First, we rewrite the inner portion to make sure it is factored, then we will determine the Period, Midline, Amplitude, and Phase Shift. \[y={\color{{red}}-4}\cos\left({\color{{green}}\frac{{1}}{{5}} }x\right){\color{{blue}}+1}\]

• Period: \(\dfrac{2\pi}{ {\color{ green } 1 / 5 } }=10\pi\)
  • Half-Period: \(5\pi\)
  • Quarter-Period: \(\frac{ 5\pi }{ 2 }\)
• Midline: \(y={\color{ blue }+1}\)
• Amplitude: \(\left|{\color{ red }-4}\right|=4\)
  • Because this value is initially  negative, the Cosine will begin at a minimum.
  • Minimum: \({\color{ blue } 1}-4=-3\)
  • Maximum: \({\color{ blue } 1}+4=5\)
• Phase Shift: 0

Because there of the negative in front of the Cosine, there will be reflected over the \(x\) axis. Think back to the previous lesson about Cosine: if the Cosine normally starts at a maximum, reflecting it over the \(x\) axis means it will now start at a minimum!

Now for the step-by-step graphing process:

Step 1: Plot the Quarter/Half-Period values offset by Phase Shift (which is 0, so that's nice). The easiest way to accomplish this is to use the quarter-period as the basis for counting. So, starting at the phase shift (again, 0), we will add multiples of the quarter period until we have on full period:

\[0, \frac{ 5\pi }{ 2 }, \frac{10 \pi}{ 2 }, \frac{15\pi}{ 2 },\frac{20\pi}{{2}}\]

\[0, \frac{ 5\pi }{ 2 }, 5\pi, \frac{15\pi}{ 2 },10\pi\]

My graph will show the fractional values (so you can see how you could do it directly in the graph) but the final answer will have the simplified fractions.

Step 1: Plot the Fractions on Input Axis:

Step 2: Plot the Midline.

Step 3: Plot the midline intersections and max/min values.

Step 4: Connect the dots!