How do you identify the vertical and horizontal translations of sine and cosine from a graph and an equation?

Question

Vedantu Content Team · Accepted Answer

Hint: For the function $$y = f\left( x \right)$$, the vertical translation is described by the equation $$y = f\left( x \right) + a$$ and horizontal translation is described by the equation $$y = f\left( {x - a} \right)$$. the vertical translation is described by the equation $$y = f\left( x \right) + a$$ such that if $$a$$ is greater than zero then the shift in the graph is upward from the original and if $$a$$ is less than zero then the shift in the graph is downward from the original. the horizontal translation is described by the equation $$y = f\left( {x - a} \right)$$ such that if $$a$$ is greater than zero then the shift in the graph is toward right from the original and if $$a$$ is less than zero then the shift in the graph is toward left from the original.Complete step by step solution:Consider the sine function as $$y = \sin x$$ and the cosine function as $$y = \cos x$$.It is known that for a function $$y = f\left( x \right)$$, the vertical translation is described by the equation $$y = f\left( x \right) + a$$ such that if $$a$$ is greater than zero then the shift in the graph is upward from the original and if $$a$$ is less than zero then the shift in the graph is downward from the original.\n    \n    \n    \n    \n  Therefore vertical translation of a sine function is written as $$y = \sin x + a$$ and the vertical translation of a cosine function is written as $$y = \cos x + a$$. Where $$a$$ cannot be zero.\n    \n    \n    \n    \n  Thus, if we are able to write the sin function in the form $$y = \sin x + a$$ and the cosine function in the form $$y = \cos x + a$$ where in both cases $$a \ne 0$$ then it represents vertical translation in both trigonometric functions from the original.In the graph, if sine functions or cosine functions wave is not symmetric to $$x$$-axis then there is a vertical shift in the graph from the original.It is known that for a function $$y = f\left( x \right)$$, the horizontal translation is described by the equation $$y = f\left( {x - a} \right)$$ such that if $$a$$ is greater than zero then the shift in the graph is toward right from the original and if $$a$$ is less than zero then the shift in the graph is toward left from the original.Therefore horizontal translation of a sine function is written as $$y = \sin \left( {x - a} \right)$$ and the horizontal translation of a cosine function is written as $$y = \cos \left( {x - a} \right)$$. Where $$a$$ cannot be zero.Thus, if we are able to write the sin function in the form $$y = \sin \left( {x - a} \right)$$ and the cosine function in the form $$y = \cos \left( {x - a} \right)$$ where in both cases $$a \ne 0$$ then it represents horizontal translation in both trigonometric functions from the original.In the graph, if sine functions or cosine functions wave is not symmetric to $$y$$-axis then there is a horizontal shift in the graph from the original.Note: For horizontal shift, the function is $$y = f\left( {x - a} \right)$$ and for vertical shift, the function is $$y = f\left( x \right) + a$$ if the original function is $$y = f\left( x \right)$$. The horizontal translation of a sine function is written as $$y = \sin \left( {x - a} \right)$$ and the horizontal translation of a cosine function is written as $$y = \cos \left( {x - a} \right)$$. Where $$a$$ cannot be zero. The vertical translation of a sine function is written as $$y = \sin x + a$$ and the vertical translation of a cosine function is written as $$y = \cos x + a$$. Where $$a$$ cannot be zero.