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.
To graph the transformed function π¦ = 3π (1/2 π₯ β 2π) β 1, we need to understand how each transformation affects the parent function f(x) = csc(x).
The transformations applied to the parent function f(x) = csc(x) are as follows:
1. Horizontal compression: The factor of 1/2 in 1/2 π₯ compresses the graph horizontally. This means the graph will be narrower than the parent function.
2. Horizontal shift: The 2π in 1/2 π₯ β 2π shifts the graph horizontally to the right by 2π units. This means the graph will be shifted to the right.
3. Vertical stretch: The factor of 3 in 3π stretches the graph vertically. This means the graph will be taller than the parent function.
4. Vertical shift: The -1 in 3π - 1 shifts the graph vertically downward by 1 unit. This means the graph will be shifted downward.
To graph the transformed function, we will focus on key points and use mapping notation to show the work. Here are the steps to find the key points:
Step 1: Determine the key points of the parent function f(x) = csc(x) within the given interval β4π β€ π₯ β€ 4π.
The parent function f(x) = csc(x) has vertical asymptotes at x = kπ, where k is an integer, and its local maximum and minimum occur at x = (2k + 1)π/2, where k is an integer.
Within the interval β4π β€ π₯ β€ 4π, the parent function has vertical asymptotes at x = -2π, -π, π, and 2π. The local maximum and minimum occur at x = -3π/2, -π/2, π/2, and 3π/2.
Step 2: Apply the transformations to the key points of the parent function.
To find the transformed key points, we take each key point of the parent function and apply the transformations in the order they were given.
For example, let's find the transformed key point for the vertical asymptote x = -2π:
1. Horizontal compression: Multiply the x-coordinate by 1/2.
Transformed x-coordinate = (-2π) * (1/2) = -π
2. Horizontal shift: Add 2π to the transformed x-coordinate.
Transformed x-coordinate = -π + 2π = π
3. Vertical stretch: Multiply the y-coordinate by 3.
Transformed y-coordinate = csc(-2π) * 3 = (1/sin(-2π)) * 3
4. Vertical shift: Subtract 1 from the transformed y-coordinate.
Transformed y-coordinate = (1/sin(-2π)) * 3 - 1
Repeat these steps for the other key points to find all the transformed key points.
Step 3: Plot the transformed key points on the graph.
Once you have the transformed key points, plot them on the graph for the given interval β4π β€ π₯ β€ 4π. Label the asymptotes, local max/min, and number each axis.
Connecting these points will give you the graph of the transformed function π¦ = 3π (1/2 π₯ β 2π) β 1 within the interval β4π β€ π₯ β€ 4π.