To find a cosine equation with the given characteristics, we need to understand how each parameter affects the cosine function. The general form of a cosine function is:
[tex]\[ y = A \cos(B(x - C)) + D \][/tex]
where:
- [tex]\( A \)[/tex] is the amplitude,
- [tex]\( \frac{2\pi}{B} \)[/tex] is the period,
- [tex]\( C \)[/tex] is the horizontal shift (phase shift),
- [tex]\( D \)[/tex] is the vertical shift.
Given:
- Amplitude, [tex]\( A = 4 \)[/tex]
- Period, [tex]\( \frac{2\pi}{5} \)[/tex]
- Horizontal Shift, [tex]\( -\frac{\pi}{10} \)[/tex]
- Vertical Shift, [tex]\( 3 \)[/tex]
1. Amplitude ([tex]\( A \)[/tex]):
The amplitude is directly given as [tex]\( 4 \)[/tex].
[tex]\[
A = 4
\][/tex]
2. Period:
The period [tex]\( T \)[/tex] of the cosine function is related to [tex]\( B \)[/tex] by the formula:
[tex]\[
T = \frac{2\pi}{B}
\][/tex]
We are given that the period is [tex]\( \frac{2\pi}{5} \)[/tex]. Setting [tex]\( T = \frac{2\pi}{5} \)[/tex], we can solve for [tex]\( B \)[/tex]:
[tex]\[
\frac{2\pi}{B} = \frac{2\pi}{5}
\][/tex]
[tex]\[
B = 5
\][/tex]
3. Horizontal Shift ([tex]\( C \)[/tex]):
The horizontal shift is given directly as [tex]\( -\frac{\pi}{10} \)[/tex].
[tex]\[
C = -\frac{\pi}{10}
\][/tex]
4. Vertical Shift ([tex]\( D \)[/tex]):
The vertical shift is given directly as [tex]\( 3 \)[/tex].
[tex]\[
D = 3
\][/tex]
Putting it all together, the cosine function with the given characteristics can be written as:
[tex]\[
y = 4 \cos(5(x + \frac{\pi}{10})) + 3
\][/tex]
So the cosine equation that satisfies the conditions is:
[tex]\[
y = 4 \cos \left(5 \left(x + \frac{\pi}{10}\right)\right) + 3
\][/tex]
