The following transformations 𝑦 = 3𝑓 (1 𝑥 − 2𝜋) − 1 were applied to the parent function f(x) = csc(x). Graph the transformed function for the interval −4𝜋 ≤ 𝑥 ≤ 4𝜋. On your graph, label the asymptotes, local max/min, and number each axis. Use mapping notation to show your work for a minimum of 5 key points for full marks.

Question

The following transformations 𝑦 = 3𝑓 (1 𝑥 − 2𝜋) − 1 were applied to the parent function f(x) = csc(x). Graph the transformed function for the interval −4𝜋 ≤ 𝑥 ≤ 4𝜋. On your graph, label the asymptotes, local max/min, and number each axis. Use mapping notation to show your work for a minimum of 5 key points for full marks.

I'm having a lot of trouble with the local max/min and the 5 key points, could anyone help me? Please and thank you.

Answer 1

csc(x) has asymptotes, with their local branch turning points at min = 1 and max = -1

so, 3f(x-2π)-1 has its turning points at 3-1 and -3-1
Note also that since the period of csc(x) is 2π, shifting right by 2π does not change the graph in any way. So your graph will still cover exactly four periods. The turning points are all just shifted down by 1. There are still no x-intercepts.

The only real key points for csc(x) are the asymptotes at x = kπ, and the turning points at x = odd multiples of π/2. That is at x = kπ + π/2

Answer 2

To graph the transformed function 𝑦 = 3𝑓(1 𝑥 − 2𝜋) − 1, we'll start by examining the parent function f(x) = csc(x).

The parent function f(x) = csc(x) has asymptotes at x = n𝜋, where n is an integer and excludes the values of x for which sin(x) = 0. In the interval −4𝜋 ≤ 𝑥 ≤ 4𝜋, the asymptotes occur at x = 0, x = 𝜋, and x = 2𝜋. We'll label these asymptotes on the graph.

To find the local max/min points, we need to identify the critical points of the transformed function. Let's find these points.

Step 1: Transform the function
𝑦 = 3𝑓(1 𝑥 − 2𝜋) − 1
Substitute the parent function f(x) = csc(x):
𝑦 = 3csc(1 𝑥 − 2𝜋) − 1

Step 2: Find critical points for the interval −4𝜋 ≤ 𝑥 ≤ 4𝜋
To find the critical points, we'll set the derivative of the function equal to zero.

The derivative of csc(x) is -cot(x)csc(x), so the derivative of the transformed function is:
𝑦′ = -3cot(1 𝑥 − 2𝜋)csc(1 𝑥 − 2𝜋)

Setting 𝑦′ equal to zero:
-3cot(1 𝑥 − 2𝜋)csc(1 𝑥 − 2𝜋) = 0

We know that cot(x) = 0 when x = 𝜋/2 and x = 3𝜋/2. Solving for the transformed function, we have:
1 𝑥 − 2𝜋 = 𝜋/2 and 1 𝑥 − 2𝜋 = 3𝜋/2

Simplifying, we get:
1 𝑥 = 2𝜋 + 𝜋/2 and 1 𝑥 = 2𝜋 + 3𝜋/2

Solving for 𝑥:
𝑥 = 2𝜋 + 𝜋/2 and 𝑥 = 2𝜋 + 3𝜋/2

Step 3: Find key points
Now, we'll find key points on the graph using mapping notation. We'll choose 5 values of 𝑥 within the interval −4𝜋 ≤ 𝑥 ≤ 4𝜋.

1. When 𝑥 = -3𝜋:
𝑓(1 𝑥 − 2𝜋) = 𝑓(-3𝜋 - 2𝜋) = 𝑓(-5𝜋) = csc(-5𝜋)
𝑦 = 3csc(-5𝜋) - 1

2. When 𝑥 = -2𝜋:
𝑓(1 𝑥 − 2𝜋) = 𝑓(-2𝜋 - 2𝜋) = 𝑓(-4𝜋) = csc(-4𝜋)
𝑦 = 3csc(-4𝜋) - 1

3. When 𝑥 = -𝜋:
𝑓(1 𝑥 − 2𝜋) = 𝑓(-𝜋 - 2𝜋) = 𝑓(-3𝜋) = csc(-3𝜋)
𝑦 = 3csc(-3𝜋) - 1

4. When 𝑥 = 0:
𝑓(1 𝑥 − 2𝜋) = 𝑓(0 - 2𝜋) = 𝑓(-2𝜋) = csc(-2𝜋)
𝑦 = 3csc(-2𝜋) - 1

5. When 𝑥 = 𝜋:
𝑓(1 𝑥 − 2𝜋) = 𝑓(𝜋 - 2𝜋) = 𝑓(-𝜋) = csc(-𝜋)
𝑦 = 3csc(-𝜋) - 1

Using these mappings, find the corresponding y-values for each key point.

Finally, plot the asymptotes, local max/min points, and the key points on the graph, and label each axis.

Answer 3

Sure, I can help you understand how to find the local max/min and determine the key points for the transformed function. Let's break it down step-by-step.

First, let's start with the transformation equation 𝑦 = 3𝑓(1𝑥 − 2𝜋) − 1. We need to find the transformed function by applying the given transformations to the parent function f(x) = csc(x).

1. Parent Function: f(x) = csc(x)

The parent function f(x) = csc(x) has vertical asymptotes at every multiple of 𝜋, where x is not equal to zero. So, let's draw the vertical asymptotes on the graph first.

Next, let's find the key points. For mapping notation, we'll use the format (x, f(x)) → (x, y).

2. Apply the transformation: 𝑓(1𝑥 − 2𝜋)
- Multiply x by 1 and subtract 2𝜋 from it.

3. Apply the next transformation: 3𝑓(1𝑥 − 2𝜋)
- Multiply the result from the previous step by 3.

4. Apply the final transformation: 3𝑓(1𝑥 − 2𝜋) - 1
- Subtract 1 from the result of the previous step.

Now, we can use these steps to find the key points for the transformed function. Remember, we need at least 5 key points for full marks. Let's choose some values of x within the interval −4𝜋 ≤ 𝑥 ≤ 4𝜋 and find the corresponding y values.

Key Point 1: Choose x = -3𝜋/2
- Apply the transformations:
(x, f(x)) → (-3𝜋/2, f(-3𝜋/2)) → (-3𝜋/2, 3𝑓((-3𝜋/2) - 2𝜋) - 1)

Evaluate the expression to find the y-coordinate of the key point.

Repeat this process for 4 more key points, using different values of x within the given interval. You can choose values like -𝜋, 0, 𝜋, 3𝜋/2, and 2𝜋.

Once you have obtained the key points, plot them on the graph. Label the asymptotes, local max/min, and number each axis. Finally, connect the points with smooth curves to represent the transformed function.

I hope this explanation helps you with finding the local max/min and key points for the transformed function. Let me know if you have any further questions!