Exploring Quadratic Models

Students explore the Fuel Efficiency Starter File, and create scatter plots to search for relationships between columns. They share and discuss their findings with the class, discovering the limitations of linear models.

These questions are intended to spark student interest in fuel efficiency, which students will explore in this lesson.

In this lesson, we’ll learn more about the relationship between fuel efficiency and the speed a car is travelling. First, we’ll explore a dataset and see if this relationship is linear.

NOTE: While dataset is based on real data obtained from the Transportation Energy Book, it has been augmented with fictional data in order to provide a sufficiently-rich dataset for student modeling. You can find out more about this augmentation in the README tab of the dataset.

There’s definitely a relationship here. But we know that the best-possible linear model isn’t very good!

…​but all of them are so far away from so many points, all three models will make very poor predictions for a large subset of x-values.

Even though some cars are more efficient than others, fuel efficiency generally increases from 15mph to around 40-55mph. After that, fuel efficiency decreases!

Review student answers to confirm that students see a nonlinear relationship in the dataset that is fairly strong. Students should generally agree that the relationship is better fit by a curve, with efficiency peaking roughly at 45mpg. Make sure students have agreed on this vertex, as its location is important for the next activity!

Have students share their curves, encouraging them to identify where they "peak" and where they cross the x-axis. These locations will be important anchor concepts on which to build in the next section. (One option for facilitating sharing is to project the scatter plot on a whiteboard and have students come up and draw all of their curves on it).

Students are introduced to quadratic functions. They become familiar with scenarios involving quadratics and terminology to describe parabolas. They also learn how to identify quadratic sequences, and then explore each of the coefficients in Vertex Form.

Linear models are great for fitting simple relationships. But the real world isn’t so simple, and linear models are often just too primitive to handle that complexity.

Relationships like the one in the fuel efficiency scatter plot are best described by a curve. Today, we will learn about one kind of curved model called a quadratic model. Graphs of quadratic relationships are often described as "u-shaped" or "looking like an arch". More formally, mathematicians and data scientists call these kinds of curves parabolas.

We observed what appeared to be a quadratic relationship between speed and fuel efficiency, where fuel-efficiency increased up to a certain speed, then decreased again.

There are lots of relationships that change direction like this! For example, when an athlete is young, they improve as they get stronger and more skilled. But as they age, they begin to lose their speed and strength. In some relationships, the curve goes the other way - decreasing, bottoming out, and then rising again. For example, a bird might swoop down to the water to catch a fish, and then fly back up to carry it away.

Although the examples we just discussed include the characteristic rising and falling or falling and rising of the parabola, we might encounter datasets that include only the rising part of the parabola or only the falling part.

But what if we only have a list of x-y pairs showing only one part of the parabola?
How can we identify quadratic relationships from a sequence of numbers?

Remember that linear functions grow by fixed intervals, so the rate of change is constant. In the table shown here, each time the x-value increased by 1, we saw that the y-value increased by 2. This is true for any set of equal-sized intervals: a line needs to slope up or down at a constant rate in order to be a straight line!

If the rate of growth is constant, the relationship is linear.

Quadratic functions grow by intervals that increase by fixed amounts! In the table to the right, the blue arrows show a differently-sized jump between identical intervals time, meaning the function is definitely not linear! However, if we take look at the difference between those differences(shown in red), we’re back to constant growth!

If the "growth of the growth" is constant, the relationship is quadratic.

Debrief with students and allow them to share the different strategies that they used. Note: When looking at real-world datasets that can be modeled by linear or quadratic functions, these patterns will not be evident in the tables, because the points won’t fit the function perfectly!

When we graph these points on a plane, they draw our parabola. As we work with parabolas - and eventually fit them to our Fuel Efficiency dataset - we need to know how to talk about and describe them. Let’s define the parts of a parabola together.

Invite students to share what they know or can infer about a parabola’s parts. If needed, provide the definitions on the table below, which refer to specific points on a parabola.

There are two more specific terms that we can use to describe a parabola’s vertex:

For parabolas representing y as a quadratic function x the axis of symmetry is an imaginary vertical line through the vertex that splits the curve into two congruent parts that mirror each other.

Students work with the vertex form to fit a quadratic model for the Fuel Efficiency dataset. They extend the model-fitting techniques from the Exploring Linear Models lesson into quadratic relationships.

We just examined a series of quadratic and linear functions that looked a lot like the ones you might find in an Algebra 2 textbook: clean and predictable. Real-world data, however, is messy! Let’s return to our Fuel Efficiency Starter File to dig into that messiness.

The existence of a "dip" like this is normal in real data, but it doesn’t mean that the overall shape of this relationship isn’t quadratic. There’s no such thing as a perfect model!

Let’s try to build the best possible quadratic model we can for our Fuel Efficiency dataset.

One form of a quadratic model looks like this: 𝑓(𝑥) = 𝑎(𝑥 − ℎ)² + 𝑘

What do each of these coefficients mean for our model?

We know that ℎ in the speed-v-fuel efficiency model is the x-coordinate of our vertex, and that the vertex of our efficiency curve is predicted at roughly 45𝑚𝑝ℎ. All we need to do is figure out 𝑎 and 𝑘!

Before we focus on that, we need to learn about a new Pyret function num-sqr.