Lesson 14

This lesson continues to look at how adding a constant or multiplying by a constant can transform the graphs of \(y = \cos(\theta)\) and \(y = \sin(\theta)\) to match a situation. Students study the equations of functions representing a situation and interpret what the different parts of the equation tell us. Where in the previous lesson students focused on the effect of a single transformation, here they consider changes to both the amplitude and midline in the same function. Students then consider horizontal translations of cosine and sine. In particular, they examine the graph of \(y = \cos(\theta+a)\) for a specific value of \(a\) determined by a context. In the next lesson, students will work with these three types of transformations at once.

Students have seen the effects of translations and scale factors on graphs previously when they studied general transformations of functions in an earlier unit and when they studied quadratic equations and their graphs in a previous course. A quadratic equation like \(y = 3(x+\pi)^2 - 5\) is very similar to \(y = 3\cos(\theta+\pi) - 5\). In both cases 3 vertically stretches the graph (increasing the "steepness" of the parabola or the amplitude of the cosine curve), -5 gives a vertical translation, and \(\pi\) produces a horizontal translation. Students also encounter the graph of sine with a negative coefficient in this lesson, making sense of it in the context of a windmill blade rotating clockwise.

Students reason abstractly and quantitatively (MP2) when they study the function \(y = \cos\left(\theta + \frac{\pi}{6}\right)\). Through the table of values and the context, they can identify that this is a cosine graph which has been translated horizontally.