Welcome to
Jeopardy!
AP Calculus Contestants!
Call 911! We
Need a
Parametric.
Too hip to
be squared.
Can You
Function in
the Morning?
Opposites
Attract
I Saw
the Sine.
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Graph the curve and
determine the initial and
terminal points, if any.
𝒙 = 𝟒 𝐬𝐢𝐧 𝒕
𝒚 = 𝟓 𝐜𝐨𝐬 𝒕
𝟎 ≤ 𝒕 ≤ 𝟐𝝅
And the Answer Is:
𝑰𝒏𝒊𝒕𝒊𝒂𝒍 𝒂𝒏𝒅 𝑻𝒆𝒓𝒎𝒊𝒏𝒂𝒍 𝑷𝒐𝒊𝒏𝒕𝒔: (𝟎, 𝟓)
Back to the Board.
Find a Cartesian equation for a curve
that contains the parametrized curve.
What portion of the graph of the
Cartesian equation is traced by the
parametrized curve?
𝟐
𝒙 = 𝒔𝒆𝒄 𝒕 − 𝟏
𝒚 = 𝐭𝐚𝐧 𝒕
𝝅
𝝅
− <𝒕<
𝟐
𝟐
And the Answer Is:
𝒙=
𝟐
𝒚
All
Back to the Board.
Find the values of t that produce
the graph in Quadrant IV.
𝒙=𝟑− 𝒕
𝒚=𝒕−𝟏
−𝟓 ≤ 𝒕 ≤ 𝟓
And the Answer Is:
−𝟑 < 𝒕 < 𝟏
Back to the Board.
Find a parametrization for the
part of the graph that lies in
Quadrant I.
𝒚= 𝒙+𝟑
Possible Answers Include:
𝒙=𝒕
𝒚= 𝒕+𝟑
𝒕<𝟎
Back to the Board.
Find a parametrization
for the left half of
the parabola
𝟐
𝒚 = 𝒙 + 𝟐𝒙
Possible Answers Include:
𝒙=𝒕
𝟐
𝒚 = 𝒕 + 𝟐𝒕
𝒕 ≤ −𝟏
Back to the Board.
Rewrite the following
expression to have
base 3:
𝟏
𝟐𝟕
𝒙
And the Answer Is:
𝟑
Back to the Board.
−𝟑𝒙
Determine how much time
is required for an
investment to triple in
value if interest is earned
at the rate of 5.75%
compounded daily.
And the Answer Is:
≈ 19.106 𝑦𝑒𝑎𝑟𝑠
Back to the Board.
If John invests $2300 in a
savings account with a
6% interest rate
compounded annually,
how long will it take until
John’s account has a
balance of $4150?
And the Answer Is:
≈ 𝟏𝟎. 𝟏𝟐𝟗 𝒚𝒆𝒂𝒓𝒔
Back to the Board.
The half life of a certain radioactive
substance is 12 hours. There are 8
grams present initially.
a) Express the amount of
substance remaining as
a function of time.
b) When will there be 1
gram remaining?
And the Answer Is:
a) Amount = 𝟖
𝟏
𝟐
b) After 36 hours
Back to the Board.
𝒕
𝟏𝟐
The population of Glenbrook
is 375,999 and is increasing
at the rate of 2.25% per
year. Predict when the
population will be 1 million.
And the Answer Is:
After about
44.081 years
Back to the Board.
Write an equation for the
lines parallel and
perpendicular to the line
𝟐𝒙 + 𝒚 = 𝟒 and contains the
point 𝑷(−𝟐, 𝟐).
And the Answer Is:
𝑷𝒂𝒓𝒂𝒍𝒍𝒆𝒍: 𝒚 = −𝟐𝒙 − 𝟐
𝟏
𝑷𝒆𝒓𝒑𝒆𝒏𝒅𝒊𝒄𝒖𝒍𝒂𝒓: 𝒚 = 𝒙 + 𝟑
𝟐
Back to the Board.
Find the domain and
range of the following
function:
𝟏
𝒚= +𝟑
𝒙
And the Answer Is:
𝑫𝒐𝒎𝒂𝒊𝒏: 𝒙 ≠ 𝟎
𝑹𝒂𝒏𝒈𝒆: 𝒚 ≥ 𝟎
Back to the Board.
Determine if the following
function is even or odd:
𝟑
𝒙
𝒚= 𝟐
𝒙 −𝟏
And the Answer Is:
Odd function
Back to the Board.
Find the formula of
the piecewise function
displayed on
the following
graph:
And the Answer Is:
𝟑 − 𝒙,
𝒇 𝒙 =
𝟐𝒙,
Back to the Board.
𝒙≤𝟏
𝟏<𝒙
Find the composition of
functions 𝒇(𝒇 𝒙 ), 𝒇(𝒈 𝒙 ),
𝒈(𝒈 𝟓 ), and 𝒈(𝒇 −𝟏 ) when
𝒇 𝒙 =
𝟏
𝒙𝟐
and 𝒈 𝒙 =
𝒙−𝟏
And the Answer Is:
𝒇 𝒇 𝒙
𝟒
=𝒙
𝟏
𝒇 𝒈 𝒙 =
𝒙−𝟏
𝒈 𝒈 𝟓 =𝟏
𝒈 𝒇 −𝟏 = 𝟎
Back to the Board.
Is the function 𝐲 =
𝟏
𝟐
𝒙 +𝒙
one-to-one?
Explain why or why not.
And the Answer Is:
No because not every
output has only one
input. (Does not pass the
horizontal line test.)
Back to the Board.
Does the function
𝟐𝒙
𝒇(𝒙) = 𝒆 − 𝟐 have
an inverse?
1
If yes, find f If not,
explain why.
Back to the Board.
And the Answer Is:
Yes,
𝐥𝐧
𝒙
+
𝟐
1
f ( x) =
𝟐
Back to the Board.
Find the inverse of
the following function
and verify that
−𝟏
−𝟏
𝒇 𝒇 𝒙 = 𝒇 𝒇 𝒙 =𝒙:
𝟐
𝒇 𝒙 =𝒙 +𝟏
𝒙≥𝟎
And the Answer Is:
𝒇
−𝟏
𝒇 𝒇
𝒙
−𝟏
=
𝒇−𝟏 𝒇 𝒙
Back to the Board.
𝒙 = 𝒙−𝟏
𝒙−𝟏
=
𝟐
+𝟏= 𝒙−𝟏 +𝟏=𝒙
𝒙𝟐 + 𝟏 − 𝟏 =
𝒙𝟐 = 𝒙
Solve the following
equation algebraically and
support your answer
graphically.
𝒙
−𝒙
𝟐 +𝟐
=𝟓
And the Answer Is:
𝒙 = 𝐥𝐨𝐠 𝟐
𝟓 ± 𝟐𝟏
𝟐
≈ −𝟐. 𝟐𝟔 𝒐𝒓 𝟐. 𝟐𝟔
Back to the Board.
−𝟏
Graph 𝒇 𝒙 , 𝒇
𝒙
and 𝒚 = 𝒙 on the
same screen.
𝒇 𝒙 = 𝐥𝐨𝐠 𝒙
What do you notice?
And the Answer Is:
𝒇−𝟏 𝒙 𝒊𝒔 𝒂 𝒓𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏 𝒐𝒇 𝒇 𝒙 𝒐𝒗𝒆𝒓 𝒕𝒉𝒆 𝒍𝒊𝒏𝒆 𝒚 = 𝒙
Back to the Board.
Determine the period
and amplitude and
draw the graph of the
following function:
𝒚 = 𝟑 𝐜𝐬𝐜 𝟑𝒙 + 𝝅 − 𝟐
And the Answer Is:
𝟐𝝅
𝟑
Period:
Amplitude: 3
Back to the Board.
Find the value of the
six trigonometric
functions at 𝜽 given that
𝜽 = 𝐜𝐨𝐬
−𝟏
𝟗
𝟒𝟏
And the Answer Is:
𝟒𝟎
𝐬𝐢𝐧 𝜽 =
𝟒𝟏
𝟒𝟏
𝐜𝐬𝐜 𝜽 =
𝟒𝟎
𝟗
𝐜𝐨𝐬 𝜽 =
𝟒𝟏
𝟒𝟏
𝐬𝐞𝐜 𝜽 =
𝟗
𝟒𝟎
𝐭𝐚𝐧 𝜽 =
𝟗
𝟗
𝐜𝐨𝐭 𝜽 =
𝟒𝟎
Back to the Board.
Evaluate the following
expression:
𝐬𝐢𝐧 𝐜𝐨𝐭
−𝟏
𝟏𝟐
𝟓
And the Answer Is:
𝐬𝐢𝐧 𝐜𝐨𝐭
Back to the Board.
−𝟏
𝟏𝟐
𝟓
𝟓
=
𝟏𝟑
Show that 𝐜𝐬𝐜 𝒙 is an odd
function of x.
Using this, show that the
reciprocal of an odd
function is also odd.
And the Answer Is:
𝒓
𝒓
𝐜𝐬𝐜(−𝜽) =
=−
= − 𝐜𝐬𝐜 𝜽
−𝒚
𝒚
The reciprocal of cosecant is the sine function:
𝟏
−𝒚
𝒚
= 𝐬𝐢𝐧 −𝜽 =
=−
= −𝐬𝐢𝐧(𝜽)
𝐜𝐬𝐜 −𝜽
𝒓
𝒓
Back to the Board.
Solve the following
equation in the
specified interval:
𝐜𝐬𝐜 𝒙 = 𝟐
𝟎 < 𝒙 < 𝟐𝝅
And the Answer Is:
𝝅
𝟓𝝅
𝒙 = 𝐚𝐧𝐝
𝟔
𝟔
Back to the Board.