (Show asymptote)
f’
y = 3√x + 4
g) y = 3x2 – 1
g’
y = -2x2
h) y = ln (x + 2) (Show asymptote)
h’
y = x2 – 4x
i) y = – ex – 1 (Show asymptote)
i’
y = -3√x
j) y = – ½x
j’
y = -2x1/3
k) y = 4-x + 2 (Show asymptote)
k’
y = x2/3
l) y = -log(x – 2) (Show asymptote)
l’
y = x5/3
m) y = -5ex + 1 (Show asymptote)
m’
y = -2x4/3
n) y = –ln x + 2 (Show asymptote)
n’
y = -sin x
o) y = 3 cos 2x
o’
y = –log (-x) (Show asymptote)
p) y = ln (-x) (Show asymptote)
p’
y = ln (x – 1) (Show asymptote)
q) y = sin |x|
q’
y = ¼|x| + 2
r) y = | ln x |
r’
y = | x2 + 3x|
s) y = |cos x|
s’
y = e|x|
t) y = √-x
t’
y = |3√x|
u) y = | sin x|
u’
y = ln |x|
2. <10min. Solving Exponential, Logarithmic and Trigonometric Equations Review
Some equation-solving tips – study if you CARE enough…
NOTES

If youre in log form, go to expo form…and vice versa

When solving trig equations, use x-y-r formulas 1st to determine if a triangle can be
formed…since the angle might simply lie on the quadrant

Quadrant angle equations might yield 1 or 2 answers; triangle equations always yield
2 answers.

30-60-90 triangle ratios are 1-√3-2 and 45-45-90 triangle ratios are 1:1:√2

Use All-Students-Take-Calculus to determine which quadrants work

Adjust for the sign of the quadrant when writing ratios
Copy the Q so I can do a quick assessment: use RADIAN MODE.
a) e½x = 10
b) ln 3x = -1
c) 25x = 16
d) 10x – 2 = 4
e) log (2x – 3) = 13
f) sin θ = -½
g) cos θ = -1
h) tan θ = -1
i) sin θ = -√3/2
j) sin θ = 1
3. <10min. Solving inequalities…using the Number-Line trick!
Some Domain-solving tips – study if you CARE enough…

Polynomial function D: ALL real numbers since there are no breaks

[Even-] Radical function e√m D: set m > 0 since m must be non-negative

[Odd-] Radical function o√m D: ALL real numbers because we can find the oddroots of negative numbers!

Rational function, f = p(x)/q(x) D: exclude roots of q(x) since q(x) ≠ 0

Logarithmic functions log m D: set m > 0 since log of 0 or negative numbers are
undefined

Mixed Problems D: apply as many rules as they apply...and "merge" the results!

Caution! For log expressions in the denominator (1/log m), observe that log 1 = 0 →
insure that m ≠ 1 by solving m = 1

Caution! (x2 + k) does not have any real solutions i.e. (x2 + k) is never 0.

To solve inequalities, ALWAYS use the Number-line trick, ZOMG!
NOTES

start with + or – on the right-most end of the number-line depending on whether the
leading term is + or -

examine the exponent of the right-most root-term of the function and switch the sign
on the number-line if the exponent is ODD, else the sign stays if the exponent of the
corresponding root-term is EVEN

proceed left-wards, repeating the 2nd step for each root on the number line

Caution! After drawing the number-line and writing in the signs [+, -] we darken the
region on the number-line corresponding to the inequality in the Q

Write your final answers in interval notation.
Copy the Q so I can do a quick assessment.
a) Solve: x2 – 2x – 3 < 0
b) Solve: -x2 < -9 Tip! Move terms to left-side 1st!
c) Solve: (x + 4)2(x – 4)(x + 3) > 0
d) Solve: –x2(x + 1)(x – 2)3 < 0
e) Find the D of: f = √(x2 – 5x)
f) Solve: (x – 2)/(x + 1) < 0 Tip! Watch the V.A!
g) Find the D of: f = log [(x – 2)/(x + 4)2
h) Find the D of f = √(9 – 4x2) Tip! Do some algebra 1st.
i) Solve: (x – 4)(x – 2)3 /[(x + 1)(x + 2)2] > 0 Tip! Watch the V.A!
j) Find the D of: f = log √(x – 2)2
4. <15min. Graph the following Rational functions after finding the xI, yI, V.A. and H.a.
→ using the Number-Line trick!
NOTES

For xI, y = 0 ⟹ set the top = 0 and solve for x. It may be No Solution.

For yI, x = 0 ⟹ set all x-s = 0 and simplify. It may be No Solution.

For V.A., find values of x such that f is undefined ⟹ set the bottom = 0 and solve. It
may be No Solution.

For H.A., examine end-behaviour of the function as x →±∞ while only considering the
leading terms of the top and bottom. In case of 0, determine 0+ or 0-.
Copy the Q so I can do a quick assessment.
a) y = 1/x
b) y = (x2 + 4x – 5)/(x2 + 4x – 12)
c) y = 3/ (4 – x)
d) y = 5/(x – 3)2
e) y = 6x/ (x – 3)2
f) y = -2x/ (x + 1)
g) y = (x2 – 9)/(x2 – 1)
h) y = 3x/(x2 – x – 6)
i) y = 9x/(16 – x2)
j) y = -5/(x2 + 10)
k) y = 5x2/(x2 – 9)
l) y = (x – 1) / [x(x2 – 4)]
m) y = (x2 + 3)/(x2 – 9)
n) y = (x2 – 9)/(x2 + 4)
5. <5min. Find the following values without a calculator.
Copy the Q so I can do a quick assessment.
a) cos(2π/3)
b) tan (7π/6)
c) sin(3π/2)
d) cos(5π/4)
e) cos (5π/6)
f) ln (1/e2)
g) ln e
h) log 0.01
i) log √10
j) log5 (125)
k) ln 1
l) log √10
m) ln e-3