Vertical Translations! Please help!

Question

A cosine function has an amplitidue of 1/3, a period of pi/3, horizontal shift of -2pi, and a vertical shift of -3.

What is the y-value of the negative function when x=pi?

Answer 1

The amplitude of 1/3 is the coefficient of the cosine function like so y = (1/3)cos(t)

For the period, remember that the "angular velocity" (w) = 2*pi*f or 2*pi/P where pi = 3.14159, f = the frequency, but f = 1/P where P = the period. So, y = (1/3)cos(2*pi*t/pi/3) = (1/3)cos(6t)

In order to shift horizontally, you let y = (1/3)cos{6(t+2*pi)} = (1/3)cos(6t + 12*pi) = (1/3)cos(6t + 37.7)

Then to shift vertically down 3 units y = -3 + (1/3)cos(6t + 37.7)

Answer 2

Hey Dee,

Yeah, these can definitely get confusing. There are patterns that we can use to help us recognize exactly how to solve this, though.

Let's just start with the generic equation for a cosine function, which looks like this:

f(x) = A cos(B(x - C)) + D

To cut to the chase, here's what each of those letters (A,B,C and D) do to the equation f(x) = cos(x):

1. |A| is the amplitude (or height) of the graphed equation
2. 2π/|B| is the period (of a cosine or sine function -- that's important to realize! It doesn't work for tangent and cotangent because their periods are only defined on a period of π, not 2π. So for tangent and cotangent the period is π/|B|)
3. C is the horizontal (or phase) shift
4. D is the vertical shift

We're given pretty much each of the pieces of this equation:

1. Our amplitude is 1/3, so that means A = 1/3
2. We know our period is π/3, so that means 2π/B = π/3 --> B = 6
3. We're also given our horizontal (or phase) shift, so we can say C = -2π (This means the entire graph is shifted to the left by 2π)
4. And we know we shift our equation by -3 (meaning the entire graph is moved down by 3), so D = -3

So now we put it all together:

f(x) = A cos(B(x - C)) + D

f(x) = (1/3) cos (6(x - (-2π)) + (-3)

f(x) = (1/3) cos(6(x + 2π)) - 3 --> This equation describes every shift in the graph that we were told!

Now we can solve the original question: What is f(x) when x = π?

To solve this, we can now just punch x = π into our equation:

f(π) = (1/3) cos(6(π + 2π)) - 3

f(π) = (1/3) cos(6(3π)) - 3

f(π) = (1/3) cos(18π) - 3 --> Here we need to recall that cos(π) = -1, and cos(2π) = 1. That means the cosine of any even number multiplied by π will be 1. So cos(18π) = 1

f(π) = (1/3) (1) - 3

f(π) = (1/3) - 3

= (1/3) - (9/3)

f(π) = -8/3 <-- This is our final answer!

**It's important, though, that you go through an understand why each of those letters (A,B,C and D) do what they do.

Try a simple equation on your graphing graphing calculator (or use this website: https://www.desmos.com/calculator)

1. Start by graphing the equation f(x) = cos(x)
2. Now let's try and figure out what A does to our function by making our function f(x) = A cos(x). Let's say A=2 so our function looks like f(x) = 2cos(x). Try graphing that. What does it do? What if A = -2 so f(x) = -2cos(x) -- what does that do?
3. Now let's try D by itself -- let's say D = 5 so f(x) = cos(x) + 5. What does that do? What happens if instead D = 1/2? What happens if instead D = -3?
4. Now let's try B by itself -- let's say B = 2 so f(x) = cos(2x). What does that do? Can you solve for the graph's period (Hint: Remember the period is 2π/|B|). What if B = 1/2? What if B = -3? What does each of those do?
5. Now let's try C by itself -- let's say C = 3 so f(x) = cos(x-3). What does that do? What if C = -5? What if C = 1/2? What do those do?

Go through that process so you begin to see patterns. It will really help solidify what each piece of the function does. Then try it with a sine function. Then try it with a tangent function. Write down what patterns you see.