Game Changer
Simplify: (2x3y-5)-3
Write your answer using positive exponents
y15 / 8x9
Graph the inequality
b less than or equal to -8
Closed circle at -8, shaded to the left
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Solve w2 = 9, where w is a real number.
Simplify your answer
w = 3 or -3
Solve (u + 3)2 - 64 = 0, where u is a real number. Simplify your answer.
u = 5 or -11
Draw a sample graph f(x) that shows the graph increasing from point A to point B.
See sample graphs
Draw the graph of y = |x| translated to produce y = |x+5| - 2.
See drawn graphs
Notes
Translations
(Example below with y = |x|
Vertical Shift
A) |x| + 3 Shift down 3 units
B) |x| + 3 Shift up 3 units
Horizontal Shift
A) |x - 3| Shift right 3 units
B) |x + 3| Shift left 3 units
Reflections
- To find f(-x) reflect across the y – axis (multiply x – coordinates by -1)
- To find -f(x) reflect across the x-axis (multiply by y – coordinates by -1)
Rewrite the expression without a negative exponenet
-4n-4
-(4/n4)
Note.
x-3 = 1/x3
Calculate log (37/8)
0.665
Find the horizontal asymptote(s) of f(x) = (2x - 1)/(x² - 4)
y = 0
Notes
Horizontal Asymptotes
The three rules that horizontal asymptotes follow are based on the degree of the numerator, n, and the degree of the denominator, m.
1. If n < m, the horizontal asymptote is y = 0.
2. If n = m, the horizontal asymptote is y = a/b.
3. If n > m, there is no horizontal asymptote.
Given g(x) = (1/7)x
Find g(0)
g(0) = 1
Find the range of f(x) = (2x - 1)/(x² - 4)
Use interval notation
(-infinity, infinity)
Note.
Range = y - coordinates
Find the vertical asymptote(s) of f(x) = (2x - 1)/(x² - 4)
x = 2 and x = -2
Note.
To find the vertical asymptotes, set the denominator equal to zero and solve for x.
Suppose
q(x) = 2x + 1
r(x) = -2x2 - 1
Find (r of q)(4)
(r of q)(4) =-163
Note. (r of q)(4) = r(q(4))
Graph the inequality
y<-1/2 x + 2
See individual graphs
Note.
< use dashed line and shade after test point
Find the domain of f(x) = (2x - 1)/(x² - 4)
Use interval notation
(-infinity, -2) U (-2, 2) U (2, infinity)
Note.
Domain = x - coordinates
Set denominator = 0 to get the limits
Solve for x. Round the answer to the nearest hundredth.
e-3x = 6
x = -0.60
Note.
ln is the inverse of e
Suppose
q(x) = 2x + 1
r(x) = -2x2 - 1
Find (q of r)(4)
(q of r)(4) = -65
Note.
(q of r)(4) = q(r(4))
Solve the system of equations.
-1/3x + 1/6y = 7
3/4x + 1/2y = -14
x = -20
y = 2
Use the change of base formula to compute log (base 5) of 4. Round your answer to the nearest thousandth.
0.861
Note.
Change of Base Formula
log(base b) of a = (log (base c) of a)/(log (base c) of b)
Find all excluded values for the expression. Find all values of y for which the expression is undefined.
(y + 3)/(y2 - 5y - 24)
y = -3, 8
Note.
Values that make the rational expression undefined are those that make the denominator = 0.
Graph the parabola
y = (x+3)2 - 4
See individual graphs
Notes
Translations
(Example below with y = |x|
Vertical Shift
A) |x| + 3 Shift down 3 units
B) |x| + 3 Shift up 3 units
Horizontal Shift
A) |x - 3| Shift right 3 units
B) |x + 3| Shift left 3 units
Given g(x) = (1/7)x
Find g(-1)
g(-1) = 7
-3 if -2.5 < x < -1.5
g(x) = -2 if -1.5 < x < -0.5
-1 if -0.5 < x < 0.5
0 if 0.5 < x < 1.5
1 if 1.5 < x < 2.5
See drawn graphs
How much should be invested now at an interest rate of 6.5% per year, compounded continuously, to have $3500 in four years?
Do not round the intermediate computations, and round the final answer to the nearest cent.
$2698.68
Note.
Continuous Compounding Formula
P = P0ert
P = Balance
P0=Initial Principal
r = Annual Interest Rate (decimal form)
t = Time (in years)
So, P0 = P/(ert)
Solve for x
log (base 9) of x = 2
x = 81
Note.
log(base b) of x = a [logarithmic form] equivalent to ba = x