Find the equation for the curve in its final position. The graph y=sin (x) is shifted a distance of pi/4 to the left, reflected in the x-axis, translated 2 units downward, then stretched by a factor of 7.
Let's go step by step to determine the equation for the curve in its final position.
1. Shifted to the left by π/4:
To shift the graph π/4 units to the left, we need to replace x with (x + π/4) in the equation y = sin(x).
2. Reflected in the x-axis:
When the graph is reflected in the x-axis, the sign of the equation changes from positive to negative. Therefore, we need to negate the equation: -sin(x + π/4).
3. Translated 2 units downward:
To translate the graph 2 units downward, we need to subtract 2 from the equation: -sin(x + π/4) - 2.
4. Stretched by a factor of 7:
To stretch the graph by a factor of 7, we need to multiply the equation by 7: 7(-sin(x + π/4) - 2).
So, the equation for the curve in its final position is 7(-sin(x + π/4) - 2).