# 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*}. If needed, graph \begin{align*}y = a(x)^2\end{align*} with various values of *a* and explore.

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

Enter the lists shown at the right. Create a scatter plot of \begin{align*}L1\end{align*} and \begin{align*}L2\end{align*}. Then, enter the vertex form of the parabola in \begin{align*}Y1\end{align*} with an initial guess for each value for \begin{align*}a, h\end{align*}, and \begin{align*}k\end{align*}. 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*} 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*} and \begin{align*}L2\end{align*}.

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

## Problem 3 – Match the Double Arches

Change \begin{align*}L1\end{align*} and \begin{align*}L2\end{align*} 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*}

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*}. 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*} and \begin{align*}B\end{align*} meet is \begin{align*}(0, 3.2)\end{align*}.

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

## 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*} and \begin{align*}L2\end{align*} 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*} by changing the \begin{align*}Y=\end{align*} equation. Important things to remember are; what does the value of \begin{align*}a\end{align*} do to the graph, and what would your \begin{align*}y-\end{align*}intercept be (\begin{align*}c\end{align*} 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.