Precalc Q2 Bootcamp Jeopardy

Slope

100

Find the slope of the perpendicular line to:

x + 2y = 8

Answer: 2

y = -x/2 + 4  --> m = -1/2  --> slope = -1/-1/2 = 2

200

Find the slope of the perpendicular line to:

x - (1/3)y = 1

Answer: -1/3

y = 3x - 3  --> m = 3  --> slope = -1/3

300

Find the slope of the perpendicular line to:

(1/2)x + 3y = 6

Answer: 6

y = (-1/6)x + 2  --> m = -1/6  --> slope = -1/-1/6 = 6

400

Find the slope of the perpendicular line to:

(1/4)x - (1/8)y = (1/4)

Answer: -1/2

y = 2x - 2  --> m = 2  --> slope = 1/2

500

Find the slope of the perpendicular line to:

(2/3)x + (5/8)y = 2

Answer: 15/16

y = (-16/15)x + (16/5) --> m = -16/15  --> slope = -1/-16/15 = 15/16

Exponents/Logarithms

100

$5,400 was deposited into a bank account and invested at a rate of 4.1%. How much money is in the account after 14 years?

Answer: $9,477.72

A = 5400(1.041)¹⁴

200

Solve for x:

log₂(x - 2) = 3

Answer: x = 10

₂log₂(x - 2) = ₂3  --> x - 2 = 8  --> 

x = 10

300

Expand the logarithm:  log(x² - 25)

Answer: log(x + 5) + log(x - 5)

log((x+5)(x -5)) --> log(x+5) + log(x-5)

*Difference of two squares! 

400

Solve for x: 

log₃(x² - 13) = 2

Answer: x = root(22)

₃log₃(x² - 13) = ₃2  --> x² - 13 = 9 --> x² = 22  --> x = root(22)

500

Solve for x:

log₆(x² + 8x + 15) - log₆(x + 3) = 2

Answer: 31

₆log₆((x² + 8x + 15)/(x+3)) = ₆2

((x+3)(x+5)/(x+3)) = 36  --> x + 5 = 36  --> x = 31

Rational Functions/Remainders

100

Divide x² + 5x + 3 by x + 2 and find the remainder

Answer: -3   

200

Find the positive rational root:

2x³ - 7x² + 2x - 7

Answer: x = 7/2

x²(2x - 7) + 1(2x - 7) = 0

(x² + 1)(2x - 7) = 0   --> 2x - 7 = 0  --> x = 7/2

300

Divide x⁴ - 9x² + 8 by x² - 1

(Use Long Division)

Answer: x² - 8

400

Find the vertical and horizontal asymptotes:

y = (x - 8)/(x² - 4)

Answers: 

Vertical: x = -2 and x = 2

Horizontal: y = 0

500

Find the x and y intercepts:

y = (x² - 1)/(x² - 6x + 5)

Answers: 

x-intercept: x = -1   or  (-1,0)

y-intercept: y = -1/5  or  (0, -1/5)

Trigonometry

100

Convert 144 degrees to radians.

Answer: (4/5)*pi

200

Identify the amplitude, period, and phase shift for:

f(t) = (-1/4)sin(3t - 2*pi)

Answers:

amplitude = |a| = 1/4     side note: b = 3

period = 2*pi/b = 2*pi/3

phase shift = c/b = 2*pi/3

300

If tan(θ) = 5/12, then what is csc(θ)?

Answer: 13/5

opposite = 5 ; hypotenuse = 13

sin(θ) = 5/13  --> csc(θ) = 1/sin(θ) = 1/5/13 = 13/5

400

Simplify the trig expression:

(tan²x - sin²x)/(sec²x)

Answer: sin⁴x

(sin²x/cos²x) - (sin²xcos²x/cos²x) = (sin²x(1 - cos²x)) = sin²xsin²x = sin⁴x/cos²x

(sin⁴x/cos²x)/sec²x = sin⁴xcos²x/cos²x = sin⁴x

500

Solve for x:

root(2)/2 = sin(x - pi/4)

*There are 2 solutions.

Answers: x = pi/2  and  x = pi

x - pi/4 = pi/4  -->  x = pi/2

x - pi/4 = 3pi/4  -->  x = pi

Transformations and Inverses

100

f(x) = x² and g(x) is the transformed function.

What is the transformation function if f(x) moves 5 units to the left and 9 units down?

Answer: g(x) = f(x + 5) - 9

200

Find the inverse function of: 

y = lnx +2

Answer: y = e^x-2

x = lny + 2  --> x - 2 = lny --> _e(x-2) = _elny

e^(x-2) = y

300

Find the inverse function of:

y = 5e^2x

Answer: y = (1/2)ln(x/5)

x = 5e^2y  -->  x/5 = e^2y  --> ln(x/5) = ln(e^2y) --> ln(x/5) = 2y  --> (1/2)ln(x-5) = y

400

Let's say you are looking at number of households over the years in a region. 

Here are some points: (2010,40000) and (2015,60000).

What linear function illustrates this scenario?

Answer: y = 4000(x - 2010) + 40000

40000 = y-intercept

slope = (60000-40000)/(2015-2010) = 4000

x-variable is in terms of years so 2010 is your starting point.

500

Number of Households over the years since 2010 is illustrated by 

y = 4000(x - 2010) + 40000.

If y^-1 is the inverse function, then evaluate: 

y^-1(72000)

Answer: 2018

y^-1 = (x - 40000)/(4000) + 2010

y^-1(72000) = 8 + 2010 = 2018