for graphing basic trig functions such as y=2sinx or y=1/3cosx, how do you know what the points are to graph with out using a calculator?
You can pick as many x points as you want. The more you plot, the easier it is to plot a smopoth and accurate curve.
For the y values that go with each x, you need a calculator, a special slide rule or a table of trig functions. Hardly anyone uses tables or slide rules anymore.
but for my quiz, i am not allowed a calculator
In that case, pick points that you know. For 2sinx, say, you know that
sin 0 = 0
sin pi/6 = .5
sin pi/4 = .707
sin pi/3 = .866
sin pi/2 = 1
and so on
If you know a few key values, you can sketch the graph easily. If you don't happen to know the value of pi/3, etc, use degrees. It's easy to plot x=0,30,45,60,90, etc.
How to find general solutions
To graph basic trigonometric functions like y = 2sin(x) or y = (1/3)cos(x) without using a calculator, you can rely on the properties of these functions and some key points on their graphs. Here's a step-by-step explanation:
1. Start by understanding the basic shape of the trigonometric functions:
- The sine function (sin(x)) has a periodic shape that oscillates between -1 and 1. Its graph starts at the origin (0, 0), rises to its maximum at (π/2, 1), returns to the x-axis at (π, 0), reaches its minimum at (3π/2, -1), and completes the cycle at (2π, 0). This pattern repeats for all multiples of 2π.
- The cosine function (cos(x)) also has a periodic shape that oscillates between -1 and 1. Its graph starts at (0, 1), reaches a minimum at (π/2, 0), returns to the x-axis at (π, -1), reaches its maximum at (3π/2, 0), and completes the cycle at (2π, 1). Again, this pattern repeats for 2π, 4π, and so on.
2. Adjust the amplitude (coefficient in front of the function):
- For y = 2sin(x), the coefficient of 2 indicates that the amplitude is multiplied by 2. This means the graph will reach a maximum of 2 and a minimum of -2.
- For y = (1/3)cos(x), the coefficient of (1/3) indicates that the amplitude is reduced to one-third. Hence, the graph will reach a maximum of (1/3) and a minimum of (-1/3).
3. Determine the period of the function:
- The period of a function is the distance it takes for one complete cycle. For sin(x) and cos(x), the period is 2π. However, if there is a coefficient in front of x, it will affect the period. For example, for y = a sin(bx), the period is (2π) / |b|, and for y = a cos(bx), the period is (2π) / |b|.
4. Calculate additional key points and sketch the graph:
- Determine the period and divide it into equal intervals (e.g., 2π/4 = π/2 for four intervals).
- Plug in values of x into the function and calculate the corresponding y-values.
- Plot these points on your graph, considering the amplitude and period described earlier.
- Connect the points smoothly to form the graph. Keep in mind the shape and periodicity of the trigonometric function.
By applying these steps, you can graph basic trigonometric functions without the need for a calculator. However, it's always beneficial to verify your graph using a graphing tool or calculator if available.