Determining the equation of a trig function | Graphs of trig functions | Trigonometry

Understanding Periodic Functions: A Fun Guide

Hey there! Today, we’re diving into the world of periodic functions, which are super cool because they repeat their patterns over and over. We’ll focus on how to write the equation of a periodic function by looking at its graph. This function behaves like sine or cosine functions, but with its own unique twist. Let’s break it down step by step!

Finding the Midline and Amplitude

First things first, we need to figure out the midline and amplitude of our function. The midline is like the average height of the wave, sitting right in the middle between the highest and lowest points.

• Maximum Value: ( y = 1 )
• Minimum Value: ( y = -5 )

To find the midline, we calculate:

[
ext{Midline} = frac{1 + (-5)}{2} = frac{-4}{2} = -2
]

So, the midline is ( y = -2 ).

Next, let’s find the amplitude, which tells us how tall the waves are from the midline. The function goes 3 units above the midline (from -2 to 1) and 3 units below it (from -2 to -5). Therefore, the amplitude is:

[
ext{Amplitude} = 3
]

Creating the Function

Now that we know the midline and amplitude, we can start forming the function. It can be either a sine or cosine function:

1. Cosine Form:
   [
   f(x) = 3 cos(kx) – 2
   ]
2. Sine Form:
   [
   f(x) = 3 sin(kx) – 2
   ]

Next, we need to figure out which one it is: sine or cosine.

Checking the Function at ( x = 0 )

To decide between sine and cosine, let’s see what happens at ( x = 0 ):

• Cosine Function:
  [
  f(0) = 3 cos(0) – 2 = 3(1) – 2 = 1
  ]
• Sine Function:
  [
  f(0) = 3 sin(0) – 2 = 3(0) – 2 = -2
  ]

Since the function is at the midline when ( x = 0 ), it can’t be the cosine function because it doesn’t give us the midline value. So, it’s a sine function!

Finding the Period

Now, let’s determine the period, which is how long it takes for the function to complete one full cycle. From the graph, we see that the function crosses the midline with a positive slope every 8 units.

The period of a sine function is:

[
ext{Period} = frac{2pi}{k}
]

Setting this equal to 8, we get:

[
frac{2pi}{k} = 8
]

Solving for ( k ), we find:

[
k = frac{2pi}{8} = frac{pi}{4}
]

The Final Equation

With the amplitude, midline, and ( k ) value in hand, we can write the final equation of our function:

[
f(x) = 3 sinleft(frac{pi}{4}xright) – 2
]

And there you have it! This equation perfectly describes the periodic function based on its graph. Now you’re all set to tackle periodic functions with confidence!