Parent Functions (and Conic Sections)

Front

Domain: x-values, left-to-right

Range: y-values, bottom-to-top

Back

Attributes of Functions

• Increasing: rises from

left to right (Positive slope)

• Decreasing: falls from

left to right (Negative slope)

•Write using the domain•Always use parenthesis

•Write using the domain•Exclude y-values of zero (parenthesis where graph ends on x-axis)

Front

• Positive: Above the x-axis

• Negative: Below the x-axis

Front

Back

Strategies for working with Inequalities

Table

Graph

Solve algebraically

Graph right side in y1, Graph the left side in y2, look at the graph to answer Turn inequality into words to find the part of the graph that fits Write your answer as the x-values (domain) for the correct section of the graph

Inequality Strategies

Systems of Inequalities

Transformations

g(x) =A f(B (x - C)) + D

Vertical Dilation by a factor of A

Horizontal Dilation by a factor of

Horizontal Translation of C

Vertical Translation of D

Vertical are “outsiders” and they “tell the truth”

Horizontal are “insiders” and they “lie”:

• Horizontal Translations move opposite the sign

• Horizontal Dilations stretch/shrink by the reciprocal

1

B

Front

Back

, 0,Ax By C A no fractions or decimals

2 1

1 1

y ym

x x

1 1( )y y m x x

3 2 3 3 2 3

2 1 4 2 4

x x yis sameas

y x y

Using Matrices to Solve Systems of Equations

Slope Formulay mx b

Standard Form of a Line

1

A B

A B

Slope-Intercept Form

Point-Slope Form

f(x)=x Domain Range

Linear (-∞,∞) (-∞,∞)

f(x)=C

Constant (-∞,∞) C

Linear

(0,0)

Front

(0,C)

Back

Solving Absolute Value Equations

Isolate the Absolute Value Bars, then: Re-Write the equation twice:

Once just removing the absolute value bars Once removing the absolute value bars and multiplying the other side of the equation by a negative

Solve each new equationCheck for extraneous solutions by plugging back into the absolute value equation

Isolate the Absolute Value Bars, then: Re-Write the inequality twice:

Once just removing the absolute value bars Once removing the absolute value bars, and changing the direction of the inequality symbol, and multiplying the other side of the equation by a negative Separate the two new inequalities with “AND” if the original Abs. Val. was less than the other side; Separate with “OR” if the original Abs. Val. was greater than the other side

Solve each new inequality

Solving Absolute Value Inequalities

Absolute Value

f(x) = |x|

Domain

(-∞,∞)

Range

[ 0,∞)

(0,0)

Front

Complex #s:

2

3

4

1

1

1

i

i

i i

i

Back

Vertex Form:

Quadratic Function:

Standard Form:

Intercept (root) Form:

Vertex?

( ) ( )( )f x a x p x q

2( ) ( )f x a x h k

2( )f x ax bx c

,h k

,

2 2

b bf

a a

,

2 2

p q p qf

Graphing by starting at vertex:

• Move horiz. a # of units • Square that #;

• Multiply by the dilation

• Result is # to move vertically

Quadratic Formula

2 4

2

b b acx

a

Complete the Square:

2x bx

2

x

1

2

f(x) = xn Polynomial even

*n is even

F(x) = x2

Quadratic

f(x) = xn Polynomial odd

*n is odd and n>1

F(x) =x3

Cubic

Polynomial

Domain Range

(-∞, ∞) [0, ∞)

Domain Range(-∞, ∞) (-∞,∞)

(0,0)

(0,0)

Polynomial Rules:

See back of this pageFront

Back

Polynomials

Maximum # Turns: One less than the degree

End Behavior:

Total Number of Solutions (Real & Imaginary) = Degree

Behavior at X-intercepts(root): Look at the factor that each came from. The exponent of that factor (Multiplicity) indicates the behavior at the root:

Domain

[0, ∞)

Range

[0, ∞)

Square Root

x

y

(0,0)

f(x) = x

Front

Back

Check each Root for Factors with Multiplicities of 2: TANGENT to x-axis so graph bounces at that x-int.Check each Asymptote for Factors with Multiplicities of 2: Graph will come TOGETHER at that asymptote

ROOTS-Solution to TOP: Set numerator = 0 and solve(x-intercepts) Write answers as points: (#, 0)

END BEHAVIOR – Look at RATIO of FIRST TERMSUse the degree of the top and bottom to decide:Balanced: Top Heavy: Bottom Heavy:

y = # Oblique Asymptote y = 0(ratio reduced) (may need to divide) Consider Parent:

x

y

x

y

1

x 21

x

Y-INTERCEPT: Plug in 0 for x, simplify and write as a point (0, #)

ASYMPTOTES (VERTICAL) - Solution to BOTTOMSet denominator = 0 and solveWrite answers as lines: x = #

2

(0,0)

y

x

y

x

F(x) =

F(x) = (0,0)

(0,0)

Domain(-∞,0)U(0, ∞)

Range(-∞,0)U(0, ∞)

1

x

(Rational)

2

1

x

Rational

(RationalSquared)

Domain(-∞,0)U(0, ∞)

Range(0, ∞)

Front

(0,0)

Back

b c b ca a a

n n nab a b

cb bca a

b

b c

c

aa , whenb c

a

b

c c b

a 1, whenc b

a a

n n

n

a a

b b

bb

1a

a

bb

1a

a

nn n

n

a b b

b a a

Expeonent Properties

Uninhibited Growth and Decay

Exponential Growth and Decay

f(x) = 10x

Exponential(Base 10)

f(x) = ex

Exponential(Base e)

ExponentialFront

Domain Range(-∞, ∞) (0, ∞)

(0,1)

Back

Logarithmic

f(x) = log x Logarithmic

(Base 10)

f(x) = ln x Logarithmic

(Base e)

Domain

(0, ∞)

Range

(- ∞, ∞)(1,0)

Front

Back

Conic Sections

x

y

x

y

x

y

Conic SectionsFront

x

y

x

y

x

y

x

y

Back

Unit Circle

Sine and CosineFront

A: Amplitude (Vertical Dilation by a factor of A)B: Horizontal Dilation by a factor of 1/BC: Horizontal TranslationD: Vertical Translation

y Asin(B( C)) D

( )old parent periodB

new period

Hi-Mid-Lo-Mid-Hi

(Hungry-Men-Like-McDonald’s-Hamburgers)

Mid-Hi-Mid-Lo-Mid

(My-Happy-Mother-Loves-Me)Sine:

Cosine:

Transformations of

Sine and Cosine: