# 11.4: Manual Fit

*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 . If needed, graph with various values of *a* and explore.

- In vertex form or in standard form, what happens when ?
- If , the graph will be narrow and open up. If , the graph will be what?

Enter the lists shown at the right. Create a scatter plot of and . Then, enter the vertex form of the parabola in with an initial guess for each value for , and . 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 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 and .

- To make the parabola open down, what must be true about the value of ?
- To make the parabola wide, what must be true about the value of ?
- What is the equation of your parabola that fits the data?

## Problem 3 – Match the Double Arches

Change and to match the screenshot shown a the right. Now graph,

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 . 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 and meet is .

- What is the equation of the piece of the graph labeled ?
- What is the equation of the piece of the graph labeled ?

## 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 and shown and create a scatter plot with an appropriate window.

- What is the equation?

Using the same data, match the graph in standard form by changing the equation. Important things to remember are; what does the value of do to the graph, and what would your intercept be ( 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.