Hi could you help me with this question on graphing ssinusoidal functions.
"Remember how we saw a relationship between y=sin x and y=cos x...what is the equivalent equation of
y=sin0.5(x-45)if the function before transformations was y=cos x?
To find the equivalent equation, you need to determine the relationship between the given equation y = sin(0.5(x - 45)) and the original equation y = cos x.
1. Start with the original equation y = cos x.
2. Notice that in the given equation, the input x is modified by (x - 45), which means the graph is shifted right by 45 units.
3. Next, observe that the input x is also multiplied by 0.5, which means the graph is compressed horizontally by a factor of 2 (1/0.5 = 2).
4. Finally, the original equation y = cos x is changed to y = sin 0.5(x - 45).
To summarize:
- The graph is shifted right by 45 units.
- The graph is compressed horizontally by a factor of 2.
Therefore, the equivalent equation is y = cos(2(x - 45)).
Sure, I can help you with that question! To find the equivalent equation of y = sin(0.5(x - 45)) if the function before transformations was y = cos(x), we need to understand the effect of each transformation and apply them to the original equation.
Let's break down the original equation y = cos(x) and understand its properties first. The cosine function has a period of 2π, meaning it repeats itself every 2π units along the x-axis. The maximum value of cos(x) is 1, and the minimum value is -1. It also has a phase shift of 0, which means it starts at its maximum value at x = 0.
Now, let's analyze the given equation y = sin(0.5(x - 45)) step by step:
1. Inside the parentheses, there is a transformation of (x - 45). This causes a horizontal shift to the right by 45 units. So, the function starts at its maximum at x = 45.
2. The coefficient of 0.5 outside the parentheses influences the period of the function. Since the coefficient is less than 1, it elongates the period to 2π/0.5 = 4π. This means the function will complete one full cycle every 4π units along the x-axis.
3. The sine function ranges between -1 and 1, just like the cosine function. However, there is no vertical transformation in this equation.
Combining all the above transformations, we can rewrite the equation y = sin(0.5(x - 45)) as y = sin(4π(x - 45)), since the period transformation (0.5) and the phase shift transformation (45) are accounted for.
Therefore, the equivalent equation of y = sin(0.5(x - 45)) if the function before transformations was y = cos(x) is y = sin(4π(x - 45)).
I hope this explanation helps you understand how to find the equivalent equation of a sinusoidal function after transformations. Let me know if you have any further questions!