Quadratic Equations

Lesson

- To model real-world quadratic equations.
- To practice identifying key features of a quadratic function as well as interpreting them in terms of the real-world situation.

- Funnel
- Duct tape
- Tape
- Construction paper (optional)
- 3 Unused pencils
- Baking soda
- White vinegar
- Plastic soda or water bottle
- Toilet paper
- Phone camera or video camera
- Measuring tape
- Stopwatch (optional)
- Cork
- Measuring cup

Work on your own or in small groups.

- Unscrew the top of the bottle and put it to the side.
- Flip the bottle upside down. Duct tape the pencils eraser side down to the rocket to make feet. Make sure the pencils enable the bottle’s mouth to stand level about 2 inches/5 cm above the ground.
- Cut out and tape a cone using your construction paper. Duct tape the cone to the bottom of the water bottle.
- Take a piece of toilet paper about a foot long and place one tablespoon of baking soda in the middle.
- Fold in the sides of the toilet paper and roll it so you make a tiny tight roll of toilet paper with the baking soda inside. Your rolled up baking soda should be able to fit through the mouthpiece of your water bottle.
- Take your cork and add enough duct tape around it so that the cork fits snuggly in the mouth of the water bottle.
- Use the funnel to fill the water bottle with 8 oz/1 cup of white vinegar.
- Go outside in an open place. It is best to be near a place where you have points of reference you can use to tell how high the rocket goes (for example, you could go near a basketball hoop or a wall).
- Use duct tape to mark on the floor where the launch pad of your rocket will be. Make sure it is on a flat surface.
- Ready your camera. It might be good to have a friend help you with this part.
- Carefully drop the toilet paper with the baking soda in the mouth of the rocket and quickly stick the cork into the mouth of the water bottle securely. Flip the bottle so that the mouth of the water bottle is upside down and give the bottle two hard shakes very quickly.
- Quickly stand the rocket on the launch pad. Videotape the result. Either use the video to time how long the rocket is in the air or have a stopwatch on the side to time the flight of the rocket.
- Measure from the landing pad to the place where the rocket landed.
- Look at the video and see what looks like highest height that the rocket reached. Measure to this height and record it.

- How far from the landing pad did your rocket land?
- How long did it take for the rocket to land?
- Divide the amount of time the rocket traveled from its launch to its landing in half. This will be the amount on either side of the $x$
`x`-axis that you plot your quadratic. For example, if the rocket was in the air for $3$3 seconds, the graph would begin at $x=-1.5$`x`=−1.5 and end at $x=1.5$`x`=1.5. - Think of the place where your rocket was launched from as the point $(a,0)$(
`a`,0) where $a$`a`is the number you calculated in question 3. The $x$`x`-axis will represent the time it takes the rocket to complete its path in seconds. The $y$`y`-axis represents the vertical height of the rocket. What point on the graph would represent where the rocket landed? - How many zeros should the function representing the rocket’s path have? Why does this make sense in terms of the situation?
- What would be the zeros of the function that represents the path of the rocket?
- Plug the zeros into the function $y=a(x-m)(x-n)$
`y`=`a`(`x`−`m`)(`x`−`n`), where $m$`m`and $n$`n`represent the zeros of the function. - Solve for the constant $a$
`a`by plugging in the $y$`y`-intercept for $y$`y`and the appropriate values for $x$`x`. - From your findings in questions 7 and 8 what is the equation that represents your rocket’s path?
- Graph the equation you created. Time in seconds should be on the $x$
`x`-axis and vertical distance should be on the $y$`y`-axis. Be sure to label the axes and title the graph. - What is the maximum height that your rocket reached?
- What is the equation for the vertex, or turning point, of this function? Interpret it in terms of the situation.
- What is the range of the function you created? Interpret it in terms of the situation.
- On what interval is the function increasing? On what interval is it decreasing? Interpret these values in terms of the situation.
- Use your equation to determine how high the rocket was after $1$1 second.
- Use your equation to determine how high the rocket was after $1.5$1.5 seconds.

- Work with a partner to determine ways your rocket can be improved. Explain why you think the modifications would change the trajectory of the rocket. How will this affect the graph?
- Gather the necessary materials and make the alterations. Test out the rocket and create a new equation to model the path of the new rocket. Compare it to the trajectory of the old rocket. How did the graph change?

Manipulate rational, exponential, and logarithmic algebraic expressions

Apply algebraic methods in solving problems