Building Quadratic Functions to Describe Situations (Part 1)

ID: karop-tupuv
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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Algebra, Algebra 2
Grade: 8-9
Standards: HSF-BF.A.1HSF-BF.A.1.aHSF-IF.A.2
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17 questions

Building Quadratic Functions to Describe Situations (Part 1)

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Notice and Wonder: An Interesting Numerical Pattern

1) Study the table.

x012345y01664144256400\begin{array}{|c|c|c|c|c|c|c|} \hline \\[-1em] x & 0 & 1 & 2 & 3 & 4 & 5 \\[-1em] \\ \hline \\[-1em] y & 0 & 16 & 64 & 144 & 256 & 400 \\[-1em] \\ \hline \end{array}

What do you notice? What do you wonder?

Falling from the Sky

A rock is dropped from the top floor of a 500-foot tall building. A camera captures the distance the rock traveled, in feet, after each second.

A template for answering this question. Ask your instructor for an alternative.

2) How far will the rock have fallen after 6 seconds?

3) Show your reasoning.

Jada noticed that the distances fallen are all multiples of 16.

She wrote down:

16=16164=164144=169256=1616400=1625\begin{array}{rcl} \\[-1em] 16 & = & 16 \cdot 1 \\[-1em] \\ \\[-1em] 64 & = & 16 \cdot 4 \\[-1em] \\ \\[-1em] 144 & = & 16 \cdot 9 \\[-1em] \\ \\[-1em] 256 & = & 16 \cdot 16 \\[-1em] \\ \\[-1em] 400 & = & 16 \cdot 25 \\[-1em] \end{array}

Then, she noticed that 1, 4, 9, 16, and 25 are 12,22,32,421^2,2^2,3^2,4^2 and 525^2.

4) Use Jada’s observations to predict the distance fallen after 7 seconds. (Assume the building is tall enough that an object dropped from the top of it will continue falling for at least 7 seconds.)

5) Show your reasoning.

6) Write an equation for the function, with dd representing the distance dropped after tt seconds.

Galileo and Gravity

Galileo Galilei, an Italian scientist, and other medieval scholars studied the motion of free-falling objects. The law they discovered can be expressed by the equation d=16t2d=16 \cdot t^2, which gives the distance fallen in feet,dd, as a function of time, tt, in seconds.


An object is dropped from a height of 576 feet.

7) How far does it fall in 0.5 seconds?

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To find out where the object is after the first few seconds after it was dropped, Elena and Diego created different tables.


Elena’s table:Diego’s table:time (seconds)distance fallen (feet)0011626434ttime (seconds)distance from the ground (feet)05761560251234t\begin{array}{cc} \text{Elena's table:} & \text{Diego's table:} \\ \begin{array}{|c|c|} \hline \\[-1em] \textbf{time (seconds)} & \textbf{distance fallen (feet)}\\[-1em] \\ \hline \\[-1em] 0 & 0 \\[-1em] \\ \hline \\[-1em] 1 & 16 \\[-1em] \\ \hline \\[-1em] 2 & 64 \\[-1em] \\ \hline \\[-1em] 3 & \\[-1em] \\ \hline \\[-1em] 4 & \\[-1em] \\ \hline \\[-1em] t & \\[-1em] \\ \hline \end{array} & \begin{array}{|c|c|} \hline \\[-1em] \textbf{time (seconds)} & \textbf{distance from the ground (feet)}\\[-1em] \\ \hline \\[-1em] 0 & 576 \\[-1em] \\ \hline \\[-1em] 1 & 560 \\[-1em] \\ \hline \\[-1em] 2 & 512 \\[-1em] \\ \hline \\[-1em] 3 & \\[-1em] \\ \hline \\[-1em] 4 & \\[-1em] \\ \hline \\[-1em] t & \\[-1em] \\ \hline \end{array}\end{array}

8) How are the two tables alike?

9) How are they different?

Here is Elena's table.

time (seconds)distance fallen (feet)001162643Cell A4Cell BtCell C\begin{array}{|c|c|} \hline \\[-1em] \textbf{time (seconds)} & \textbf{distance fallen (feet)}\\[-1em] \\ \hline \\[-1em] 0 & 0 \\[-1em] \\ \hline \\[-1em] 1 & 16 \\[-1em] \\ \hline \\[-1em] 2 & 64 \\[-1em] \\ \hline \\[-1em] 3 & \text{Cell A} \\[-1em] \\ \hline \\[-1em] 4 & \text{Cell B} \\[-1em] \\ \hline \\[-1em] t & \text{Cell C} \\[-1em] \\ \hline \end{array}

Determine the values that belong in the missing cells.

10) Cell A:

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11) Cell B:

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12) Cell C:

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13) Explain your reasoning.

Here is Diego's table.

time (seconds)distance from the ground (feet)0576156025123Cell A4Cell BtCell C\begin{array}{|c|c|} \hline \\[-1em] \textbf{time (seconds)} & \textbf{distance from the ground (feet)}\\[-1em] \\ \hline \\[-1em] 0 & 576 \\[-1em] \\ \hline \\[-1em] 1 & 560 \\[-1em] \\ \hline \\[-1em] 2 & 512 \\[-1em] \\ \hline \\[-1em] 3 & \text{Cell A} \\[-1em] \\ \hline \\[-1em] 4 & \text{Cell B} \\[-1em] \\ \hline \\[-1em] t & \text{Cell C} \\[-1em] \\ \hline \end{array}

Determine the values that belong in the missing cells.

14) Cell A:

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15) Cell B:

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16) Cell C:

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17) Explain your reasoning.