# 11.4: Manual Fit

**At Grade**Created by: CK-12

*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*}*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*}

- 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*}

- 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*}

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*}

- 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*}

- What is the equation?

Using the same data, match the graph in standard form \begin{align*}(y = ax^2 + bx + c)\end{align*}

- 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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