Translations and stretches of trigonometric functions Translations We

Translations and stretches of trigonometric functions

Translations We have seen before that the following are translations of a function f (x) + d translates f (x) vertically a distance of d units upward f (x) – d translates f (x) vertically a distance of d units downward f (x + c) translates f (x) horizontally c units to the left f (x – c ) translates f (x) horizontally c units to the right. These functions 1 y y = sin x have a period of 0 -2 p -p 2 p x These functions have an amplitude of 1 -1 1 -2 p -p 0 -1 y y = cos x p 2 p x

Translations The graphs of these function can be transformed in the same way that you transformed the graphs of other functions If f(x) = sin (x) f (x) + d translates f (x) vertically a distance of d units upward Draw f(x) = sin (x) +2 y 3 f(x) = sin (x) + 2 2 1 -2 p -p 0 -1 -2 f(x) = sin x p x 2 p

Translations The graphs of these function can be transformed in the same way that you transformed the graphs of other functions If f(x) = sin (x) f (x) – d translates f (x) vertically a distance of d units downward Draw f(x) = sin (x) – 1 y 3 2 1 -2 p -p 0 f(x) = sin x p x 2 p -1 -2 f(x) = sin (x) – 1

Translations The graphs of these function can be transformed in the same way that you transformed the graphs of other functions If f(x) = sin (x) f (x + c) translates f (x) horizontally c units to the left y 3 2 1 -2 p -p 0 -1 -2 f(x) = sin x p x 2 p

Translations The graphs of these function can be transformed in the same way that you transformed the graphs of other functions If f(x) = sin (x) f (x – c) translates f (x) horizontally c units to the right y 3 2 1 -2 p -p 0 -1 -2 f(x) = sin x p x 2 p

Translations The graphs of these function can be transformed in the same way that you transformed the graphs of other functions If f(x) = cos (x) f (x) + d translates f (x) vertically a distance of d units upward Draw f(x) = cos (x) +2 y 3 f(x) = cos (x) + 2 2 1 -2 p -p 0 -1 -2 f(x) = cos x p x 2 p

Translations The graphs of these function can be transformed in the same way that you transformed the graphs of other functions If f(x) = cos (x) f (x) – d translates f (x) vertically a distance of d units downward Draw f(x) = cos (x) – 1 y 3 2 1 -2 p -p 0 f(x) = cos x p x 2 p -1 -2 f(x) = cos (x) – 1

Translations The graphs of these function can be transformed in the same way that you transformed the graphs of other functions If f(x) = cos (x) f (x + c) translates f (x) horizontally c units to the left y 3 2 1 -2 p -p 0 -1 -2 f(x) = cos x p x 2 p

Translations The graphs of these function can be transformed in the same way that you transformed the graphs of other functions If f(x) = cos (x) f (x – c) translates f (x) horizontally c units to the right y 3 2 1 -2 p -p 0 -1 -2 f(x) = cos x p x 2 p

Translations We can combine horizontal and vertical translations by looking at equations in the form f(x) = sin (x + c) + d f(x) = cos (x + c) + d f(x) = tan (x + c) + d Sketch f (x) = sin x In the same set of axis sketch the graph (a) f(x) = sin (x) + 1 y The function is shifted 1 unit upward. 3 2 f(x) = sin x 1 x -2 p -p 0 -1 -2 p 2 p

Translations Write a sine equation for the following function The function f(x) = sin (x) is shifted 3 units down. So, the equation is f(x) = sin (x) – 3 y 3 -2 p -p 0 -1 -2 -3 -4 p 2 p x

Translations Write a cosine equation for the following function y 3 2 1 x -2 p -p 0 -1 -2 p 2 p

Vertical stretches We have seen before that when a function undergoes a stretch af (x) stretches f (x) vertically with scale factor a The function f(x) = asin x is the vertical stretch of f(x) = sin x The function f(x) = acos x is the vertical stretch of f(x) = cos x If a > 1, the function will appear to stretch away from the x -axis. If 0 < a < 1, the function will appear to compress closer to the x-axis. If a < 0, the function will also be reflected over the x-axis. With the vertical stretch, the amplitude of the sine or cosine function will change from 1 to |a|. The period of the function will not change.

Vertical stretches When the graph of a function undergoes a vertical stretch, every y-value in the original function is multiplied by the value of a. If f(x) = sin (x) Draw f(x) = 3 sin (x) The sine curve has been stretched vertically by a factor of 3 y 3 The maximum f(x) = 3 sin (x) values are at 2 y=3 f(x) = sin (x) The minimum 1 values are at y x = -3 -2 p -p p 0 2 p The amplitude -1 of the new function is 3 -2 The period is -3 2 p

Vertical stretches When the graph of a function undergoes a vertical stretch, every y-value in the original function is multiplied by the value of a. If f(x) = sin (x) Draw f(x) = 0. 5 sin (x) The sine curve has been stretched vertically by a factor of 0. 5 y 3 The maximum values are at 2 y = 0. 5 f(x) = sin (x) The minimum 1 f(x) = 0. 5 sin (x) values are at y x = -0. 5 -2 p -p p 0 2 p The amplitude -1 of the new function is 0. 5 -2 The period is -3 2 p

Vertical stretches When the graph of a function undergoes a vertical stretch, every y-value in the original function is multiplied by the value of a. If f(x) = sin (x) Draw f(x) = -2 sin (x) The sine curve has been stretched vertically by a factor of -2 y 3 The maximum f(x) = -2 sin (x) values are at 2 y=2 f(x) = sin (x) The minimum 1 values are at y x = -2 -2 p -p p 0 2 p The amplitude -1 of the new function is 2 -2 The period is -3 2 p

Horizontal stretches We have seen before that when a function undergoes a stretch The functions f(x) = sin (bx), f(x) = cos (bx), f(x) = tan (bx) represent horizontal stretches of sine, cosine and tangent functions. We can also say that every x-value in the original function is divided by b.

Horizontal stretches Multiplying or dividing the x-values by a number in this way changes the period of a trigonometric function. If b > 1, the period will be shorter, and the function will appear to compress toward the y-axis. If 0 < b < 1, the period will be longer, and the function will appear to stretch away from the y-axis. If a < 0, the function will also be reflected over the y-axis.

Vertical stretches If f(x) = sin (x) Draw f(x) = sin (2 x) The sine curve has been stretched horizontally by a factor of 0. 5 The maximum values are at y=1 The minimum values are at y = -1 The amplitude has not changed y 3 2 f(x) = sin (2 x) f(x) = sin (x) 1 x -2 p -p 0 -1 -2 -3 p 2 p

Vertical stretches If f(x) = cos (x) Draw f(x) = 2 cos (3 x) and vertically by a factor of 2 The maximum values are at y=2 The minimum values are at y = -2 The amplitude has changed, it is 2 y 3 2 f(x) = 2 cos (3 x) f(x) = cos (x) 1 x -2 p -p 0 -1 -2 -3 p 2 p

Vertical stretches If f(x) = tan (x) Draw f(x) = tan (2 x) The tangent curve has been stretched horizontally by a factor of 0. 5 f(x) = tan (x) y f(x) = tan (2 x) 3 2 1 x -2 p -p 0 -1 -2 -3 p 2 p