<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />

# 11.4: Manual Fit

Difficulty Level: 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 y=a(xh)2+k\begin{align*}y = a(x - h)^2 + k\end{align*}. If needed, graph y=a(x)2\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 0<a<1\begin{align*}0 < a < 1\end{align*}?
• If a>1\begin{align*}a > 1\end{align*}, the graph will be narrow and open up. If a<1\begin{align*}a < -1\end{align*}, the graph will be what?

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

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

## Problem 3 – Match the Double Arches

Change L1\begin{align*}L1\end{align*} and L2\begin{align*}L2\end{align*} to match the screenshot shown a the right. Now graph, Y1=(1.5(X+2)2+5.5)(4X and X0)\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 (2,5.5)\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 A\begin{align*}A\end{align*} and B\begin{align*}B\end{align*} meet is (0,3.2)\begin{align*}(0, 3.2)\end{align*}.

• What is the equation of the piece of the graph labeled A\begin{align*}A\end{align*}?
• What is the equation of the piece of the graph labeled B\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 L1\begin{align*}L1\end{align*} and L2\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 (y=ax2+bx+c)\begin{align*}(y = ax^2 + bx + c)\end{align*} by changing the Y=\begin{align*}Y=\end{align*} equation. Important things to remember are; what does the value of a\begin{align*}a\end{align*} 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)?

• 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.

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

Show Hide Details
Description
Tags:
Subjects: