Math 30-1

Vertical Translations Can be represented as

y-k = f(x) or y= f(x)+k

• If k > 0 the translation is

• If k < 0 the translation is

• up

• Down

• With a vertical translation the domain?

• stays the same and the range may change

• vertical translation Mapping Notation:

• (x, y) → (x, y+ k)

• Replacements: y → y-k

Horizontal Translations

Can be represented as
y = f(x-h)

If h > 0 the translation is • If h <0 the translation is

Right, left

With a horizontal translation the range

stays the same and the domain may change

Horizontal

Mapping Notation:

x, y) → (x+h, y)

what are Vertical Reflection about the x-axis?

x); if a <0 the graph has a vertical reflection about the x-axis > Mapping Notation: (x.y) → (x.-y) • Invariant Points): x-intercepts

what are horizontal Reflection about the y-axis?

y = f(bx); if b < 0 the graph has a horizontal reflection about the y-axis > Mapping Notation: (x.y) → (-x, y) > Invariant Points): y-intercepts

What are Vertical Stretch about the x-axis by a factor of 'a'?

• y=af (x)

• Mapping Notation:
  (x, y) → x, ay)

What are Horizontal Stretch about the y-axis by A factor of 1/b

• f(bx)

• Mapping Notation:

(x, y) → (1/bx, y)

> Invariant Points): y-intercepts

What is function notation?

y=af (b(x-h)) +k

• Make sure that 'y' is isolated and

'b' is factored out

How to Describing Transformations

Must be in SRT (order matters)

What is mapping notation for a full function transformation?

Mapping Notation

• (x. x)→ x+h, ay +k

What is inverse as a transformation?

**Inverse means we interchange 'x' And 'y'.**

• Reflection about the line y = x

• Invariant Points): anything on the
  line y = x (ex. (1,1), (-2,-2))

Mapping Notation

• (x, y) → (y,x)

Function Notation

• y = f-1(x)

• Can only be used if the inverse is a function

What is vertical line test?

Used on the graph of your original relation to determine if it is a function (can only pass through one point

What is horizontal line test.

• Used on the graph of your original relation to determine if the inverse is a function (can only pass through one point

• If the inverse is not a function, you can restrict the domain of the original so that its inverse is a function (think quadratic).

What is the equation of a circle/unit circle?

x² + y²=r²

x² + y²=1²

What is the arc length formula?

a = r0

• O must be in radians

Convertin degrees/radians

Degree Radian: multiply by pi/180

Radian Degree: multiply by 180/pi

What is the cast rule?

C: cos positive at q4

A: all positive at q1

S: sin positive at q2

T: tan positive at q3

What are the trig. Ratios?

Sin: y/r or just y

Cos: x/r or just x

Tan: y/x or sin/cos

Csc: r/y or 1/sin

Sec: r/x or 1/cos

Cot: x/y or cos/sin

What are function notations for trig function?

y=asin (b(x-c) +d or

y= acos(b(x-c)) + d

• Make sure that 'y' is isolated and

'b' is factored out

Basically same as regular transformations just that c replaces the h.

How to Find Equation of Graph

a= |max-min|/2

d= max + min)/2

b= 2pi/period, p= 2pi/b

Period is amount of time to complete one cycle.

c' value for:

• For a sine curve → find the closest midline and see where it has phase shifted from the y-axis

• For a cosine curve → find the closest maximum and see where it has phase shifted from the y-axis

How to approach trig function word problems?

• Applications always use radians*
  Sketch graph over one cycle.
  You will need the maximum, minimum, and the period to do this.

• You need to have your five key points (and four sections) of your graph shown and labeled).

• Don't forget to label your axes, scale and key points.

2. Determine the equation of the graph using the formulas for 'a', b', and 'd', as well as

strategies to determine the 'c' value (depends on whether you are using sine or cosine).

Range of trig function graphs?

Range of Sinusoidal Graphs

If a >0:

• {yld-asy≤a+ayeR)

> Min = d-a, Max = d+a

• If a <0:

> Lyld+asysd-a,yeR)

• Min = d+a, Max = d-a

What are tan functions?

• default period of pi

• Asymptotes at every 90, by default

• No defined range.

What are first degree trig functions?

Steps:

1. Isolate for the trig function

2. Use the CAST Rule

3. Find the reference angle

4. Use domain to determine solutions for terminal arms.

What are second degree trig functions?

1. Do a substitution for the trig function to make factoring easier

2. Factor the equation.

3. Set factors equal to zero and solve

4. Substitute the trig function back in

5. Follow same steps as solving first degree trig equation

How to solve Trig Equations Involving Identities?

Steps:

1. Use an identity from the formula sheet (typically Pythagorean identity or double angle identity) to write the equation involving only one trig function

2. Follow the same steps as first or second degree trig equation

What are general solutions?

Gives the solutions for all terminal angles over the set of real numbers

• For example, it is written in the form:

x = Angle + 2pin, n e i

What and how to verify?

• Does not prove an identity, just shows that it works for a specific value or set of values.

• Graphical ~ put left side of equation into y, and right side of equation into 2. Both graphs should be identical.

• Algebraic ~ substitute in a value

LS = RS. Use a T-Chart to keep

things organized.

What are NPVs?

• Come from setting the denominators) equal to zero and solving for variable x or 0).

• Be careful with sneaky
  denominators (reciprocal identities, quotient identities)

• Always stated in general solution form.

What are things to remember when proving identities?

• You CANNOT move things across the equal sign (or the line in the T-Chart)

• Start with the more complex side

• Look for identities on formula sheet

• Write in terms of sine and cosine

• Make a common denominator to add or subtract fractions

• Factoring

> if you have a binomial ~ common factor or difference of squares

if you have a trinomial ~ factor using decomposition