Welcome to Chapter 6, Lesson 2! In this lesson, we move from the unit circle to the coordinate plane. We are going to visualize trigonometric functions as waves (for sine and cosine) and asymptotic curves (for tangent and cotangent). Understanding how to sketch these graphs and identifying their key features is crucial for modeling periodic phenomena in physics and engineering.

The General Sinusoidal Equation

Most of our work will focus on transforming the parent functions $y = \sin(x)$ and $y = \cos(x)$. The general forms you will encounter are:

$$y = a \sin(bx + c) + d$$ $$y = a \cos(bx + c) + d$$

Each variable in these equations controls a specific transformation:

  • Amplitude ($|a|$): This determines the vertical stretch or compression. It tells us how far the graph goes up and down from its center (midline). If $a$ is negative, the graph is reflected across the x-axis.
  • Period ($\frac{2\pi}{b}$): The variable $b$ affects the horizontal stretch or compression. While the standard period is $2\pi$, the new period becomes $\frac{2\pi}{|b|}$. This tells us how long it takes for the function to complete one full cycle.
  • Phase Shift ($-\frac{c}{b}$): This is the horizontal shift. Be careful here! If the equation is $bx + c$, the shift is to the left. If it is $bx - c$, the shift is to the right. To find the exact value, we set the argument to zero ($bx + c = 0$) and solve for $x$.
  • Vertical Shift ($d$): This shifts the entire graph up or down. The line $y = d$ becomes the new midline or average value of the function.

Tangent and Cotangent

We will also look at Tangent and Cotangent functions. Remember that these graphs look different because they have vertical asymptotes where the functions are undefined. For $y = \tan(x)$, the period is $\pi$ (not $2\pi$), and asymptotes occur at odd multiples of $\frac{\pi}{2}$.

Lesson Topics

Here is the breakdown of the specific skills and problem types we will cover in this lesson:

  • Graphing Base Transformations:
    • Sketching $y = a \sin(x)$ or $y = a \cos(x)$ (Amplitude changes)
    • Sketching $y = \sin(bx)$ or $y = \cos(bx)$ (Period changes)
    • Combining transformations: $y = a \sin(bx)$
  • Analyzing Function Properties:
    • Determining Amplitude and Period
    • Identifying Domains and Ranges of trig functions
    • Calculating Phase Shifts
  • Advanced Graphing (Shifts):
    • Vertical shifts: $y = \sin(x) + d$
    • Phase shifts: $y = \sin(x + c)$
    • Complex combinations: $y = a \sin(bx + c) + d$
    • Sketching Tangent or Cotangent functions (Problem type 2)
  • Reverse Engineering:
    • Writing the equation of a sine or cosine function given its graph (Problem types 1 & 2)

Professor Baker's Tip

When graphing a complex function like $y = 3\sin(2x - \pi) + 1$, do not try to do it all at once! Start by sketching the parent function $y = \sin(x)$, then adjust the period, then the amplitude, and save your vertical and horizontal shifts for the very last step. It is much easier to move the graph once you have the correct shape!