This activity is intended to supplement Algebra I, Chapter 10, Lesson 7.
Time required: 20 minutes
In this activity, students will manipulate parabolas in vertex form so that the curve matches a set of data points graphed as a scatter plot. This activity will serve to reinforce understanding of the vertex form for a parabola. In the extension, students will find an equation in vertex form and standard form that matches points from the Gateway Arch in St. Louis.
Topic: Quadratic Functions
Graph the parabola so that its vertex and shape match a set of plotted points.
Understand the value of a and its contribution to shape and direction of opening.
Apply knowledge of parabolas to parabolic shapes in real world problems.
Teacher Preparation and Notes
This activity is intended for an Algebra 1 or Algebra 2 class.
Students will need to be able enter data into lists and graph as a scatter plot. They will also need to be able to graph a function and adjust window settings.
Students will answer questions about the vertex, direction of opening, and the relative width of opening for a particular shape.
Problem 1 – Match the graph, Part 1
Vertex form for the equation of a parabola is shown on the worksheet. Students will use this information in the next few questions to help them answer questions.
Notice that the students may not get the “exact” answer that they wish. Tell them that they will find more exact methods of finding matching equations for data in later classes.
You may also want to cover the effects of changing the window on the appearance of the graph. Stress the importance of knowing the minimum, maximum, and scale to determine the equation.
Problem 2 – Match the graph, Part 2
Problem 3 – Match the Double Arches
Problem 4 – The Main Cables of a Suspension Bridge
Several loops of cable are represented here. Students will be matching an equation to a particular piece of the graph. What the students have learned about vertex form should be of help in this problem.
To graph the given screen, see the equations to the right. Conditional statements are used to limit the domain of the function.
Extension – The Gateway Arch in St. Louis
The Gateway Arch in St. Louis, the “Gateway” to America, is a shape that looks like a parabola to the casual observer (It is actually called a catenary curve.).
Discussion that follows includes how the equations are the same, and different. Assist the students in expanding the vertex form so that a direct comparison can be made for the two equations.
This section gives students a few real-world situations where they can find parabolas. Students can find the equations that model these situations.
- Hang a chain (or necklace) against a piece of graph paper and trace its graph (or take a digital photo). Write an equation in vertex form to match the shape of the curve.
- Place a laminated piece of graph paper behind a drinking fountain and take a digital photo. Write an equation to match the shape of the curve.