Thursday’s Lesson
Warming Up to Quadratics with the Parabola Web

Learning a new technology while grappling with new concepts or skills can be confusing. Getting students comfortable with the technology first makes it easier for them to focus on the learning, rather than on the tool. A common strategy is to simply allow students to play with the new tool for a few minutes before starting an activity. A different approach is to take students through a structured exploration, or “software warm-up.”

Figure 1.The RTT Quadratic Grapher allows students to explore the relationship between the symbolic and graphic representations of quadratic functions.

The Ready to Teach (RTT) Quadratic Grapher (Figure 1) allows students to explore the relationship between symbolic and graphic representations of quadratic functions. The Software Warm-Up introduces students   to the Quadratic Grapher and its features. The warm-up is designed to get students thinking about the ways that changes in the symbolic representation of a quadratic function mirror the changes in its graph (a parabola) and prepare them for an activity we call the Parabola Web.

Software Warm-Up: The RTT Quadratic Grapher

To access the Quadratic Grapher, go to: http://rtt.pbs.org/rtt/interactives.cfm

Note: You need Java 1.3.1 or higher to run the interactives. Check your Java by using the RTT wizard (http://rtt.concord.org/Wizard), or install or update Java Software for the Desktop (http://java.com).

The first part of the warm-up familiarizes students with the features of the Quadratic Grapher and emphasizes making observations.

Warm-Up 1: Try It and See

1. Quadratic Equation. Locate the symbolic expression at the bottom of the screen. This is the symbolic representation of a quadratic function. What function does the program show when it opens?
2. The Slider. Move the slider (the inverted triangle at the bottom of the graph). What does it do? Use the slider to find the vertex of the starting parabola and a few points on it.
3. New Function. Click the “New Function” button and you’ll see a new parabola. Where did the first one go? In the table on the right-hand side, can you tell which expression represents which parabola? What happens to a parabola when you click on its graph? Or when you click on its expression in the table?
4. The Equation. Look at the symbolic expression at the bottom of the screen. Click the box in front of x² (the coefficient) to select it. Notice what happens when you click the up and down arrows a few times. Which parabola changes? How?
   Choose a parabola you want to change (by clicking the corresponding color in the table at right). Click the box in front of the x term to select it. Notice how the  parabola and the symbolic expression change when you click the up and down arrows a few times. Do the same with the last box (the constant term). What do you notice now?
5. Experiment. Use the other buttons. Each time you try  a new button, see what changes you notice – in the parabola, in the symbolic expression (at the bottom), and in the table (at right). Watch what the slider does, too. (There is a lot to notice – look around!)

Warm-Up 2: Make It and Check It

The second part of the software warm-up lets students glimpse what the Quadratic Grapher can do. The warm-up hints at software features and tasks that students will need for the Parabola Web activity, but avoids revealing the  “Aha!” moments that the activity is designed to generate.

1. Make three parabolas whose vertices are all on the   y-axis. What do their symbolic expressions have in common?
2. Make two parabolas that are mirror images of each other. What do you notice about their symbolic  expressions?
3. Graph the parabola y = 2x² + 8x + 4. What are the coordinates of its vertex? (Hint: There are two ways check if you are correct. What are they?)
4. Graph the parabola y = 2(x – 4)² – 8. Then graph another parabola that crosses the x-axis at the same points as that parabola does. Compare the vertices of these two parabolas.
5. Make a parabola with its vertex at (12, 0). Use the “Change x Scale” button. Use the “Duplicate Function” button to make another parabola that has a different vertex, but crosses the original parabola. Then find the intersection point(s). (Hint: How can the slider help you?)

The Parabola Web Activity

In Wednesday's Lesson: The Starburst Activity (page 10), you transformed linear functions. The Parabola Web activity is based on the same idea – transforming a family of functions on the coordinate plane – but this time it’s quadratic functions. Now that students have done the warm-up activity, try out the real thing.

Challenge A: Comparing Graphs and Symbols 

Figure 2. This family of four quadratic functions forms a “parabola web.”  The function rules that describe each parabola have the form y = ax², where a is a constant.

Figure 3.The Quadratic Grapher challenges students to think about the graph of a function described by a symbolic expression in the form 0x² + bx + c, highlighting the difference between linear and quadratic functions.

Look at the Parabola Web in Figure 2. All the parabolas pass through (0, 0). The function rules that describe each parabola have the form y = ax², where a is a constant.

1. Use the Quadratic Grapher to help you write the function rule for each of the parabolas in the web. Now, look closely at the rules, and in the following problems, try to figure out how the coefficient of x² (the letter a) is related to the graph of the corresponding parabola.
2. Write a function rule to describe a parabola with the same vertex, whose other points lie somewhere between parabolas A and B.
3. Write a function rule to describe a parabola that looks like parabola B reflected about the x-axis.
4. Write a function rule to describe a parabola that looks like parabola D reflected about the x-axis.
5. How does the number you choose for the coefficient of x² (the letter a) change the shape of a parabola? Write your conclusions and explain your reasoning.

Challenge B: Moving the Parabola Web

The parabolas in Challenge A form a web with vertices at (0, 0). Try moving them.

1. Show how you would move the vertex of each parabola from (0, 0) to (0, 1) by changing the function rule. Moving a graph in this way is called a vertical translation.
2. Show how you would move the vertex of each parabola from (0, 0) to (2, 0). Moving a graph in this way is called a horizontal translation.
3. Show how you would move the vertex of each parabola from (0, 0) to (-1, 2). Demonstrate and explain your reasoning using one of the parabolas. How does the function rule for the parabola change?

From Seeing to Understanding

The Quadratic Grapher is a powerful tool that helps   students see what is difficult to show on paper. For example, a fundamental difference between linear and quadratic functions is the x² term – the defining attribute of quadratic functions. But once students begin studying quadratics, they may never revisit a symbolic expression without a coefficient of x². On the Quadratic Grapher, watch what happens in the polynomial form of the equation when you vary only the coefficient of x². What does the graph look like when the coefficient of x² is 0? A symbolic expression in the form 0x² + bx + c does not generate the graph of a parabola, but a line – it’s a linear function (bx + c is analogous to mx + b) (Figure 3)!

Observing the relationship between the graphic and symbolic representations can increase the students’ familiarity with the characteristics of quadratic functions and improve their general understanding of functions and the relationships that functions can describe.

In the case of the Parabola Web, feeling comfortable with using the RTT Quadratic Grapher is key to students exploring the characteristics of quadratic functions from multiple perspectives.