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# 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 $y = a(x - h)^2 + k$. If needed, graph $y = a(x)^2$ with various values of a and explore.

• In vertex form or in standard form, what happens when $0 < a < 1$?
• If $a > 1$, the graph will be narrow and open up. If $a < -1$, the graph will be what?

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

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

## Problem 3 – Match the Double Arches

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

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 $(-2, 5.5)$. 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 $A$ and $B$ meet is $(0, 3.2)$.

• What is the equation of the piece of the graph labeled $A$?
• What is the equation of the piece of the graph labeled $B$?

## 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 $L1$ and $L2$ shown and create a scatter plot with an appropriate window.

• What is the equation?

Using the same data, match the graph in standard form $(y = ax^2 + bx + c)$ by changing the $Y=$ equation. Important things to remember are; what does the value of $a$ do to the graph, and what would your $y-$intercept be ($c$ 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.

Feb 22, 2012

## Last Modified:

Oct 31, 2014
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TI.MAT.ENG.SE.1.Algebra-I.11.4