The equation y = a sin(bx + c) + d represents a sine function that has been transformed in several ways, including amplitude, period, phase shift, and vertical shift. The variable 'a' affects the amplitude, which is the height of the wave; 'b' determines the frequency and period of the wave; 'c' shifts the graph horizontally, and 'd' shifts it vertically. These transformations allow for a wide variety of sine curves that can model real-world periodic phenomena.
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The amplitude is given by |a|, which means if 'a' is negative, the graph will be reflected across the x-axis but still maintain the same amplitude.
The period of the sine function changes with different values of 'b'. A larger 'b' results in a shorter period and more cycles within a given interval.
The phase shift is determined by how much 'c' moves the graph left or right, affecting where one complete cycle starts on the x-axis.
The vertical shift represented by 'd' moves the entire graph up or down, changing the midline from y=0 to y=d.
When graphing, it's important to find key points such as maximums, minimums, and zeros to accurately portray the transformed sine function.
Review Questions
How does changing the value of 'a' affect the graph of y = a sin(bx + c) + d?
Changing the value of 'a' directly affects the amplitude of the sine function. If |a| is increased, the peaks and troughs of the wave become higher and lower respectively, making it taller. If 'a' is negative, it flips the graph over the x-axis. Understanding this helps visualize how varying 'a' alters wave characteristics without changing its basic shape.
What impact does adjusting 'b' have on both frequency and period in y = a sin(bx + c) + d?
Adjusting 'b' modifies both frequency and period. The frequency increases as 'b' becomes larger, meaning more cycles are completed in a given interval. The period, calculated as $$\frac{2\pi}{b}$$, decreases with increasing 'b', leading to a more compressed wave. This relationship is crucial for understanding how quickly or slowly a wave oscillates.
Analyze how changing 'c' influences the overall behavior of the sine function in y = a sin(bx + c) + d.
Changing 'c' shifts the entire graph horizontally due to its role as a phase shift. Specifically, it translates left or right based on its value when divided by 'b'. This shift alters where peaks and troughs occur on the x-axis, affecting how we interpret periodic phenomena over time. Additionally, it impacts intersections with other functions if analyzed together.
The amplitude is the maximum distance from the midline to the peak or trough of the sine wave, determined by the absolute value of 'a' in the equation.