Precalc Test 2

What quadrant is each function positive?

All Students Take Classes In Q1, all the functions are positive. In Q2, only the sine and cosecant functions are positive. In Q3, only the tangent and cotangent functions are positive. In Q4, only the cosine and secant functions are positive.

How to understand why the cosine function looks the way it does?

As the cosine function travels around the unit circle, the order that it is positive and negative goes as follows. Q1: + , Q2: - , Q3: - , Q4: + This follows the way the cosine function moves on a real graph. It starts at a maximum, being positive yet going down, then it reaches the midline, turning negative. It then rounds the minimum and starts going up again, yet still being negative. Finally it crosses the midline, becoming positive again and reaches the maximum again.

How to understand the way the negative cosine function looks the way it does?

As the negative cosine function travels around the unit circle, the order that it is positive and negative goes as follows: Q1: - , Q2: + , Q3: + , Q4: - This follows the way the negative cosine function moves on a real graph. It starts at the minimum and precedes moving up towards the midline. As it crosses the midline it becomes positive. It stays positive as it rounds the maximum and travels back down to the midline. After it passes the midline, it becomes negative again and ends as it reaches the minimum.

How to understand the way the negative sine function looks the way it does?

As the negative sine function travels around the unit circle, the order that it is positive and negative goes as follows: Q1: - , Q2: - , Q3: + , Q4: + This follows the way the negative sine function moves on a real graph. It starts at the midline and precedes moving down. It stays negative as it rounds the minimum and starts it's movement up. It becomes positive as it passes the midline again and goes up to the maximum. It stays positive as it goes back down to the midline.

How to understand why the sine function looks the way it does?

As the sine function travels around the unit circle, the order that it is positive and negative goes as follows: Q1: + , Q2: + , Q3: - , Q4: - This follows the way the sine function moves on a real graph. It starts going positive, then reaches a maximum, as it starts its decent, it becomes negative and then goes back up, still negative, back to the midline.

Where to put dots on a function?

At least 5 points. It should follow the curve of the graph. Start with your starting point, then put a point where it either begins to change direction or when it passes the midline. continue this strategy on the rest of the line for at least one period

How to find the domain?

Easy, it is always ( negative infinity, positive infinity ) or "All real numbers" Be sure to use parentheses when including infinity and brackets when including non-infinity numbers

How to write out transformations.

First begin with anything that involves multiplication or division. - these will be your vertical or horizontal compressions / stretches - be sure to include the words "by a factor of" in your wording Next, go on to anything that involves addition or subtraction - these will be your vertical or horizontal shifts - be sure to include which direction (left, right, up, or down) in your wording to be precise

How do you find the period?

First, take out the middle equation that is surrounded in parentheses: "(x-c)" Next, set up a double sided inequality equation with the parent function period on the outside like so: 0 \< (x - c) \< 2pi (Hint, \< means "less than or equal to") Next, solve the inequality: 0 + c \< x \< 2pi + c This is your starting an ending point of your new period! Finally, to find out how long the period is, subtract the starting point from the ending point.

How should I divide the points according to the period?

If the period is 2pi: The midway point would be pi, and the quarter point would be pi/2 If the period is pi: The midway point would be pi/2, and the quarter point would be pi/4 If the period is pi/2: The midway point would be pi/4. and the quarter point would be pi/8 If the period is 4pi: The midway point would be 2pi, and the quarter point would be pi REMEMBER: these basic points are ONLY if the graph does not have a horizontal shift. If it does, move your starting point to that location and add/subtract the shift from each point from these basic periods.

How to find the range of a function?

It is the amplitude combined with the vertical shift. First shift the graph, then add and subtract the maximum from the midline point. Parent function: [-1,1] Example of a different range: [-5,1] - Vertical shift down of 2 - amplitude of 3

How do you find the amplitude?

Pick one of the maximums (or minimums) and measure how many units it is from the tip of the maximum (or minimum) to the midline

How to determine the labeled points of a graph?

Start with labeling your starting and ending points of your period. Then, divide that space into 4 equal parts. Label the first tick mark to show what measurement you are going by for the rest of the tick marks. Next, label the halfway point Make sure you have at least these 4 marks labeled, and more if she asks you for more than just one period.

What does a negative sine graph look like?

Starts at the origin and starts going down (Reflected across the x-axis in the picture, I couldn't find a picture for this graph)

What does a sine graph look like?

Starts at the origin and starts going up

What does a negative cosine graph look like?

Starts on the negative y-axis and starts going up (Reflected across the X-axis in the picture, I couldn't find a picture for this graph)

What does a cosine graph look like?

Starts on the positive y-axis and starts down

What is an amplitude?

The distance from the midline to a maximum (or minimum). ALWAYS POSITIVE

What is the range?

The highest and lowest point reached by the function according to the y-axis

What is a period?

The length, in radians, it takes to complete one full cycle/rotation of a function.

How to measure arch lengths?

The marks made on the x-axis tell you the arc lengths (as long as they are in radians (pi)) Example: If a question asks you what the arch length of the maximum in a regular parent function sine graph, it would be pi/2. Example: if a question asks you to label 3pi/4 on a cosine graph, you would label the point on the graph where it crosses the midline a second time (as it goes positive again).

What is a midline?

The middle line of the function equally between the maximums and minimums

30-60-90 Triangle

The side opposite the 30 would be "x" The side opposite the 60 would be "x times the square root of 3" The side opposite the 90 would be "2x"

45-45-90 Triangle

The sides opposite the 45s would be "x" The side opposite the 90 would be "x times the square root of 2"

How to find the period on a graph?

When the graph starts to repeat itself. Sine: origin when it starts to go up to the next origin where it starts to go up Cosine: maximum to maximum -Sine: origin where is starts to go down to the next origin where it starts to go down -cosine: minimum to minimum

How to find the phase shift (horizontal shift) of a function?

where the start of your function is. (sine: origin, up. cosine: max. -sine: origin, down. -cosine: min.) Also, look at your equation, it is letter C

y = -173 cos (534x + 9002pi/612)) - 120 1. What is the domain of this function? 2. how does the new domain effect the graph? 3. What are the endpoints for the function?

y= (A)cos(B(x-C))-D Don't get tricked, it is simple! 1. ALL REAL NUMBERS! 2. It doesn't, it is the same as before. The since it is all real numbers, it just keeps going. 3. (negative infinity, positive infinity)

y = 7cos (4x - 3pi/2)) - 1 1. What is the period of this function? 2. Is it a stretch or compression? 3. What is the new beginning and end of the period?

y= (A)cos(B(x-C))-D Look in the "B" and "C" spots for this problem. 1. 0 \< 4x-3pi/2 \< 2pi 0 + 3pi/2 \< 4x \< 2pi + 3pi/2 3pi/8 \< x \< 7pi/8 7pi/8 - 3pi/8 = 4pi/8 = 1/2pi 2. Compression because 4>1 3. The new beginning is 3pi/8 The new end is 7pi/8

y = 9 sin(3x + pi/2)) - 1 1. What is the amplitude of this function? 2. Is it a stretch or compression? 3. How does it affect the look of the graph?

y= (A)sin(B(x-C))-D 1. "4" is the amplitude because it is in the "A" spot. 2. It is a stretch because 9>1 3. It makes the graph look taller.

y = -2sin (8x - 3pi/4) + 5 1. What is the range of this function? 2. Is it a positive or negative shift? 3. Does it matter that the 2 is negative?

y= (A)sine(B(x-C))-D Look in the "A" and "D" spots for this problem. 1. the new range is (3,7) - this is because there was a vertical shift up of 5, and the amplitude added 2 in each direction of the middle line. 2. It is a positive shift because the 5 is positive. 3. No! You will always take the absolute value of the number in the "A" spot ( |A| ). This just tells you how far up and down you will need to go.

Where does the Amplitude go when writing a function?

y= (A)sine(B(x-C))-D in "A" |A|>1 = vertical compression |A|<1 = vertical stretch A=1 = regular amplitude (like parent function)

What will affect the period of the function?

y= (A)sine(B(x-C))-D in "B" |B|>1 = horizontal compression |B|<1 = horizontal stretch B=1 = regular period (like parent function)

Where do you find the phase shift (horizontal shift) of a function in the equation?

y= (A)sine(B(x-C))-D in "C" C>0 = shift in left (negative) direction C<0 = shift in right (positive) direction C=0 = regular position (like parent function) Do not be confused when looking at the equation! (x-C) "X MINUS C"

What is an example of finding the period using real numbers?

y=3sin(2x+pi/2)-2 0 \< 2x + pi/2 \< 2pi -pi/2 \< 2x \< 3pi/2 (-pi/2) /2 \< x \< (3pi/2) /2 -pi/4 \< x \< 3pi/4 Starting point: -pi/4 Ending point: 3pi/4 Period length: pi