Experience First:

Today students continue their exploration of functions by analyzing graphs representing contextual scenarios. Note that a huge theme of the course is covariation, which focuses on how variables change with respect to each other, rather than just looking at individual input-output pairs. Two covarying relationships are discussed in this lesson: price and demand (page 1) and price and revenue (page 2). 

 

On page 1, students compare two supermarket purchases (baby formula and soft drinks) and consider how the demand for each product changes as the price changes. Students should notice from the slopes of both graphs that customers who buy soft drinks are more likely to give up this purchase when the price increases, but people who buy baby formula, while decreasing their consumption slightly, can not just give up buying formula for their children. There are many ties to economics here. Consider consulting your school’s econ teacher to make cross-curricular connections to price elasticity.

 

The second page of the activity has students thinking about graph behavior, specifically intervals of increasing/decreasing, maxima, and x-intercepts. We want students to connect the idea of optimal with an extrema as this is an important tie in to AP Calculus once they learn about optimization. The idea of not yet having reached optimal conditions until we can no longer squeeze out any more revenue is an important one. Informally, we want students to arrive at the idea that they will reach optimal conditions once revenue has stopped increasing and is just about to start decreasing. Note that students may be using some estimation to find the maximum revenue and this is okay.

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Monitoring Questions:

• Why are these graphs downward sloping?

• How does an increase in price affect the number of formula customers? How does an increase in price affect the number of soft drink customers?

• Do you think the relationship between price and revenue is also linear? Why or why not?

• Can you think of another price that a business analyst would say is “undercharging”?

• Why is charging $50 still not optimal?

• Can you think of another price that a business analyst would say is “overcharging”?

• How can we use the graph to figure out when the company makes no money?

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Formalize Later:

​When debriefing the lesson, be sure to use precise language that clearly communicates if we’re talking about inputs, outputs, or the function itself. Avoid using vague words like “it”, “the function”, or “the graph”. Using the variables of the activity context is helpful, rather than more generic phrases like “the y’s are going up”. Note that we define what it means for a function f to be increasing on a given interval of input values if as the input values increase, the output values increase as well. This is subtly different than defining what it means for the outputs to be increasing.

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It is also important to emphasize to students that intervals of increasing/decreasing are talking about intervals of the domain. Students should not be giving the y-values from lowest to highest. Conversely, when talking about a maximum or minimum value, the y-value is needed. Discourage students from giving the location of the extreme value (the x-value) or even giving an ordered pair (we consider this students “covering their bases” and it’s hard to tell whether they truly understand that an extreme value is an output).

 

A final note about intervals of increasing/decreasing: Technically at a point with a horizontal tangent line (at a max/min), the function is neither increasing nor decreasing. For this reason, we tend to use open intervals when discussing intervals of increasing/decreasing. However, that does not mean that the x-value of the horizontal tangent line needs to be excluded from an interval on which the function is otherwise increasing, since it is still true that f(a)≤f(b) whenever a≤b. For example, it is okay to say that the cubic function is increasing on its entire domain (-∞,∞), even though at x=0, the function is neither increasing nor decreasing. For the same reason, it is okay to include endpoints on intervals of increasing/decreasing.

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