Chapter 6 Part 3: Graphing Trigonometric Functions

Welcome back to Professor Baker's Math Class! Today, we're continuing our exploration of Chapter 6, focusing on the graphs of trigonometric functions and how to manipulate them. Get ready to sketch, analyze, and understand the core concepts behind these important functions.

Topics Covered:

  • Sketching Sine and Cosine Functions: We'll start with the basics, understanding how to sketch $y = a \sin(x)$ and $y = a \cos(x)$. Remember, 'a' represents the amplitude, which affects the height of the graph.
  • Period Changes: Next, we'll explore the effect of 'b' in $y = \sin(bx)$ and $y = \cos(bx)$. The period of these functions is given by $2\pi/b$. A larger 'b' compresses the graph horizontally, while a smaller 'b' stretches it out.
  • Transformations: We'll learn how to graph functions like $y = \pm \sin(bx)$ or $y = \pm \cos(bx)$. The $\pm$ sign reflects the graph across the x-axis.
  • Amplitude and Period Together: Combining these, we'll tackle $y = a \sin(bx)$ and $y = a \cos(bx)$, understanding how both amplitude and period affect the graph.
  • Amplitude and Period: Knowing how to identify the amplitude ($\vert a \vert$) and period ($2\pi/b$) of sine and cosine functions is key.
  • Reciprocal Functions: We'll dive into matching graphs and equations for secant, cosecant, tangent, and cotangent functions, paying attention to asymptotes and key points. Remember these functions are reciprocals of cos, sin, and tan respectively.
  • Sketching Reciprocal Graphs: We will cover the process of sketching secant and cosecant graphs (Problem Type 1 & 2), as well as tangent and cotangent graphs (Problem Type 1 & 2). Understanding the asymptotes is crucial here!
  • Vertical Shifts: Learn how to graph $y = \sin(x) + d$ or $y = \cos(x) + d$, where 'd' represents a vertical shift of the graph. A positive 'd' moves the graph up, and a negative 'd' moves it down.
  • Combining Transformations: We'll explore more complex transformations, including vertical shifts, reflections, and period changes, expressed as: $y = a \sin(bx+c) + d$ or $y = a \cos(bx+c) + d$.
  • Phase Shift: The phase shift $-c/b$ shifts the graph horizontally.
  • Putting it all together: Grasp how the amplitude, period, phase shift, and vertical shift affect a sinusoidal graph. The general form is $y = a\sin(bx+c) + d$ or $y = a\cos(bx+c) + d$.

Remember, practice is key to mastering these concepts! Keep sketching those graphs, and don't hesitate to ask questions. Good luck, and happy graphing!