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11.4: Manual Fit

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Algebra I, Chapter 10, Lesson 7.

ID: 12274

Time required: 20 minutes

Activity Overview

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.

• 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.

Associated Materials

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.

Instruct students that to change the graph, they can substitute different values into the Y=\begin{align*}Y=\end{align*} equation. Students should enter initial values (non-zero) into the vertex form of the equation to begin their exploration.

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.

Acceptable equation: y=2(x1)2+3\begin{align*}y = 2(x - 1)^2 + 3\end{align*}

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

Students are given another set of plotted points and asked to grab and move the parabola to match the graph. Encourage discussion of the placement of the vertex, and the relative width of the curve. This time, a negative value for a\begin{align*}a\end{align*} is required. y=0.25x2\begin{align*}y=-0.25x^2\end{align*}

Problem 3 – Match the Double Arches

After matching the data well, the M\begin{align*}“M”\end{align*} double arches appear quite nicely. Discussion could follow about reflections, symmetry, and the design of company logos using mathematical or geometric figures that are pleasing to the eye.

Note: Students can enter the less than or equal to ()\begin{align*}(\le)\end{align*} and the greater than or equal to ()\begin{align*}(\ge)\end{align*} symbols by pressing 2nd [0]\begin{align*}2^{nd} \ [0]\end{align*}, the Catalog menu, and selecting from the list.

To enter the word and, students can press 2nd\begin{align*}2^{nd}\end{align*} [MATH] and move to the LOGIC menu.

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.

Section A: y=0.2(x+4)2Section B: y=0.2(x4)2\begin{align*}\text{Section A:} \ y = 0.2(x + 4)^2\\ \text{Section B:} \ y = 0.2(x - 4)^2\end{align*}

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.).

Students will create an equation in vertex form to match the data given in L1\begin{align*}L1\end{align*} and L2\begin{align*}L2\end{align*}.

Using the same data, students are asked to match the graph in standard form. Important things to remember are; what does the value of a do to the graph, and what would your y\begin{align*}y-\end{align*}intercept be (c\begin{align*}c\end{align*} in the equation)?

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.

Other Arches

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.

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