Trigonometry Worksheet & Quiz

What's the Equation?

A new graph is shown below. Can you find the equation y = a f(b(x-c)) + d?

 

Worksheet Activity

Follow these steps. You will need to use the Graph Explorer (in the other file) to discover the answers and build your understanding.

  1. Step 1: The 'a' value (Amplitude)
    In the Graph Explorer, set b=1, c=0, d=0. Now, change the 'a' value from 1 to 2, then to 3. What happens? Try -1 and -2. 'a' controls the vertical stretch (height) of the graph, known as the amplitude. If 'a' is negative, it also flips the graph over the x-axis.
  2. Step 2: The 'd' value (Vertical Shift)
    Set a=1, b=1, c=0. Now, change 'd' from 0 to 2. What happens to the whole graph? Try -2. 'd' moves the entire graph up or down. This is the vertical shift, and the line y=d is the new "midline" of the graph.
  3. Step 3: The 'b' value (Period)
    Set a=1, c=0, d=0. The normal graph (where b=1) completes one full cycle in 360° (or 2π). Now, change 'b' to 2. How long does it take to complete a cycle? (Answer: 180°). What if b=0.5? (Answer: 720°). 'b' controls the horizontal stretch or shrink. The period (length of one cycle) is 360° / b.
  4. Step 4: The 'c' value (Phase Shift)
    Set a=1, b=1, d=0. Now, change 'c' from 0 to 90. Which way did the graph move? (Answer: Right by 90°). What about c=-90? (Answer: Left by 90°). 'c' controls the horizontal shift, known as the phase shift.
  5. Step 5: Combine & Create!
    Try to create your own graph in the Explorer. Can you create the equation y = 2 * sin(0.5 * (x - 90°)) + 1? Set the sliders and see if it matches what you expect.
  6. Step 6: Test Your Knowledge
    Once you feel confident, use the What's the Equation? quiz above. Click "New Question" and try to find the correct parameters for the graph shown. Good luck!