Answer
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Step 1:: Identify the shape of the graph
To determine whether a graph represents a sine or cosine function, first look at the shape of the graph. Both sine and cosine graphs are periodic (repeating) curves. The key difference between their shapes is the position of the highest and lowest points.
Step 2:: Locate the maximum and minimum points
On the other hand, for a cosine function, the maximum points are at $$y=1$$ and the minimum points are at $$y=-1$$.
However, these maximum and minimum points occur at different locations along the x-axis for sine and cosine functions.
Step 3:: Determine the location of maximum and minimum points
For a sine function, the maximum and minimum points occur at the midpoints of the graph. That is, if you draw a horizontal line through the highest or lowest point, it will intersect the x-axis at the midpoints. For a cosine function, the maximum and minimum points occur at the endpoints of the graph. That is, if you draw a horizontal line through the highest or lowest point, it will intersect the x-axis at the endpoints.
Step 4:: Check for phase shifts
If the graph is shifted horizontally (to the left or right), it may represent a sine or cosine function with a phase shift. In this case, you can either translate the function back to its original position (by adding or subtracting the phase shift value from the input) or identify the type of the graph based on its new position.
Step 5:: Conclusion
By following these steps, you can determine whether a graph represents a sine or cosine function. The key differences between the two are the position of the maximum and minimum points and the location of the endpoints and midpoints.
Final Answer
A graph represents a sine function if its maximum and minimum points occur at the midpoints of the graph. A graph represents a cosine function if its maximum and minimum points occur at the endpoints of the graph.
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