5.4 Investigate Transformations of Sine Curves

Prerequisite Skills

1. Write an equation for the quadratic function that results from each transformation.

a) The graph of y = x is translated 2 units downward and 1 unit to the right. 

b) The graph of y = x2 is reflected in the x-axis and then compressed vertically by a factor of  ⅓

c) The graph of y = x2 is stretched vertically by a factor of 4 and then translated 2 units to left and 1 unit up.

2. Write the meaning of each statement.

3. Sketch a graph of each function. Then, write the domain and range for each function.

a) y = 3(x - 1)2

b) y = -2x2 + 3

1. Sketch a graph of each function for 0° _<x_< 360°. Determine the period, amplitude, domain, and range.

a) y = 2 sin x     b) y = -¼ sin x

2. Compare the graphs of each pair of functions for 0° _<x_< 360°. Determine the period, amplitude, domain, range, phase shift, and equation of the horizontal axis.

a) y = sin x - 2.5 and y = sin x + 2.5

b) y = sin(x + 30°) and y = sin(x - 90%)

3. Draw a sketch of y = -5 sin x for one period.

a) Locate all the points where y = 5 and give the values of x.

b) Locate all the points where y = -5 and give the values of x.

4. Draw a sketch of y = sin(x + 180°) for one period.

a) Locate all the points where y = 1 and give the values of x.

b) Locate all the points where y = -1 and give the values of x.

5. Write an equation for each sine function

6. Graph one cycle of each function. Label the x-intercepts, the maximum points, the minimum points, and the equation of the horizontal axis. Write the domain and range of the cycle.

a) f(x) = 2 sin x-1

b) f(x) = -3 sin x

c) S(x) = sin(x – 60°)

7. Without graphing, consider the function y = sin x + 55. Identify the period, amplitude, phase shift, domain, range, and the equation of the horizontal axis.

8. For each of the functions, find the coordinates of the maximum and minimum points.

a) y = 2 sin x-5

b) y = -sin(x + 60°)

9. Write an equation for each sine function. Indicate the intervals in which the function is increasing and decreasing over one period. 

a) amplitude = 5, horizontal axis along y = 2 

b) amplitude = 1, horizontal axis along the x-axis, phase shift of 45° to the right

10. A windmill with 3-m long blades has a centre 6 m above the ground. One of the blades starts out parallel to the ground.

a) Sketch a graph that represents the height of the tip of the blade relative to the angle it forms with the horizontal as it rotates once fully. The blades rotate counterclockwise,

b) Determine an equation that represents the height of the tip of the blade with respect to the ground.

11. The graph of a sine curve passes through the points (0°, 12), (90°, 10), (180°. 8), and (270°, 10). Determine an equation that represents this function.

12. A sine function of the form 

y = sin(x - 2) has the same graph as y = -sin x. Find all values of d.