. The following transformations y=3f(1/2x-2pi)-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
you know that csc(x) has turning points at all odd multiples of pi/2
so, csc(1/2 x - 2pi) has turning points where x/2 - 2pi is an odd multiple of pi/2. They occur with period pi, and are at y = 3-1 and -3-1
The asymptotes are midway between the turning points
To graph the transformed function, we will first determine the effect of each transformation on the parent function f(x) = csc(x).
1. Vertical Stretch/Compression:
The factor of 3 in the transformation equation, y = 3f(1/2x - 2π) - 1, indicates a vertical stretch by a factor of 3.
2. Horizontal Stretch/Compression:
The factor of 1/2 in the transformation equation, y = 3f(1/2x - 2π) - 1, indicates a horizontal compression by a factor of 1/2.
3. Horizontal Shift:
The term -2π in the transformation equation, y = 3f(1/2x - 2π) - 1, indicates a horizontal shift to the right by 2π units.
4. Vertical Translation:
The term -1 in the transformation equation, y = 3f(1/2x - 2π) - 1, indicates a vertical translation downward by 1 unit.
Now, let's find key points to plot on the graph using mapping notation. We'll focus on the interval -4π ≤ x ≤ 4π.
1. The parent function f(x) = csc(x) has asymptotes at x = 0, π, 2π, -π, -2π, etc.
Transformations applied:
- Vertical stretch by a factor of 3: y = 3csc(x)
- Horizontal compression by a factor of 1/2: y = 3csc(1/2x)
- Horizontal shift to the right by 2π units: y = 3csc(1/2x - 2π)
- Vertical translation downward by 1 unit: y = 3csc(1/2x - 2π) - 1
To find key points, we can evaluate the transformed function at specific x-values:
a) For x = -4π, we get y = 3csc(1/2(-4π) - 2π) - 1
= 3csc(-2π - 2π) - 1
We know csc(-x) = -csc(x), so csc(-2π - 2π) = -csc(4π). Hence, y = -3csc(4π) - 1.
b) For x = -3π, we get y = 3csc(1/2(-3π) - 2π) - 1
= 3csc(-3π/2 - 2π) - 1
We know csc(x + π) = -csc(x), so csc(-3π/2 - 2π) = -csc(5π/2). Hence, y = -3csc(5π/2) - 1.
c) For x = -2π, we get y = 3csc(1/2(-2π) - 2π) - 1
= 3csc(-π - 2π) - 1
We know csc(-x) = -csc(x), so csc(-π - 2π) = -csc(3π). Hence, y = -3csc(3π) - 1.
d) For x = -π, we get y = 3csc(1/2(-π) - 2π) - 1
= 3csc(-π/2 - 2π) - 1
We know csc(x + π) = -csc(x), so csc(-π/2 - 2π) = -csc(3π/2). Hence, y = -3csc(3π/2) - 1.
e) For x = 0, we get y = 3csc(1/2(0) - 2π) - 1
= 3csc(-2π) - 1
We know csc(-x) = -csc(x), so csc(-2π) = -csc(2π). Hence, y = -3csc(2π) - 1.
You can find more key points by substituting different x-values within the given interval and following the same process.
2. Now, let's plot the graph with the labeled asymptotes, local max/min, and numbered axes:
(Note: The exact coordinates will depend on the values of csc at specific points.)
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--------|--------|--------|--------|--------|--------|--------|--------|--------|-------- x-axis
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--------------------------------------- x-axis
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Local Min: | Local Max: |
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Asymptote: |
x = 0, π, 2π, -π, -2π, etc. |
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--------------------------------------- x-axis
I hope the above information helps you understand how to graph the transformed function based on the given transformations.
To graph the transformation y=3f(1/2x-2π)-1 for the interval -4π ≤ 𝑥 ≤ 4π, we need to follow these steps:
1. Start with the parent function f(x) = csc(x).
- The graph of the parent function csc(x) is the reciprocal of the sine function sin(x).
- It has vertical asymptotes where the sine function crosses the x-axis (at x = π, 2π, -π, -2π, and so on).
2. Apply the transformations to the parent function:
a. The function is vertically stretched by a factor of 3. This means that the y-values are multiplied by 3.
b. The x-values are condensed horizontally by a factor of 1/2. This means that the x-values are multiplied by 2.
c. The function is shifted to the right by 2π units horizontally. This means that we add 2π to the x-values.
d. The function is shifted downward by 1 unit vertically. This means that we subtract 1 from the y-values.
3. To find key points, we can choose x-values from the interval -4π to 4π, and map them using the transformations.
Let's find the key points with the mapping notation:
a. For x = -4π:
- Apply the transformations: 1/2(-4π - 2π) = -3π, 3csc(-3π) = 3/csc(3π) = 3/0 (undefined).
- Note that csc(3π) is undefined as it corresponds to one of the vertical asymptotes.
- So, the point (-4π, undefined) is a vertical asymptote.
b. For x = -3π:
- Apply the transformations: 1/2(-3π - 2π) = -7π/2, 3csc(-7π/2) = 3/csc(7π/2).
- Find the value of csc(7π/2) by finding the reciprocal of sin(7π/2).
- Note that sin(7π/2) = -1, so csc(7π/2) = 1/-1 = -1
- Therefore, the point (-3π, -3) is a key point.
c. For x = -2π:
- Apply the transformations: 1/2(-2π - 2π) = -3π, 3csc(-3π) = 3/csc(3π) = 3/0 (undefined).
- Similar to point A, (-2π, undefined) is a vertical asymptote.
d. For x = -π:
- Apply the transformations: 1/2(-π - 2π) = -5π/2, 3csc(-5π/2) = 3/csc(5π/2).
- The value of csc(5π/2) is undefined, so (-π, undefined) is a vertical asymptote.
e. For x = 0:
- Apply the transformations: 1/2(0 - 2π) = -π, 3csc(-π) = 3/csc(π) = 3/0 (undefined).
- Similar to points A and C, (0, undefined) is a vertical asymptote.
By following a similar process for other points within the given interval, we can determine the rest of the key points.
Once we have found enough key points, we can plot them on the graph and connect them smoothly. Remember to label the asymptotes, local max/min points, and number each axis.