Write a cosine function that has a midline of $$ y=5 $$, an amplitude of 4 , a period of 2 , and a horizontal shift of 1 to the left.

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To write a cosine function with the specified characteristics, we can use the general form of the cosine function:

$$
y = A \cos(B(x - C)) + D
$$

Where:
- $$ A $$ is the amplitude,
- $$ B $$ is related to the period,
- $$ C $$ is the horizontal shift,
- $$ D $$ is the midline.

Given the conditions:
- Midline $$ D = 5 $$
- Amplitude $$ A = 4 $$
- Period $$ P = 2 $$
- Horizontal shift $$ C = 1 $$ (to the left, so we use $$ -1 $$)

First, we need to find $$ B $$ using the formula for the period:

$$
P = \frac{2\pi}{B}
$$

Setting $$ P = 2 $$:

$$
2 = \frac{2\pi}{B}
$$

To find $$ B $$, we can rearrange the equation:

$$
B = \frac{2\pi}{2} = \pi
$$

Now we can substitute $$ A $$, $$ B $$, $$ C $$, and $$ D $$ into the general form:

$$
y = 4 \cos(\pi(x + 1)) + 5
$$

Thus, the cosine function that meets all the specified conditions is:

$$
y = 4 \cos(\pi(x + 1)) + 5
$$
The cosine function is $$ y = 4 \cos(\pi(x + 1)) + 5 $$.

Write a cosine function that has a midline of \( y=5 \), an amplitude of 4 , a period of 2 , and a horizontal shift of 1 to the left.

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