Simplifying the Graphing and Transformation of Trig Functions

Question 1

Construct the parent graph for sine and cosine function

1) What is the domain and range of both functions?
2) What is x and what is y values?
3) Demonstrate how to graph both functions!

Answer 1

1) Domain = (–∞, ∞), because θ can rotate around the unit circle in either direction an infinite amount of times. Remember, you can also have 1080 degrees …

Range= (–1, 1), because the radius of the unit circle is 1, the y values can’t be more than 1 or less than -1

2) sin θ is the y-value, cos θ is the x-value

3)
- Draw the unit circle.
- Make a sine and a cosine table. On the left side of these tables, put in the milestones of the circle. 0, π/2, π, 3π/2, 2π.
- At all corners, put in cartesian pair: (1,0),(0,1),(-1,0)(0,-1)
- Remember that sin is y. Take the sin table, go through the milestones and note which y value the cartesian pair has at that place.
- Remember that cos is x. Take the cos table, go through the milestones and note which x value the cartesian pair has at that place.
- Draw

Note:

• Graphs are periodic, they repeats every 2π radians
• Sine = odd function, because it’s symmetrical about the origin. Cos = even function, because it is symmetrical about the y-axis

Question 2

Name the steps to graph a tangent.

Name the steps to graph a cotangent.

Answer 2

1. Draw the unit circle with the cartesian pairs, and the quarters between the milestones π/4, 3π/4, 5π/4, 7π/4 (two quarters are sufficient)
2. For the two quarters, write the cartesian coordinates (Always SR2/2, SR2/2, with +- distribution like other cartesians)
3. Make a table with X and Y values. As X-values, you take the milestones. As Y-values, you take the result of the division of y/x of the cartesian pairs. 0 are x intercepts, undefined are asymptotes
4. Graph, and note: a tangent has no max or min points

Cotangent: all the same steps, but divide x/y, not y/x.

Question 3

Name the steps to graph a secant!

Name the steps to graph a cosecant!

Answer 3

Remember, secant is 1/cos

1) Make a cos table (0, π/2, π, 3π/2, 2π and their cosine values). Remember, cosine are the X-values!
2) Make a sec table with 0, π/2, π, 3π/2, 2π and the sec values as the corresponding values (so you calculate 1/cos-value)
3) Graph in the points. Undefined = asymptotes. From these points, the lines approximate the asymptotes, so in every section divided by asymptotes you get U-like graphs.

Cosecant is the reciprocal of sine (1/sin), so do the same steps just with 1/sin

Question 4

Transforming sine and cosine graphs

1) State the formula which includes all the changes!
2) What is the sinusoidal axis and what is the amplitude?

3)
- What is the period of the parent graphs of sine and
cosine?
- What does a change mean?

4) How are domain and range of the transformed graph affected?
5) What to mind on y = sin(2πx + π/2) ?

Answer 4

1) f(x) = a · sin[p(x – h)] + v
f(x) = a · cos[p(x – h)] + v

a = Amplitude: >1 makes graph taller, <1 so fraction,
smaller, negatives put graph upside down
p = Period. Take 2π and divide by p. Result tells you
where the graph finishes.
h = Horizontal shift
v = Vertical shift

2) Sinusoidal axis = horizontal line, like x-axis.
Amplitude = distance from sin axis to max/min of graph

3) - The period is 2pi (full circle unit).
- A change means that graph moves faster (>1) or slower (0-1): more or less waves in same distance. A negative affects speed and moves in opposite direction.

4) Domain not, but range, because you transformed vertically. If you put it up by 3, the new range is (2,4)

5) Mind in f(x) = a · sin[p(x – h)] + v , p is factored out. If you get y = sin(2πx + π/2), you have to factor out to
y = sin[2π(x + 1/4)] before doing anything.