Math Unit 6 Test Flashcards
Periodic Function:
- a consistent, repetitive pattern of y-values for incremental changes in x-values
- repeats at regular intervals
Cycle:
ONE concrete pattern (the portion of the graph that repeats)
Period:
the horizontal length of one cycle (p=360°/k for the Trig Graphs)
Amplitude:
- the height going from “the middle horizontal line” (equation of the axis) to your maximum or minimum value
- A = (max value - min value) ÷ 2
- or A = max value - # equation of the axis
Equation of the Axis:
- the horizontal line that divides the graph in half, notice it does not have to be symmetrical
- y = max value + min value/2
Periodic:
- The shape of the graph repeats over some interval
- Repetitive y-values for consistent increases of x-values
Non-periodic:
No repeated pattern on y-values
y = sin x
middle, up, middle, down, middle
Key Properties: f(x) = sin x
Amplitude: 1 - (1-(-1)/2)
y-intercept: (0,0)
Maximum: 1
Domain: {x ∈ R}
Equation of the axis: y = 0
Period: 360°
x-intercept: (0°,0), (180°,0), (360°,0)
Minimum: -1
Range: {y ∈ R | -1≤ y ≤ 1}
f(x) = sin x, five key points:
(0°,0) (90°,1) (180°,0) (270°,-1) (360°,0)
y = cos x
up, middle, down, middle, up
Key Properties: f(x) = cos x
Amplitude: 1 (1-(-1)/2)
y-intercept: (0,1)
Maximum: 1
Domain: {x ∈ R}
Equation of the axis: y = 0
Period: 360°
x-intercept: (90°,0), (270°,0)
Minimum: -1
Range: {y ∈ R | -1≤ y ≤ 1}
f(x) = cos x, five key points:
(0°,1) (90°,0) (180°,-1) (270°,0) (360°,1)
How are these functions alike and different? (sin & cos)
Same curve pattern, same period, the same maximum and minimum value
y = cos x is a 90° shift to the left of the y = sin x
Amplitude:
(lesson 3)
the height from the center line to the peak (or to the trough)
the height from highest to lowest points divided by 2
Period:
(lesson 3)
goes from one peak to the next (or from any point to the next matching point)
Summary - The graph y = a sin kx will have:
Amplitude: a
Period: 360°/k
Range: {yER| -a ≤ y ≤ a}
will have… amplitude:
a
will have… period:
360/k
Range:
{yER| -a ≤ y ≤ a}
Vertical Stretch and Compression will affect the ? and the ?
amplitude
range
Horizontal Stretch and Compression will affect the ? (the x-values)
period
Summary - The graph y = a cos kx will have:
Amplitude: a
Period: 360°/k
Range: {yER| -a ≤ y ≤ a}
mapping notation
(1/kx + d, ay + c)
formula
fx = asin(k(x-d)) + c
fx = acos(k(x-d)) + c
factor out
k
if (x-d) d is..
if (x+d) d is..
positive
negative
When graphing determine the intervals of the key points using the PERIOD
Determine the PERIOD (360/k)
DIVIDE the PERIOD by 4 to determine the intervals along the x-axis
Use the ‘a’ value to determine the value on the y-axis
The “a”
- Vertical stretch or compression
- If ‘a’ is negative, there is a reflection along the x-axis
The “k”
- Horizontal stretch or compression
- If ‘k’ is negative, there is a reflection along the y-axis
The “d’
- Horizontal translation (shift left or right) ** phase shift **
The “c”
- Vertical translation (shift up or down)
Which variable represents the amplitude?
a (always positive)
Which variable represents the equation of the axis?
c
Which variable affects the period?
k
Which variable(s) affect the domain?
d and k
Which variable(s) affect the range?
a and c
Domain of one cycle
{xER|phaseshift ≤ x ≤ phaseshift + period}
(d) (d + p)
Range:
{yER| c - amplitude ≤ y ≤ c + amplitude}
y = -sinx
(Begins on the equation of the axis and goes DOWN)
y = -cosx
(Begins on the MINIMUM point)
Modelling Sinusoidal Functions
When given graph Steps:
Find the max
Find the min
Find the a (max-min/2)
Find the period
Find the k (360/p)
Find the eq/n of axis - middle point
D will depend on if its cos or sin
Modelling Sinusoidal Functions
When given words steps:
Write down a
Find eq’n of the axis by doing (max - a)
Find k by doing (360/period)
D again depends
Modelling Sinusoidal Functions
When given table steps:
Max
Min
A (max-min/2)
Period
K (360/p)
eq’n of the axis by doing (max - a)
D depends
For Ferris wheel:
Max
Min
Period = how long it takes for one full rotation
Interval = divide period by 4
Use how many revolutions to see the ending x value and how many times the cycle should repeat
K - 360/period
Y = (max + min/2)
period in ferris wheel
P = minutes x seconds
k in ferris wheel
K = 360/p
If k increases, horizontal ? for blank minutes
If k decreases, horizontal ?for blank minutes
from og equation
compression
stretch
What does the amplitude represent in Ferris wheel?
The amplitude represents the RADIUS of the ferris wheel
How fast is someone sitting on the Ferris Wheel moving in m/s? (d is distance around/circumference)
d = 2PiR
D = circumference
S = d/t
S = circumference/period
To figure out the depth of the water sub in the
t value
To find period when not mentioned do
2 x the bigger seconds - the smaller 2(bs - ss)
