Answer :
Sure, let's go through each part step-by-step:
### Part a)
Given the properties:
- Midline: [tex]\(5\)[/tex]
- Amplitude: [tex]\(13\)[/tex]
- Period: [tex]\(2\pi\)[/tex]
- Phase shift: [tex]\(0\)[/tex]
For a sinusoidal function, the general forms are:
- Sine: [tex]\(y = a \sin(b(x - c)) + d\)[/tex]
- Cosine: [tex]\(y = a \cos(b(x - c)) + d\)[/tex]
Where:
- [tex]\(a\)[/tex] is the amplitude
- [tex]\(b\)[/tex] is related to the period by [tex]\(2\pi/b\)[/tex]
- [tex]\(c\)[/tex] is the phase shift (horizontal shift)
- [tex]\(d\)[/tex] is the midline (vertical shift)
Given:
- Amplitude ([tex]\(a\)[/tex]) is [tex]\(13\)[/tex]
- Midline ([tex]\(d\)[/tex]) is [tex]\(5\)[/tex]
- Period ([tex]\(2\pi\)[/tex]) means [tex]\(2\pi / b = 2\pi\)[/tex], so [tex]\(b = 1\)[/tex]
- Phase shift ([tex]\(c\)[/tex]) is [tex]\(0\)[/tex]
Substituting these values, the sine function equation is:
[tex]\[ y = 13 \sin(1 \cdot (x - 0)) + 5 \][/tex]
So, the sine function is:
[tex]\[ y = 13 \sin(x) + 5 \][/tex]
### Part b)
Given the trigonometric equation:
[tex]\[ y = 15 \tan\left(\frac{\pi x}{3} + 2\right) \][/tex]
We need to determine:
1. Stretching factor:
- The coefficient [tex]\(15\)[/tex] in front of the [tex]\(\tan\)[/tex] function represents the stretching factor.
2. Period:
- The general form is [tex]\(y = a \tan(bx + c)\)[/tex]
- The period of a tangent function is [tex]\(\pi/b\)[/tex]
- Here, [tex]\(b = \pi/3\)[/tex], thus the period is [tex]\(\pi / (\pi/3) = 3\)[/tex]
3. Phase shift:
- To find the phase shift, solve for [tex]\(x\)[/tex] in [tex]\( \frac{\pi x}{3} + 2 = 0 \)[/tex]
- [tex]\( \frac{\pi x}{3} = -2 \)[/tex]
- [tex]\( x = -2 \cdot \frac{3}{\pi} = -1.909859317102744\)[/tex]
4. Vertical asymptotes:
- Vertical asymptotes occur where the tangent function is undefined.
- For [tex]\( y = a \tan(bx + c) \)[/tex], vertical asymptotes are at [tex]\( \frac{\pi}{2} + k\pi \)[/tex], for integer values [tex]\(k\)[/tex]
- Solve [tex]\( \frac{\pi x}{3} + 2 = \frac{\pi}{2} \)[/tex]
- [tex]\( \frac{\pi x}{3} = \frac{\pi}{2} - 2 \)[/tex]
- [tex]\( x = \frac{\pi/2 - 2}{\pi/3} = -0.40985931710274415 \)[/tex]
- As the intervals between asymptotes are [tex]\(k\pi/b\)[/tex], i.e., [tex]\( k \cdot 3 \)[/tex]
- Thus, vertical asymptotes occur at [tex]\( x = -0.40985931710274415 + k\cdot 3 \)[/tex]
5. Domain:
- The domain of the tangent function is all real numbers except where the function is undefined (at vertical asymptotes).
- Hence, the domain exclusion can be written as:
[tex]\[ x \neq -0.40985931710274415 + k\cdot 3, \; \text{for integer values } k \][/tex]
Summarizing, we have:
- Stretching factor: [tex]\(15\)[/tex]
- Period: [tex]\(3\)[/tex]
- Phase shift: [tex]\(-1.909859317102744\)[/tex]
- First vertical asymptote: [tex]\(-0.40985931710274415\)[/tex]
- Vertical asymptote intervals: [tex]\(3\)[/tex]
- Domain exclusion: [tex]\( x \neq -0.40985931710274415 + k \cdot 3, k \in \mathbb{Z} \)[/tex]
### Part c)
To determine the coordinates of points [tex]\(a, b, c, d, e,\)[/tex] and [tex]\(f\)[/tex] on the graph of the given function, we need a visual graph or a detailed description of the graph provided. Since neither is available right now, specifying the coordinates for these points is not feasible.
Therefore, without the graph, we cannot provide specific coordinates.
I hope this breakdown helps you understand each step!
### Part a)
Given the properties:
- Midline: [tex]\(5\)[/tex]
- Amplitude: [tex]\(13\)[/tex]
- Period: [tex]\(2\pi\)[/tex]
- Phase shift: [tex]\(0\)[/tex]
For a sinusoidal function, the general forms are:
- Sine: [tex]\(y = a \sin(b(x - c)) + d\)[/tex]
- Cosine: [tex]\(y = a \cos(b(x - c)) + d\)[/tex]
Where:
- [tex]\(a\)[/tex] is the amplitude
- [tex]\(b\)[/tex] is related to the period by [tex]\(2\pi/b\)[/tex]
- [tex]\(c\)[/tex] is the phase shift (horizontal shift)
- [tex]\(d\)[/tex] is the midline (vertical shift)
Given:
- Amplitude ([tex]\(a\)[/tex]) is [tex]\(13\)[/tex]
- Midline ([tex]\(d\)[/tex]) is [tex]\(5\)[/tex]
- Period ([tex]\(2\pi\)[/tex]) means [tex]\(2\pi / b = 2\pi\)[/tex], so [tex]\(b = 1\)[/tex]
- Phase shift ([tex]\(c\)[/tex]) is [tex]\(0\)[/tex]
Substituting these values, the sine function equation is:
[tex]\[ y = 13 \sin(1 \cdot (x - 0)) + 5 \][/tex]
So, the sine function is:
[tex]\[ y = 13 \sin(x) + 5 \][/tex]
### Part b)
Given the trigonometric equation:
[tex]\[ y = 15 \tan\left(\frac{\pi x}{3} + 2\right) \][/tex]
We need to determine:
1. Stretching factor:
- The coefficient [tex]\(15\)[/tex] in front of the [tex]\(\tan\)[/tex] function represents the stretching factor.
2. Period:
- The general form is [tex]\(y = a \tan(bx + c)\)[/tex]
- The period of a tangent function is [tex]\(\pi/b\)[/tex]
- Here, [tex]\(b = \pi/3\)[/tex], thus the period is [tex]\(\pi / (\pi/3) = 3\)[/tex]
3. Phase shift:
- To find the phase shift, solve for [tex]\(x\)[/tex] in [tex]\( \frac{\pi x}{3} + 2 = 0 \)[/tex]
- [tex]\( \frac{\pi x}{3} = -2 \)[/tex]
- [tex]\( x = -2 \cdot \frac{3}{\pi} = -1.909859317102744\)[/tex]
4. Vertical asymptotes:
- Vertical asymptotes occur where the tangent function is undefined.
- For [tex]\( y = a \tan(bx + c) \)[/tex], vertical asymptotes are at [tex]\( \frac{\pi}{2} + k\pi \)[/tex], for integer values [tex]\(k\)[/tex]
- Solve [tex]\( \frac{\pi x}{3} + 2 = \frac{\pi}{2} \)[/tex]
- [tex]\( \frac{\pi x}{3} = \frac{\pi}{2} - 2 \)[/tex]
- [tex]\( x = \frac{\pi/2 - 2}{\pi/3} = -0.40985931710274415 \)[/tex]
- As the intervals between asymptotes are [tex]\(k\pi/b\)[/tex], i.e., [tex]\( k \cdot 3 \)[/tex]
- Thus, vertical asymptotes occur at [tex]\( x = -0.40985931710274415 + k\cdot 3 \)[/tex]
5. Domain:
- The domain of the tangent function is all real numbers except where the function is undefined (at vertical asymptotes).
- Hence, the domain exclusion can be written as:
[tex]\[ x \neq -0.40985931710274415 + k\cdot 3, \; \text{for integer values } k \][/tex]
Summarizing, we have:
- Stretching factor: [tex]\(15\)[/tex]
- Period: [tex]\(3\)[/tex]
- Phase shift: [tex]\(-1.909859317102744\)[/tex]
- First vertical asymptote: [tex]\(-0.40985931710274415\)[/tex]
- Vertical asymptote intervals: [tex]\(3\)[/tex]
- Domain exclusion: [tex]\( x \neq -0.40985931710274415 + k \cdot 3, k \in \mathbb{Z} \)[/tex]
### Part c)
To determine the coordinates of points [tex]\(a, b, c, d, e,\)[/tex] and [tex]\(f\)[/tex] on the graph of the given function, we need a visual graph or a detailed description of the graph provided. Since neither is available right now, specifying the coordinates for these points is not feasible.
Therefore, without the graph, we cannot provide specific coordinates.
I hope this breakdown helps you understand each step!
