Precalc flashcards 1-3

Solving Linear equations & inequalities

isolate the variable

solving quadratic equations & inequalities

isolate zero

solving polynomial equations & inequalities

isolate zero

solving rational equations & inequalities

multiply by the LCD

Solving absolute value equations & inequalities

isolate the absolute value

Solving absolute value equations l a l =

a, if a > 0 ; -a, if a < 0

Solving absolute value inequalities l f(x) l > c means…

f(x) > c OR f(x) < -c

Solving exponential equations

isolate the power

Solving radical equations

isolate the radical

Solving logarithmic equations

Write as the log of one expression and then isolate the log

Solving trig equations

1) isolate the trig ratio 2) use zero product property 2) use a trig identity

To translate a graph up…

add a number

To translate a graph down…

subtract a number

To translate a graph right…

subtract a number “within”

To translate a graph left…

add a number "within"

To vertically stretch a graph

multiply by a number c (c>1)

To vertically shrink a graph

multiply by a number c (0<c<1)

f(x) + k

translates graph up k units

f(x) - k

translates down k units

k * f(x) , where k > 1

vertical stretch

k * f(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 4) domain & range are interchanged

f(kx), where k >1

horizontal shrink

f(kx), where 0 < k <1

horizontal stretch

f(-x)

reflection across the y-axis

-f(x)

reflection across x-axis

-f(-x)

reflection through origin

f(lxl)

reflection of QI and QIV through y-axis (lose QII and QIII)

lf(x)l

Reflection of QIII and QIV through x-axis

1/f(lxl)

y→0⁺ ←→ y→+∞

y→0^-←→ y→ -∞

y=0 ←→ y is undefined

f(h-x)

f(x+h) then replace x by -x (reflection of f(x+h) through y-axis)

lf(x)l 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) = x^1/2, x^1/4, x^1/6…

square root family

f(x) = x^1/3, x^1/5, x^1/7 …

cubic root family

f(x) = x^-1, x^-3, x^-5, …

rational function

f(x) = x^-2, x^-4, x^-6…

bell curve family

f(x)= x^2/3, x^4/5, x^6/7, ….

bird family

f(x) = [lxl]

greatest integer function

f(x) = lxl

absolute value

f(x) = ax²+bx+c

parabola family

vertex of f(x) = ax² +bx+c

Vertex → (h,k)

h = -b/2a

k = f(-b/2a)

f(x)= a_nxⁿ + a_n-1x^n-1+…+ a₀ ; n is even

1) end behavior → parabola 2) intercepts 3) relative extrema (n-1) 4) symmetry

f(x) = a_nxⁿ + a_n-1x^n-1 + …+ a₀ ; n is odd

1) outside behavior → cubic 2) intercepts 3) relative extrema (n-1) 4) symmetry

1) asymptotes 2) intercepts 3) symmetry 4) plot points, if needed

To find vertical asymptotes…

Set denominator of simplified rational expression equal to 0

To find horizontal asymptotes of rational functions…

n=m, horizontal asymptote at y = a/b

n<m, horizontal asymptote at y=0

n>m, no horizontal asymptote

f(x) = √(c-x²), c>0

circular function

f(x) = √(x²-c), c>0

hyperbolic function

f(x) = √(x²+c)

hyperbolic function