What are the phase shift and the vertical shift for the function f(x) = cos 4(x+5) + 3? And what direction is each shift going in (up or down)?
f(x-a) is f(x) shifted right by a. So, yours is shifted left by 5.
f(x)+k is f(k) shifted up by k.
To determine the phase shift and vertical shift for the function f(x) = cos 4(x+5) + 3, let's break it down:
1. Phase Shift: The phase shift represents the horizontal shift of the graph. It shows how much the graph is shifted to the right or left. In this case, the phase shift is -5. To find the phase shift, we need to determine the value inside the parentheses of the cosine function, which is (x+5). Since the general form is (x - h) for a horizontal shift of h units, we see that (x+5) equals (x-(-5)).
In this equation, h is -5, which means the graph is shifted 5 units to the left. Therefore, the phase shift of f(x) = cos 4(x+5) + 3 is -5 to the left.
2. Vertical Shift: The vertical shift represents the upward or downward shift of the graph. It indicates how much the graph is shifted veritcally along the y-axis. In this case, the vertical shift is +3. To find the vertical shift, we look at the constant term outside the cosine function, which is +3.
Since the cosine function oscillates between -1 and +1, adding a constant term shifts the entire graph upward or downward. In this case, +3 shifts the graph 3 units up. Therefore, the vertical shift of f(x) = cos 4(x+5) + 3 is +3 units up.
So, to summarize:
- The phase shift of f(x) = cos 4(x+5) + 3 is -5 units to the left.
- The vertical shift of f(x) = cos 4(x+5) + 3 is +3 units up.