The following transformations π¦ = 3π (1/2 π₯ β 2π) β 1 were applied to the parent function f(x) =csc(x). Graph the transformed function for the interval β4π β€ π₯ β€ 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. [6C]
To graph the transformed function, we need to apply each transformation step by step on the parent function f(x) = csc(x).
The given transformations are:
y = 3f(1/2x - 2Ο) - 1
Step 1: Starting with the parent function f(x) = csc(x), let's plot the graph of the parent function over the given interval β4Ο β€ x β€ 4Ο.
To map the key points, we need to determine the values of x that correspond to certain key points on the graph.
For the parent function f(x) = csc(x), the asymptotes occur at x = 0, Β±Ο, Β±2Ο, Β±3Ο, ....
Now, let's focus on the interval β4Ο β€ x β€ 4Ο and find the values of x for the selected points of interest.
1. Local Maximum:
The first local maximum occurs at x = Ο/2.
2. Local Minimum:
The first local minimum occurs at x = -Ο/2.
3. Increasing Interval:
Between the local minimum and the local maximum, we can choose a point.
4. Decreasing Interval:
Between the local maximum and the local minimum, we can choose a point.
5. A point on an asymptote:
Let's choose a point near the asymptote x = Ο.
Now that we have selected the key points, we can plug these values of x into the given transformation to find their corresponding y values.
Step 2: Apply the transformations to the parent function:
For each key point, we will apply the transformations one by one to determine the corresponding y values.
1. Local Maximum:
x = Ο/2
First, apply the transformation f(1/2x - 2Ο) - 1:
f(1/2(Ο/2) - 2Ο) - 1 = f(-3Ο/4) - 1
Find the corresponding y value by evaluating the parent function at the transformed x value:
f(-3Ο/4) - 1 = csc(-3Ο/4) - 1
2. Local Minimum:
x = -Ο/2
Apply the transformations:
f(1/2x - 2Ο) - 1 = f(1/2(-Ο/2) - 2Ο) - 1 = f(-5Ο/4) - 1
Find the corresponding y value:
f(-5Ο/4) - 1 = csc(-5Ο/4) - 1
3. Increasing Interval:
Choose a point such as x = βΟ/4.
Apply the transformations:
f(1/2x - 2Ο) - 1 = f(1/2(-Ο/4) - 2Ο) - 1 = f(-9Ο/8) - 1
Find the corresponding y value:
f(-9Ο/8) - 1 = csc(-9Ο/8) - 1
4. Decreasing Interval:
Choose a point such as x = 3Ο/4.
Apply the transformations:
f(1/2x - 2Ο) - 1 = f(1/2(3Ο/4) - 2Ο) - 1 = f(-Ο/4) - 1
Find the corresponding y value:
f(-Ο/4) - 1 = csc(-Ο/4) - 1
5. A point on an asymptote:
Choose a point near x = Ο.
Apply the transformations:
f(1/2x - 2Ο) - 1 = f(1/2(Ο) - 2Ο) - 1 = f(-3Ο/2) - 1
Find the corresponding y value:
f(-3Ο/2) - 1 = csc(-3Ο/2) - 1
Now that we have determined the corresponding y values for the key points, we can plot them on the graph. We will label the asymptotes, local maxima and minima, and number each axis accordingly.