Fungsi 12: Fungsi Periodik dan Grafik Fungsi Trigonometri Kelas 10

Summary

This educational video script introduces the concept of periodic functions, specifically focusing on trigonometric functions. It defines a periodic function and explains its properties using the sine and cosine functions as examples. The script discusses amplitude and period, illustrating these with graphical representations. It guides viewers through determining the period, amplitude, and shifts of trigonometric functions, providing step-by-step instructions for graphing. The tutorial is designed to be an introductory lesson, with more detailed exploration in future trigonometry lessons.

Takeaways

• 📘 The video is a continuation of a mathematics class discussing periodic functions, specifically focusing on Subsection 4.12.
• 🔄 A function is defined as periodic with a period P if it satisfies the equation f(x + P) = f(x), indicating it repeats every P units.
• 📊 The concept of amplitude and period in trigonometric functions is introduced, with amplitude being the maximum distance from the function's graph to the x-axis and the period being the distance over which the function repeats.
• 📈 The video discusses the basic graphs of the sine and cosine functions, explaining that the sine function has a wave-like graph with peaks and troughs, while the cosine function has a similar shape but starts at the peak.
• 🌊 The standard form of a sinusoidal function is given as y = a * sin(Bx + C) + D, where 'a' is the amplitude, 'B' affects the period, 'C' is the horizontal shift, and 'D' is the vertical shift.
• 🔢 The amplitude of a trigonometric function is represented by the formula (maximum value - minimum value) / 2, and the period is given by 2π / the coefficient of 'x'.
• ⏳ The video provides a step-by-step guide on how to graph sinusoidal functions, including how to adjust for amplitude, period, and shifts.
• 📐 Examples are given to illustrate how to determine the period, amplitude, and shifts of a function, and then graph it accordingly.
• 📚 The video concludes with a call to action for viewers to practice determining the period, amplitude, and shifts of given trigonometric functions and to graph them.
• 🎓 The instructor encourages viewers to like and subscribe to the channel for more educational content.

Q & A

What is a periodic function?

A function is considered periodic if it repeats its values at regular intervals or periods. It is defined by the equation f(x + P) = f(x), where P is the period of the function.

What is the relationship between the period of a function and its coefficient in the equation?

The period of a function is inversely proportional to the coefficient of the variable in the function's equation. Specifically, for a function in the form f(x) = a * sin(b * x + c) or f(x) = a * cos(b * x + c), the period is given by 2π/b.

What is amplitude in the context of trigonometric functions?

Amplitude refers to the maximum distance from the graph of a trigonometric function to the x-axis. It is represented by the coefficient 'a' in the equations y = a * sin(bx + c) or y = a * cos(bx + c).

How do you determine the phase shift of a trigonometric function?

The phase shift of a trigonometric function is determined by the value of 'c' in the equations y = a * sin(bx + c) or y = a * cos(bx + c), divided by the coefficient 'b'. A positive value of 'c/b' indicates a shift to the right, while a negative value indicates a shift to the left.

What is the period of the sine function y = sin(x)?

The period of the sine function y = sin(x) is 2π, as it completes one full cycle from 0 to 2π.

How does the amplitude of a function change when the coefficient 'a' is doubled?

When the coefficient 'a' in a trigonometric function is doubled, the amplitude of the function also doubles, meaning the maximum and minimum values of the function will be twice as far from the x-axis.

What is the effect of multiplying the variable 'x' by a coefficient 'b' in a trigonometric function?

Multiplying the variable 'x' by a coefficient 'b' in a trigonometric function, as in y = a * sin(b * x) or y = a * cos(b * x), compresses or stretches the graph horizontally, effectively changing the period of the function to 2π/b.

What is the vertical shift 'D' in the context of the function equation y = a * sin(bx + c) + D?

The vertical shift 'D' in the function equation y = a * sin(bx + c) + D moves the graph of the function up or down without affecting its period or amplitude. A positive 'D' shifts the graph up, while a negative 'D' shifts it down.

How do you calculate the phase shift in radians for a function given in degrees?

To calculate the phase shift in radians for a function given in degrees, you divide the degree value by the coefficient 'b' and then convert the result to radians by multiplying by π/180.

What is the domain and range of the basic sine and cosine functions?

The domain of the basic sine and cosine functions is all real numbers (R), and the range is from -1 to 1, inclusive. This means the functions' values oscillate between a minimum of -1 and a maximum of 1.