math help please!!!!!!!!!!!!!!!!!!

Question

Write a short essay (complete with an example that you make up) that would tell someone how to graph a sine function that is shifted vertically and horizontally and stretched vertically and horizontally. Your essay should include how they can identify each of these transformations from the equation and then how they can use their knowledge of the transformations to find points on the graph.

Answer 1

To shift f(x) = Sin(x) vertically: Sin(x) ± 5 raises or lowers it by 5

To shrink f(x) vertically: 0.5Sin(x)

To stretch f(x) horizontally: Sin(x/5)

To shift f(x) horizontally: Sin(x ± pi/5)

I suggest you graph each of these on your graphing calculator and compare against f(x) = Sin(x)

Don't forget to set the Mode to RADIAN not DEGREE.

I leave it up to you to write a short essay...!!!!!!!!!!!!!!!

Answer 2

Hello, Anthony

The key to writing this essay is understanding the sine equation and how each of these stretches occur.

So, in a very general manner a sine function can be written Asin(ωθ + α) + C. Now, this might not be a form you're familiar with. Usually, you just see sinθ, which means that the variables A and ω are 1, and the variables α and C are 0. But including all of these variables will help to explain these different processes and how they affect the sine function. So let's look at these extra variables one by one.

A

First off, we see that our sine function is being multiplied by some value A. Without doing any calculations or graphing, we can think about what that's going to do to the function. Let's just make things simple by focusing on the case when sinθ = 1.

If sinθ = 1, then Asinθ = A*(1)

That means that if A is greater than 1, the output of our sine function will grow.

For example, if A = 2, then Asinθ = (2) * (1) = 2, meaning that the coefficient caused the sine function to double in value.

On the other hand, if A is less than 1, it will reduce the output of the sine function.

For example, if A is 0.5, then Asinθ = (0.5) * (1) = 0.5, meaning that the coefficient caused the sine function to halve in value.

From these two situations we can conclude that a coefficient that is multiplied by the whole sine function (which we represented as an A here), will cause the height of the function to increase or decrease, corresponding to vertical stretching. (The reason we normally represent this coefficient with the letter A, is because it refers to a wave's amplitude.)

C

Let's continue addressing the variables outside the sine function for now. In this case, we have some variable C that is added to the sine equation.

sinθ + C

Again, just to make it easier, let's focus on the case when sinθ = 1. In that case, we can see that whatever value we choose for C will add to or subtract from the sine function. For example, C = 2.

sinθ + C = (1) + (2) = 3 The variable C has, thus, shifted the output of our function up two values.

Likewise, if C = -2:

sinθ + C = (1) + (-2) = -1 Causing the sine function to shift down two places.

Adding (or subtracting) from a sine function, therefore, causes a vertical shift in its graph.

α

Okay, things are going to get a little bit more tricky here. This variable is found within the sine function itself, so we need to think about what the notation is saying.

sin(θ + α) What this means is that you are evaluating the sine function at a value that is equal to some degree θ plus some additional amount α. To make sense of that, let's continue with our example of when sinθ = 1. The sine function equals 1 at π/2 (and 5π/2, etc.), so we'll use that value of θ. Then, let's use the same value for α as well. S0,

sin(θ + α) = sin( (π/2) + (π/2)) = sin(π) = 0

Or, let's use a different value for α, like -π. In that case,

sin(θ + α) = sin( (π/2) + (-π) = sin(-π/2) = -1

You can see, that by adding in the value α, what we did was essentially change the degree θ at which we evaluated the function. In the first case, we shifted the point at which our sine function began so that it would start its cycle - or phase - at a value π/2 greater. In the second case, we shifted the point to a value π less. Since, on a graph, radians/degrees would be marked on the x-axis, this transformation corresponds to a horizontal shift of the entire function, also know as a phase shift (for which reason, this variable α is known as the phase angle.)

ω

Finally, we can talk about the last variable in this equation, which is multiplied by θ itself. Let's examine it's role by returning again to the example in which θ = π/2, and thus sinθ = 1.

sin(ωθ) = sin(ω*(π/2))

The angle θ, in this case (π/2), will increase or decrease depending upon the value of its coefficient ω. For example, if ω is 2, then:

sin(ωθ) = sin((2)*(π/2)) = sin(π) = 0

Now, this is not the same as the phase shift we saw before. Let's try again with the same value of ω, but evaluate the function at another angle, say π/4. In that case,

sin(ωθ) = sin((2)*(π/4)) = sin(π/2) = 1

What is happening here, is that the coefficient ω is doubling the value of θ at which we are evaluating the function: If we think about how it is affecting our graph, we can imagine that it is causes the sine function to reach the height it would for a degree θ in half as many radians as it normally would.

So,the sine function is, oscillating back and forth between 1 and -1 twice as fast as it would normally, meaning it would have maxima twice as often.

Okay, let's look at this situation then with an ω value that is less than 1, like 0.5. Because this is just another coefficient, we know what's going to happen: Every θ will be halved.

So, we see that the sine function is oscillating half as quickly as it would normally, with half as many maxima as it normally would have.

The choice of ω value, therefore, causes our sine function to stretch or shrink horizontally across the graph, affecting how quickly it moves through a phase. (This property of a wave, number of cycles per time, is known as its frequency. So ω is the symbol for frequency.)

In summation, Asin(ωθ + α) + C, where the variables A, ω, α, and C cause the following transformations:

1. A: A coefficient which stretches the function vertically (known as a wave's amplitude)
2. ω: A coefficient which stretches the function horizontally (known as frequency)
3. α: A variable which shifts the function horizontally (known as phase angle)
4. C: A variable which shifts the function vertically.

Hope this helps.

Daniel