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11.4: Manual Fit

Difficulty Level: At Grade Created by: CK-12
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This activity is intended to supplement Algebra I, Chapter 10, Lesson 7.

Problem 1 – Match the Graph, Part 1

The vertex form for the equation of a parabola is \begin{align*}y = a(x - h)^2 + k\end{align*}y=a(xh)2+k. If needed, graph \begin{align*}y = a(x)^2\end{align*}y=a(x)2 with various values of a and explore.

  • In vertex form or in standard form, what happens when \begin{align*}0 < a < 1\end{align*}0<a<1?
  • If \begin{align*}a > 1\end{align*}a>1, the graph will be narrow and open up. If \begin{align*}a < -1\end{align*}a<1, the graph will be what?

Enter the lists shown at the right. Create a scatter plot of \begin{align*}L1\end{align*}L1 and \begin{align*}L2\end{align*}L2. Then, enter the vertex form of the parabola in \begin{align*}Y1\end{align*}Y1 with an initial guess for each value for \begin{align*}a, h\end{align*}a,h, and \begin{align*}k\end{align*}k. See how the equation fits and then adjust the values to make the graph fit the data.

  • What is the vertex of the parabola?
  • What was your value of \begin{align*}a\end{align*}a for the parabola?
  • What is the equation of the parabola you fit to the data?

Problem 2 – Match the Graph, Part 2

Repeat the process from Problem 1 to find the equation of a parabola that matches the data in \begin{align*}L1\end{align*}L1 and \begin{align*}L2\end{align*}L2.

  • To make the parabola open down, what must be true about the value of \begin{align*}a\end{align*}a?
  • To make the parabola wide, what must be true about the value of \begin{align*}a\end{align*}a?
  • What is the equation of your parabola that fits the data?

Problem 3 – Match the Double Arches

Change \begin{align*}L1\end{align*}L1 and \begin{align*}L2\end{align*}L2 to match the screenshot shown a the right. Now graph, \begin{align*}Y1=\frac{(-1.5(X + 2)^2 + 5.5)}{(-4 \le X\ \text{and}\ X \le 0)}\end{align*}Y1=(1.5(X+2)2+5.5)(4X and X0)

Next, match the second half of double arches.

  • What do you notice about the two parabolas that formed the double arches?

  • The vertex of the left arch is \begin{align*}(-2, 5.5)\end{align*}(2,5.5). What is the vertex of the right arch?
  • What is the equation of your parabola that matches the data?

Problem 4 – The Main Cables of a Suspension Bridge

Here is a picture of a suspension bridge. Several loops of cable are represented. See the graph below to match an equation to a particular part of the graph.

The point where pieces \begin{align*}A\end{align*}A and \begin{align*}B\end{align*}B meet is \begin{align*}(0, 3.2)\end{align*}(0,3.2).

  • What is the equation of the piece of the graph labeled \begin{align*}A\end{align*}A?
  • What is the equation of the piece of the graph labeled \begin{align*}B\end{align*}B?

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.

Use what you know about the vertex form to write an equation to match its shape and dimensions. Enter \begin{align*}L1\end{align*}L1 and \begin{align*}L2\end{align*}L2 shown and create a scatter plot with an appropriate window.

  • What is the equation?

Using the same data, match the graph in standard form \begin{align*}(y = ax^2 + bx + c)\end{align*}(y=ax2+bx+c) by changing the \begin{align*}Y=\end{align*}Y= equation. Important things to remember are; what does the value of \begin{align*}a\end{align*}a do to the graph, and what would your \begin{align*}y-\end{align*}yintercept be (\begin{align*}c\end{align*}c in the equation)?

  • What is your equation in standard form?
  • How are the two equations similar?
  • How are the two equations different?
  • Expand the vertex form and convert it to standard form to make a final comparison.

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Date Created:
Feb 22, 2012
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Oct 31, 2014
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TI.MAT.ENG.SE.1.Algebra-I.11.4