Lesson 2 “Sine” Language Solidify Understanding

Ready

For each graph, identify the domain and range. Then write the interval(s) where is positive, , and the interval(s) where is negative, .

1.

a coordinate plane with 4 line segments graphed and connected together. All 4 lines are potions of linear functions. x–5–5–5555y555000

a.

Domain:

b.

Range:

c.

Positive:

d.

Negative:

2.

a coordinate plane with 3 line segments graphed and connected together. 2 of those lines are linear functions and the other line is a portion of a quadratic function. x–4–4–4–2–2–2222444666888y–2–2–2222000

a.

Domain:

b.

Range:

c.

Positive:

d.

Negative:

3.

a curved line graphed between the points (-2 pi, 0) and (3 pi over 2, negative pi) representing the function f of x = sine of x x–2π–2π–2π–3π / 2–3π / 2–3π / 2–π–π–π–π / 2–π / 2–π / 2π / 2π / 2π / 2πππ3π / 23π / 23π / 2y–π–π–ππππ000

a.

Domain:

b.

Range:

c.

Positive:

d.

Negative:

4.

a curved line graphed between the points (-2 pi, 3) and (pi, negative 3) representing the function f of x = cosine of x x–2π–2π–2π–3π / 2–3π / 2–3π / 2–π–π–π–π / 2–π / 2–π / 2π / 2π / 2π / 2πππ3π / 23π / 23π / 2y–2–2–2222000

a.

Domain:

b.

Range:

c.

Positive:

d.

Negative:

Write the piecewise equation that describes each graph.

5.

A coordinate plane with a segment of a linear function with the slope of negative 2 between points (0,1) and (3,-5). A second segment of a different linear function with a slope of 1 over 4 between points of (4,-1) and (8,0) is also graphed. x222444666888y–4–4–4–2–2–2222000

6.

A coordinate plane with a segment of a linear function with the slope of 3 over 2 between points (-2,2) and (2,8). A second segment of a different linear function with a slope of negative 1 over 3 between points of (3,1) and (9,-1) is also graphed. x555101010y555000

Set

Recall the following facts from the lesson done in class:

  • The Ferris wheel has a radius of .

  • The center of the Ferris wheel is above the ground.

Due to a safety concern, the management of the amusement park decides to slow the rotation of the Ferris wheel from for a full rotation to for a full rotation.

A circle within a circle representing a ferris wheel. The ferris wheel is divided into 10 equal parts with corresponding points labeled with the letters A through J. A right triangle is drawn between the center point, point a, and point b. The inner angle is 36 degrees, the hypotenuse is 25 feet, and the height is 25 times sine of 36 degrees.

7.

Calculate how high a rider will now be after passing position on the diagram.

8.

a.

Calculate the height of a rider at each of the following times , where represents the number of seconds since the rider passed position on the diagram.

Elapsed time since passing position

Calculations

Height of the rider

(in feet)

b.

Plot the position you calculated on the diagram. Connect the center of the circle to the point you plotted. Then draw a vertical line from the plotted point on the Ferris wheel to the line segment in the diagram. Each time you should get a right triangle similar to the one in the figure.

A circle within a circle representing a ferris wheel. The ferris wheel is divided into 10 equal parts with corresponding points labeled with the letters A through J. A right triangle is drawn between the center point, point a, and point b. The inner angle is 36 degrees, the hypotenuse is 25 feet, and the height is 25 times sine of 36 degrees.

9.

How did the position of the triangles you drew change between and ?

10.

How did the triangles you drew change between , , and ?

11.

How did the triangles you drew change between and ?

12.

Describe a relationship between the angle the hypotenuse makes with the -axis in each triangle and the angle of rotation. Use the diagram to help you think about the question. (The dotted arc shows the angle of rotation.)

A radian circle graphed on a coordinate plane where the left side of the x axis is labeled pi and right side is labeled 0. There are 4 right triangles graphed in each quarter of the radian circle.

Go

Perform the indicated operation. Divide out all common factors in your answers.

(Assume no denominator is equal to .)

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