# Lesson 13

Amplitude and Midline

- Let's transform the graphs of trigonometric functions

### 13.1: Comparing Parabolas

Match each equation to its graph.

- \(y = x^2\)
- \(y = 3x^2\)
- \(y = 3(x-1)^2\)
- \(y = 3x^2-1\)
- \(y = x^2-1\)

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.

- Write a function describing the relationship between the height \(h\) of \(W\) and the angle of rotation \(\theta\). Explain your reasoning.
- Describe how your function and its graph would change if:
- the windmill blade has length 3 meters.
- The windmill blade has length 0.5 meter.

- Test your predictions using graphing technology.

### 13.3: Up, Up, and Away

- 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.
- Graph your function using technology. How does it compare to the graph where the center of windmill is at \((0,0)\)?
- 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.