ALEKS Math Placement
Translate this equation to the right 3 and up 4: f(x)=1/2x^2
(1/2(x-3)^2)+4
π/3
(1/2, sqrt(3)/2)
What is the vertex of this equation? -3(x-5)+2 (the parentheses are absolute val. symbols)
(5,2)
unit circle (x,y)
(cos, sin)
π/6
(sqrt(3)/2, 1/2)
π/4
(sqrt2/2, sqrt2/2)
circumference
2πr
When finding the domain of composition functions should you find the domain of the simplified function or the unsimplified function?
The unsimplified function (ESPECIALLY if there radicals)
How do you translate a graph side to side?
The value inside the parentheses (CHANGE SIGN)
How do you translate a graph up or down?
The value outside of the parentheses (DONT change sign)
x^a/b
b√x^a
y=af(x)
multiply y coordinate by a
If a question asks for all real zeroes should you include imaginary numbers?
no
Should you leave negatives in the denominator?
no (multiply by everything negative 1)
Should you round down when a problem asks for the smallest whole number?
no (round up to the next whole number regardless of the decimal size- it can be less than .5 and you'd still round up)
perimeter/ length
no power
sqrt(a/b) =
sqrt(a)/sqrt(b)
area
squared
When writing an equation or working ANY inequality problem what does it help to do first?
write out what's happening
What should you double check for EVERY time
writing problem down wrong (NEGATIVE SIGNS), uncarried negative signs, unflipped inequality signs, undotted lines, incorrect multiplication, fully simplified
x^a times x^b =
x^a+b
x^a/x^b
x^a-b
b√x^a
x^a/b
(x^a)^b
x^ab
graph ≥, ≤
solid line
π/2
(1,0)
antiderivative (x^a)
(x^a+1)/a+1
What is the quadratic formula?
-b±√b^2-4ac/2a
When you take the square root of a variable, how many answers are there?
2 answers (negative and positive)
What is the least common multiple of 25u^3v^4 and 10w^2u^5v^6
50u^5w^2v^6
log_6 36 = 2
6^2 = 36
When should you find the Lowest Common Factor
ANY TIME you are adding or subtracting fractions, regardless of whether or not there are variables
What should you always do first
Cancel/ simplify/ factor
graph >, <
DOTTED line
How do you find the slant asymptote?
Divide the ENTIRE numerator by the ENTIRE denominator, the slant asymptote is the quotient, ignore the remainder (might include division with variables raised to exponents)
What standard does an equation/ graph have to meet to be considered a function?
Each input can only have one output
How do you find the least common multiple?
Find the smallest whole number that both expressions divide evenly into. Then use the variables with the highest power
What should you always write out when doing your work?
Fraction conversions, simplification steps, line graphs (interval notation)
What are inverse functions.
Functions that switch the input and output
Can you take the log of a negative number?
NO
Can a solution make the denominator equal 0?
NO (ALWAYS CHECK FOR THIS BEFORE ANSWERING if a "solution" does this it is not a solution. it iS A FRAUD >~<)
Should you graph two separate equations on the same number line when doing your work?
No
When the number under the radical is large should you trust that it's simplified fully?
No (it's probably not fam)
When translating do you need to solve the equation?
No, just look at the numbers inside (side to side change sign) and outside (up and down DONT change sign) the parentheses
In the functions (f + g)(x), (f-g)(1), (f・g)(x), (f/g)(2) etc. is the x or the coefficient in the second parentheses being multiplied by the solution of f and g?
No, x is the input for the combined functions (just like it would be in single functions like f(x))
⋂ represents
Numbers in BOTH sets (Intersection)
U represents
Numbers in EITHER set (Union)
What should you do any time you solve by squaring?
Plug the answers back in (to double check)
How do you find the vertical asymptote?
Set the denominator equal to 0
Hw do you find the domain of an equation with a radical?
Set the value under the radical equal to 0. IGNORE values outside the radical. (The domain ONLY includes values that make the numbers under the radical greater than or equal to 0)
How do you find the inverse function of an equation?
Solve for x then swap the variables of the new solved equation
When performing operations on rational expressions, what does it help to circle?
Subtraction signs that apply to an entire expression (apply them at the very end)
When asked for the rate, should you give the rate or the rate times the time?
THE RATE
What do you need to change when graphing log transformations that involve a change in x coordinate?
The domain and vertical asymptote
What is the range of an inverse function?
The domain of the original function
What do you normally confuse the least common multiple with?
The greatest common factor (the least common multiple IS NOT the greatest common factor)
When doing polynomial long division, which terms do you use
The leading coefficients
What is the domain?
The x limits
What is the range?
The y limits
(x-6)(x+4) < 0
Where is the product of (x-6) and (x+4) less than 0?
What should you always physically mark on inequality graphs that are getting compared before attempting to solve the problem?
Where the graphs intersect
What helps when you're trying to simplify a radical with a power higher than 2?
Writing out a table with perfect roots of the power
Can you cancel out the denominator by multiplying both sides by its reciprocal regardless of whether or not there is a zero on the other side?
YES
When performing operations on rational expressions, if there is a coefficient, do you need to multiply the coefficient by the lowest common factor as well?
YES
When solving a system of equations can you add the equations together if one of the variables will cancel out?
YES
What is -log when graphing?
a reflection about the x axis (multiply y coordinates by negative 1)
log_a c = b if
a^(b) = c
d/dx (x^a)
ax^a-1
tan
cot
sin
csc
volume
cubed
What do you do when solving complex log/exponential equations?
distribute and factor
BETC
divide the top leading coefficient by the bottom leading coefficient (Bottom Equals Top Coefficient HORIZONTAL ASYMPTOTE)
y=f(ax)
divide x coordinate by a
(f og)(x)
f(g(x))
to find y intercept
find f(0)
What do you do for exponential growth or decay problems?
initial x percent^time
Translate -(x + 3)^3 - 2 from -x^3
left 3, down 2
log_e
ln
log
log_10
log_a (xy)
log_a x + log_a y
log_a(x/y)
log_a x - log_a y
y < 9
make a graph that shows where this equation is true
distance =
rate x time
every, each, per (writing equations)
rate/ set y value equal to (multiply this value by the x variable)
log_a x^r
rlog_a x
cos
sec
to find x intercept
set f(x) equal to 0
How do you find the vertex of a parabola?
take the derivative (slope) and set it equal to 0
When a problem says that one variable varies inversely with another variable what do you need to know/ find?
the constant k
BOB0
the horizontal asymptote = 0 (Bigger On Bottom 0)
What is the vertical asymptote of a normal log equation?
the y axis
BOTU
there is no horizontal asymptote, find slant asymptote (Bigger On Top Undefined)
Inverse variation
y=k/x (product remains constant (yx stays the same))
Direct variation
y=kx (ratio remains constant (y/x stays the same))
Joint variation
y=kzx (y varies jointly with x and z (y/xz stays the same))
Can a graph have more than one local maximum or minimum?
yes
When you're trying to find the width do you disregard negative answers?
yes
If two logs with the same base are on different sides of an equation can you cancel the logs?
yes (queen:3)
area of a circle
πr²
