Studied by 51 peopleSolving Linear eqns & inequalities
isolate the variable
Solving Quadratic eqns & inequalities
isolate zero
Solving polynomial eqns & inequalities
isolate zero
Solving Rational eqns & inequalities
multiply by the LCD
Solving absolute value eqns & inequalities
isolate absolute value
solving absolute value eqns
|a| =
a, if a>0
-a, if a<0
solving absolute value ineqalities
|f(x)| > c means…
f(x)>c
or
f(x)< -c
solving exponential equations
isolate the power
solving radical eqns
isolate the radical
solving log eqns
write as the log of one expression and then isolate the log
solve trig eqns
1) isolate the trig ratio
2) use zero product property
3) use a trig identity
to translate a graph up
add a #
to translate a graph down
subtract a #
to translate a graph right
subract a # “within”
to translate a graph left
add a # “within”
to vertically stretch a graph
multiply by a # c (c>1)
to vertically shrink a graph
multiply by a # c (0<c<1)
f(x) + k
translates a graph up k units
f(x) - k
translates a graph down k units
kf(x), where k>1
vertical stretch
kf(x), where 0<k<1
vertical shrink
f(x+h)
translates graph left h units
f(x-h)
translates graph right h units
to find x-intercepts
substitute 0 for y
to find y-intercept
substitute 0 for x
how to find inverse function
1) replace f(x) with y
2) switch x & y
3) solve for the new y
4) replace g(x) for the new y
properties of inverse function
1) symmetric with y=x
2) f(g(x)) = g(f(x)) = x
3) one-to-one function
4) Domain & Range are interchanged
f(kx), where k>1
horizontal shrink
f(kx), where 0<k<1
horizontal stretch
f(-x)
reflection across y-axis
-f(x)
reflection across x-axis
-f(-x)
reflection through origin
f(|x|)
reflection of QI & QIV through y-axis (lose QII & QIII)
|f(x)|
Reflection of QIII and QIV through x-axis (lose QI & QII)
1/f(|x|)
y → 0+ <-> y → +∞
y → 0- <-> y → -∞
y = 0 <-> y is undefined
f(h-x)
= f(x+h) then replace xby - x (reflection of f(x+h) through y-axis)
|f(x)| defined as a piecewise function
f(x) for all x where f(x) ≥ 0
-f(x) for all x where f(x) < 0
even function
a function that is symmetric to itself through the y-axis; f(-x) = f(x)
odd function
a function that is symmetric to itself through the origin; -f(x) = f(x)
f(x) = x
linear family
f(x) = x2, x4, x6
parabolic family
f(x) = x3, x5, x7
cubic family
f(x) = x1/2, x1/4, x1/6
square root family
f(x) = x1/3, x1/5, x1/7
cubic root family
f(x) = x-2, x-4, x-6
bell curve family
f(x) = x2/3, x4/5, x6/7
Bird Family
f(x) = [|x|]
greatest integer function
f(x) = |x|
absolute value
f(x) = ax² + bx + c
parabola family
Vertex of f(x) = ax² + bx + c?
Vertex → (h,k)
h = -b/2
k = f(-b/2a)
f(x) = anxn + an-1xn-1 + .. a0; n is odd
1) outside behavior → cubic
2) intercepts
3) relative extrema (n-1)
4) symmetry
f(x) = (anxn + an-1xn-1 + .. a0) / (bmxm + bm-1xm-1 + .. b0)
1) asymptotes
2) intercepts
3) symmetry
4) plot points if needed
to find vertical asymptotes
set denominator of simplified rational expression to 0
to find horizontal asymptotes of rational functions
n=m, H.A. @ y=a/b
n<m, H.A. @ y = 0
n>m, no H.A.
f(x) = (c-x²)1/2 , c>0
circular function
f(x) = (x²-c)1/2 , c>0
hyperbolic function
f(x) = (x²+c)1/2
hyperbolic function
Note
Flashcard