Lesson 14

The goal of this warm-up is to recall how a quadratic equation changes when its graph is translated horizontally. This will prepare students for examining this same translation when it is applied to trigonometric functions in this lesson.

Invite students to share how they made their matches. Make sure students recall:

• The three graphs have the same shape.
• They are horizontal translations of one another.
• Adding a positive number to the input translates the graph to the left and adding a negative number to the input translates the graph to the right. Display the three graphs together for all to see to highlight this.

In the previous lesson, students learned the definition of amplitude and midline by examining sine graphs produced to model the vertical position of a point on a rotating windmill. This activity keeps the windmill context and asks students to interpret both cosine and sine equations. They will need to focus on three different aspects of the equations:

• The amplitude of the sine graph (length of the windmill blade)
• The midline of the sine graph (height of the center of the windmill blades)
• The direction the windmill spins (clockwise or counterclockwise).

Arrange students in groups of 2. Display an image of the blades of a windmill for all to see, such as this one from a previous lesson:

Tell students that these blades extend 4 meters from the center and that the center of the windmill is 20 meters off the ground. The windmill rotates counterclockwise.

Ask students to write an equation to represent the height in meters of the point \(W\), relative to the ground, as a function of the angle \(\theta\) of rotation (\(h = 4\sin(\theta) + 20\)). Select 2–3 students to share how they wrote their equation. Tell them that in this activity they are going to work the opposite direction. That is, they are going to start with an equation giving the vertical position of a point \(W\) at the tip of a windmill blade and use it to describe the windmill.

If students struggle with the negative amplitude in the final question, encourage them to substitute in a few angle values \(\theta\) and plot them.

Display the 3 equations for all to see. Begin the discussion by asking, “What is the midline and amplitude of each function and what do these values mean in this context?” (The midline is the height of windmill blades’ center and the amplitude is the length of the blades.) Record student responses next to the equations.

Ask students how they interpreted the sign in front of the sine on the last equation (it means that the windmill is spinning in the clockwise direction as the angle \(\theta\) increases). To help students visualize the impact of the negative sign in front of the expression, note that while \(\sin(\theta)\) increases from 0 to 1 as \(\theta\) increases from 0 to \(\frac{\pi}{2}\), \(\text-\sin(\theta)\) decreases from 0 to -1 as \(\theta\) increases from 0 to \(\frac{\pi}{2}\). In terms of the windmill, this would mean that instead of going toward the highest spot on its circle of rotation, the point \(W\) begins by going to the lowest spot on its circle of rotation. This means that the windmill is rotating in a clockwise direction instead of a counterclockwise direction.

In this activity students investigate how adding a constant to the variable inside a trigonometric function influences the graph. They have done this work in an earlier unit, for polynomials and other non-periodic functions, and the underlying structure is the same for periodic functions. The particular focus of this activity is interpreting a horizontal translation for a periodic function representing circular motion.

There are several strategies students may use for graphing the functions. Monitor for students who

• evaluate the functions using the classroom unit circle display or technology and plot individual values
• identify expressions like \(\cos\left(\frac{\pi}{6} \right)\) and \(\sin\left(\frac{\pi}{6}\right)\) for the vertical intercepts of the graphs
• connect the different starting position to a change of input for the sine and cosine functions

Identifying specific values is helpful for plotting points and producing the graphs. The expressions using cosine and sine help to identify that the graphs are horizontal translations and to write expressions representing these functions. Highlight the importance of both strategies during the whole-class discussion, if they come up.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

For students who struggle to create a plot, or to identify the relationship between the graphs and the equations, consider asking them to complete a table with columns for \(\theta\), \(h\), and \(v\), where $\theta is from 0 to \(\frac{4\pi}{3}\) in increments of \(\frac{\pi}{3}\).

Begin the discussion by inviting students to share strategies they used for graphing the functions. Some strategies may include:

• Create a table of values and plot them.
• Use technology to calculate values and then plot them.
• Use transformations on a function to determine the new function and graph it.

Ask students how the new graphs are similar to and different from the graphs of \(h = \cos(\theta)\) and \(v = \sin(\theta)\). Students should understand that while they have the same shape and also the same period (\(2\pi\)) and amplitude (1), they are different because they don’t “start” at the same place. The \(v\) graph starts at \(\sin\left(\frac{\pi}{6}\right)\) rather than at \(\sin(0)\) and the \(h\) graph starts at \(\cos\left(\frac{\pi}{6}\right)\) rather than at \(\cos(0)\). Connect this back to the graphs of \(y = x^2\) and \(y = (x+3)^2\) from the warm up: both cases represent a horizontal translation to the left (by \(\frac{\pi}{6}\) for the trigonometric functions and by 3 for the parabola).

Display the graphs of \(v=\sin(\theta)\) and \(v=\sin\left(\theta+\frac{\pi}{6}\right)\) (the equation for the vertical position of the point \(P\) in this activity) for all to see on the same axes to help visualize the horizontal translation.

If equations for the transformations are not brought by students, invite them to explain how changing the starting position of the blade influences the graph. They should identify that the graph for this problem is a horizontal translation of the graph of \(v = \sin(\theta)\) and that the translation is to the left by \(\frac{\pi}{6}\). Tell students that an equation defining the vertical position of the point \(P\) is \(v=\sin\left(\theta+\frac{\pi}{6}\right)\) and that the \(\frac{\pi}{6}\) comes from the point \(P\) starting at the \(\frac{\pi}{6}\) position.

Time permitting, ask students what the graphs and equations for the vertical position would look like if the point \(P\) started at \((0,1)\) (They would be translated to the left by \(\frac{\pi}{2}\) so the equation would be \(v=\sin\left(\theta+\frac{\pi}{2}\right)\).) What if \(P\) started at \((0,\text-1)\)? (They would be translated to the left by \(\frac{3\pi}{2}\) or to the right by \(\frac{\pi}{2}\) so the equation would be \(v=\sin\left(\theta+\frac{3\pi}{2}\right)\) or \(v=\sin\left(\theta-\frac{\pi}{2}\right)\), respectively.)