Graph help for tranfromation, reflection stuff. PLease help out :)

Question

a graph of f(x)=1/4|cos(pi(x-1/2))-3| Can you tell me all that is happening here including the transformations, reflection, stretch/compression, shifts, etc, and the vertex and x/y values, and how to get them. i would really appreciate it :)

Answer 1

All functions can be written in "transformation format" in the form:

f(x) = a•f(b(x - c)) + d, where a, b, c, and d describe the following transformations:

a: responsible for vertically stretching (|a| > 1) or shrinking (|a| < 1) as well as vertically reflecting (a < 0)

1. this means all y-values of f(x) will get multiplied by "a"

b: responsible for horizontally stretching (|b| < 1) or shrinking (|b| > 1) as well as horizontally reflecting (b < 0)

1. this means all x-values will get multiplied by "1/b" NOTE: horizontal transformations have a 'reverse' relationship to vertical ones in how they are applied. For example, if a = 2, you multiply all the y-values by 2, but if b = 2, you multiply all of the x-values by 1/2.

c: will be your horizontal shift. Notice that in the formula you have x - c, which means if you see x - 2, you will go to the right 2 units because c is positive

d: will be your vertical shift. A positive d-value means go up, and a negative d-value means go down.

For your specific function, you only need to connect the terminology to the corresponding transformation to easily interpret what is asked:

f(x) = (1/4)cos[πx - π/2] - 3

First, let's rewrite it in the form I have listed above: f(x) = (1/4)cos[(π(x - 1/2)] - 3

We should see clearly that a = 1/4, b = π, c = 1/2, and d = -3. This means you have a vertical shrink by a factor of 1/4, a horizontal shrink by a factor of 1/π, a horizontal shift to the right 1/2 unit, and a vertical shift down 3 units.

For sine and cosine functions, the following terminology is relevant to connect: amplitude = |a|, period = 2π/ |b|, phase shift = c, and midline = d.

Answer 2

f(x) = (1/4)cos[πx - π/2] - 3

Start with the graph of y = cosx which has amplitude=1, period = 2π, phase shift= 0, and vertical shift = 0.

The graph of y = f(x) has the same shape as y = cosx but has:

Amplitude = 1/4

Period = 2π / π = 2

Phase shift (horizontal shift) = -(-π/2) / π = 1/2

Vertical shift = -3 (down 3 units)