Q. Which is the graph of \(y = \cos(x) + 3\)?

Step-by-step explanation for graphing \( y = \cos(x) + 3 \)

1. Step 1: Identify the parent function.

   The base function is the standard cosine function, \( y = \cos(x) \). This function has a period of \( 2\pi \), an amplitude of 1, and oscillates between -1 and 1 around the x-axis (the midline \( y = 0 \)).

2. Step 2: Identify the transformation.

   The equation is in the form \( y = \cos(x) + k \), where \( k = 3 \). Adding a constant outside the function results in a vertical shift. Since 3 is positive, the entire graph of \( y = \cos(x) \) shifts upward by 3 units.

3. Step 3: Determine the new midline and range.

   The original midline \( y = 0 \) moves up to \( y = 3 \). To find the range, apply the amplitude to the new midline: \( 3 \pm 1 \). The maximum value becomes \( 3 + 1 = 4 \) and the minimum value becomes \( 3 – 1 = 2 \). Therefore, the graph oscillates between \( y = 2 \) and \( y = 4 \).

4. Step 4: Locate key points for one period.

   Using the standard intervals \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \), calculate the new y-coordinates:

   • At \( x = 0 \): \( y = \cos(0) + 3 = 1 + 3 = 4 \) (Maximum)
   • At \( x = \frac{\pi}{2} \): \( y = \cos\left(\frac{\pi}{2}\right) + 3 = 0 + 3 = 3 \) (Midline)
   • At \( x = \pi \): \( y = \cos(\pi) + 3 = -1 + 3 = 2 \) (Minimum)
   • At \( x = \frac{3\pi}{2} \): \( y = \cos\left(\frac{3\pi}{2}\right) + 3 = 0 + 3 = 3 \) (Midline)
   • At \( x = 2\pi \): \( y = \cos(2\pi) + 3 = 1 + 3 = 4 \) (Maximum)
5. Final Description:

   The graph is a periodic wave identical in shape to \( y = \cos(x) \) but shifted 3 units above the x-axis. It has a y-intercept at \( (0,4) \), a midline at \( y = 3 \), and never crosses the x-axis.