# Lesson 13

Amplitude and Midline

• Let's transform the graphs of trigonometric functions

### 13.1: Comparing Parabolas

Match each equation to its graph.

1. $$y = x^2$$
2. $$y = 3x^2$$
3. $$y = 3(x-1)^2$$
4. $$y = 3x^2-1$$
5. $$y = x^2-1$$

Be prepared to explain how you know which graph belongs with each equation.

### 13.2: Blowing in the Wind

Suppose a windmill has a radius of 1 meter and the center of the windmill is $$(0,0)$$ on a coordinate grid.

1. Write a function describing the relationship between the height $$h$$ of $$W$$ and the angle of rotation $$\theta$$. Explain your reasoning.
2. Describe how your function and its graph would change if:
1. the windmill blade has length 3 meters.
2. The windmill blade has length 0.5 meter.
3. Test your predictions using graphing technology.

### 13.3: Up, Up, and Away

1. A windmill has radius 1 meter and its center is 8 meters off the ground. The point $$W$$ starts at the tip of a blade in the position farthest to the right and rotates counterclockwise. Write a function describing the relationship between the height $$h$$ of $$W$$, in meters, and the angle $$\theta$$ of rotation.
2. Graph your function using technology. How does it compare to the graph where the center of windmill is at $$(0,0)$$?
3. What would the graph look like if the center of the windmill were 11 meters off the ground? Explain how you know.

Here is the graph of a different function describing the relationship between the height $$y$$, in feet, of the tip of a blade and the angle of rotation $$\theta$$ made by the blade. Describe the windmill.

### Summary

Suppose a bike wheel has radius 1 foot and we want to determine the height of a point $$P$$ on the wheel as it spins in a counterclockwise direction. The height $$h$$ in feet of the point $$P$$ can be modeled by the equation $$h = \sin(\theta) + 1$$ where $$\theta$$ is the angle of rotation of the wheel. As the wheel spins in a counterclockwise direction, the point first reaches a maximum height of 2 feet when it is at the top of the wheel, and then a minimum height of 0 feet when it is at the bottom.

The graph of the height of $$P$$ looks just like the graph of the sine function but it has been raised by 1 unit:

The horizontal line $$h=1$$, shown here as a dashed line, is called the midline of the graph.

What if the wheel had a radius of 11 inches instead? How would that affect the height $$h$$, in inches, of point $$P$$ over time? This wheel can also be modeled by a sine function, $$h = 11\sin(\theta)+11$$, where $$\theta$$ is the angle of rotation of the wheel. The graph of this function has the same wavelike shape as the sine function but its midline is at $$h=11$$ and its amplitude is different:

The amplitude of the function is the length from the midline to the maximum value, shown here with a dashed line, or, since they are the same, the length from the minimum value to the midline. For the graph of , the midline value is 11 and the maximum is 22. This means the amplitude is 11 since $$22-11=11$$.

### Glossary Entries

• amplitude

The maximum distance of the values of a periodic function above or below the midline.

• midline

The value halfway between the maximum and minimum values of a period function. Also the horizontal line whose $$y$$-coordinate is that value.