Quadratic Equations

UK Secondary (7-11)

Catapult Quadratics (Investigation)

Lesson

- To find quadratic equations in real life.
- To identify key elements of a quadratic function.

- Popsicle sticks
- 1 Rubber band
- Hot glue gun
- 1 Binder clip
- 1 Bottle cap
- Scissors
- Tape measure
- Coloured tape
- Paper
- Video camera or cell phone
- Stopwatch (optional)

- Take one popsicle stick and glue one of the metal grips of the binder clip to the end of it. Be sure that only the metal part of the binder clip is glued to the popsicle stick.
- Place glue on the side of the popsicle stick to which the binder clip is glued.
- Put another popsicle stick on top of the glue so that the metal part of the binder clip glued to the popsicle stick is sandwiched between the two.
- Use two more popsicle sticks to do the same thing to the other metal part of the binder clip.
- Set the binder clips and popsicle sticks to the side to dry for a moment.
- Place another popsicle stick on the table and put glue on one end.
- Place two additional popsicle sticks on the glue you laid down.
- Place glue on the ends of the popsicle sticks you just added.
- Take another popsicle stick and lay it across the glue you laid.
- Flip over the triangle you have created over.
- Place glue down the middle popsicle stick that divides the triangle.
- Lay one of the popsicle sticks attached to the binder clip on the glue you just laid. The binder clip should be at the point of the triangle.
- Place a dot of glue on the front of the popsicle stick attached to the binder clip that is standing up.
- Place the bottle cap on the glue.
- Tie the rubber band around the lower part of the popsicle stick with the bottle cap on it.
- Tightly double knot the rubber band.
- Cut off one of the extra rubber band loops you created.
- Glue the remaining rubber band loop to the popsicle stick in the middle of the triangular stand.
- Press down the popsicle stick with the bottle cap and hold the rubber band loop in place while it dries.
- Take a piece of paper and fold it into quarters. Use the scissors to cut the piece of paper where you folded it.
- Crumple up each of the pieces of paper so that you have 4 paper balls.
- Repeat those steps with two more pieces of papers so that you have a total of 12 paper balls.
- Stand your catapult next to a wall so that it will throw the projectile parallel to the wall.
- Set up the camera and stopwatch (optional).
- Place a paper ball into the bottle cap and pull the catapult back. Be sure to hold down the stand of the catapult as you do this. Record the whole trajectory of the projectile. It may be best to leave the camera fairly zoomed out.
- Use the video to identify a reference point to which you can measure to find the highest point in the projectile’s trajectory.
- Also, use the video to determine how long the projectile was in the air (or use the stopwatch).

- How long was your projectile in the air after it left the catapult?
- How far from your catapult did the projectile land?
- Divide the amount of time the projectile 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 a projectile was in the air for $3$3 seconds, we would begin the graph at $x=-1.5$`x`=−1.5 and end it at $x=1.5$`x`=1.5. - Think of the place where your projectile was launched from as the point $\left(a,0\right)$(
`a`,0) where $a$`a`is the number you calculated in question 3. The $x$`x`-axis will represent the time it takes the projectile to complete its path in seconds. The $y$`y`-axis represents the vertical height of the projectile. What point on the graph would represent where the rocket landed? - How many zeros should the function representing the projectile’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 projectile?
- Substitute the zeros into the function $y=a\left(x-m\right)\left(x-n\right)$
`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 question 7 and 8 what is the equation that represents your projectile’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 projectile reached?
- What is the equation for the 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 projectile was after $1$1 second.
- Use your equation to determine how high the projectile was after $1.5$1.5 seconds.
- Use your knowledge of the path a projectile takes after being launched from the catapult to set up a target where you believe the projectile will land and test it out.
- Did you hit the target?
- Try adjusting the place to where you pull the catapult back. Change your target to match this adjustment. Test it out.
- Did you hit the target?
- What adjustments did you have to make? Why?