## Real World Applications – Algebra I

### Topic

What’s your vertical jump height? How can we represent this as a quadratic function?

### Student Exploration

One of the things that basketball players love to show off and brag about is their vertical jump height. If you walk past a basketball hoop, there will always be someone walking past or before playing basketball to jump and try to reach the rim. Some make it, and some don’t.

Vertical jump height is discussed quite frequently among NBA players and fans. Vertical jump height refers to the distance between the highest point a person can reach after a big jump and the standing reach height.

David Noel, from the NBA Draft of 2006, had a vertical jump height of 34 inches. The quadratic equation that can best describe his vertical jump height over time is \begin{align*}h = 162t - 192t^2\end{align*}, with \begin{align*}h\end{align*} representing the vertical jump height and \begin{align*}t\end{align*} represents time in seconds. We can use this equation to find how long it took for him to land on the ground.

Applying our understanding of factoring, we can factor out 6t from both terms on the right side of the equation. Once we do that, we have \begin{align*}h = 6t(27 - 32t)\end{align*}. We can use the Zero Product Principle to set each factor equal to zero and solve for \begin{align*}t\end{align*} when \begin{align*}h = 0\end{align*}.

\begin{align*}0 = 6t(27 - 32t)\end{align*} We factored \begin{align*}6t\end{align*} from both terms because \begin{align*}6t\end{align*} is the greatest common factor of both terms.

\begin{align*}6t = 0\end{align*} and \begin{align*}27 - 32t = 0\end{align*} We used the zero product principle to solve for \begin{align*}t\end{align*}.

\begin{align*}t = 0\end{align*} and \begin{align*}t = \frac{27}{32} = 0.84375\end{align*}, rounded to 0.84

What do these numbers mean?

This means that at 0 and 0.84 seconds Noel’s vertical jump height is 0ft. Why are there two numbers?

The two numbers represent the time that Noel started at 0 seconds and the time that it took for him to reach the ground after jumping.

For the sake of this activity, we’re going to look at the quadratic equation if someone were to find their vertical jump height if the person were jumping off a bench. How do you think the quadratic function would change?

Let’s say Noel jumped off a bench 1ft off the ground and had the same jump height. How would the quadratic equation that represents his jump height change?

From reading the “vertical shifts of quadratic functions” concept, we know that jumping from a bench would mean that this is a vertical shift, assuming that his vertical jump height is still the same. The new equation would add a “\begin{align*}c\end{align*}” value. In this case, we would add 12 (since \begin{align*}1ft = 12 \ inches\end{align*}) to the equation. Our new equation is \begin{align*}h = 162t - 192t^2 + 12\end{align*}.

Try graphing the two quadratic equations and see what the similarities and differences are.

### Extension Investigation

Advanced: Try finding your own vertical jump height and determine with your friends when you reach the highest height and what time you land. Using this information, try to find the quadratic equation that best represents your vertical jump height over time.

### Resources Cited

http://www.topendsports.com/testing/results/vertical-jump.htm