To transform the graph of \( f(x) = \sin x \) to obtain the graph of \( f(x) = -2\sin(x - 3) + 7 \), we will go through the following steps, noting the effects of each transformation:
-
Horizontal Shift: The term \( (x - 3) \) indicates a horizontal shift of the graph to the right by 3 units. This means every point on the graph of \( \sin x \) will be moved 3 units to the right.
-
Vertical Stretch and Reflection: The coefficient \(-2\) in front of the sine function indicates two transformations:
- The negative sign (-) reflects the graph across the x-axis. This means that all positive values of the sine function become negative and vice versa.
- The factor of \(2\) vertically stretches the graph by a factor of 2. Since we are reflecting it as well, all values of the sine function will now be multiplied by \(-2\), which affects the amplitude from 1 (the original sine amplitude) to 2.
-
Vertical Shift: The \( +7 \) at the end shifts the entire graph vertically upwards by 7 units. This means that the middle line (the axis of the sine wave) will move up from \( y = 0 \) to \( y = 7 \).
Summary of Transformations
Combining these transformations, we have:
- Shift right by 3 units.
- Reflect across the x-axis.
- Stretch vertically by a factor of 2.
- Shift up by 7 units.
The resulting graph \( f(x) = -2\sin(x - 3) + 7 \) will have a maximum value of \( 7 \) and a minimum value of \( 5 \) (since amplitude is 2, the maximum is \( 7 + 2 \) and the minimum is \( 7 - 2 \)). The key features of the sine wave are altered accordingly to reflect these transformations.
1 answer