Estimated Time
2 days
Prerequisites
- Graphing scatter plots
- Drawing line-of-best fit
- Functional notation
- Solving quadratic equations
Materials Needed
- Graphing paper or graphing calculator
Ohio Standards Alignment
Topics
Quadratic functions: graphs, intercepts, vertex, symmetry; linking representations; quadratic equations; intersections of lines and quadratic curves
Overview
Students develop a plan to maximize profits on a donut sale using data provided. The parabolic graphs of selling price vs. income and selling price vs. profit are used to explore intercepts, symmetry, and the vertex. The graphs and table allow for a side-by-side comparison of linear and quadratic functions. The goal is for students to link representations of data through tables and graphs and to see the connection to algebraic solutions. A quadratic rule is generated using the idea that income is selling price multiplied by number sold. This problem is suited to an introduction of quadratic functions after a study of quadratic equations. An extension could be for students to gather their own data through a survey for a fund-raising effort in their school and devise a selling plan.
The "Hook"
The Student Council at a large high school sells donuts on Friday mornings as a fund raiser. They are considering raising the price of a donut to make more money.
Critical Question
Will raising the price result in more income? Can you think of an argument that it will raise more money and a counter-argument that it will not?
The Investigation
Distribute student activity sheets. Graphs can be done by hand on graph paper or on a graphing calculator using the Stat List Editor and Stat Plots.
Students should complete the table and answer Question 1.
Critical Question
Which relationships are linear and which are non-linear? How can you tell from the graph? How can you tell from the table?
If not already familiar with first and second differences, students can be instructed to explore which relationships have first-order differences that are constant (linear) and which have second-order difference that are constant (quadratic).
Continue with Question 2. Follow up with a discussion on symmetry and the vertex in quadratic functions. If additional practice is needed, students can do the same for the profit function.
Critical Question
If a line of symmetry is drawn on a quadratic graph, what is its relationship with the vertex?
In Question 3, students only need to use the rule for
number sold as written in Question 1e to write the explicit rules for the quadratic functions.
Critical Question
What is the difference in the explicit rules whose solution set is a parabolic graph or a linear graph?
Question 4 requires students to interpret the graphs in the context of the problem. They may be instructed to construct the graphs on the same coordinate plane, or if using a graphing calculator, to graph the functions simultaneously to aid in analysis.
Question 5 can be optional and leads to solving systems of linear and non-linear equations. It requires that students know how to solve a quadratic equation.
Critical Question
How do these solutions connect to your selling plan?
Teaching Tips
- The question posed in the “Hook” requires students to realize that charging more may increase income, but eventually the decrease in buyers no longer willing to pay the price, leads to a decrease in income. Income is dependent on both price and number sold.
- To save class time, students can be assigned Question 1 for homework the night before.
- Students may need some discussion on how to determine income and profit and what they mean in the context of the donut sales.
- If using the Stat List Editor on the calculator, students can be encouraged to populate the lists by attaching a formula to a list name. [Assume: L1 = selling price; L2 = Number of Donuts Sold; Let: Income be L3 = L1 × L2; Cost of Donuts be L4 = L2 × .20; Profit be L5 = L3 – L4]
- The column for “Cost of Donuts” is the purchase price for the number of donuts bought at that selling price. We assume no extras are being bought.
- Comparison of linear and quadratic functions should lead to the idea that linear functions are always increasing or decreasing, while quadratic functions do both and the “turning point” is the vertex.
- In Question 5 students should be encouraged to connect the solutions to the graphs.
- If students know how to find the vertex and intercepts algebraically that idea may be added to Question 5. Or this question can be an opportunity to motivate how to find the values algebraically.
- Problem solutions are provided here.
Citation
From the teaching files of Teresa Graham.