Activity
We have seen quadratic functions modeling the height of a projectile as a function of time.
Here are two ways to define the same function that approximates the height of a projectile in meters, seconds after launch:
Which way of defining the function allows us to use the zero product property to find out when the height of the object is 0 meters (when the object is on the ground)?
Compare your answer:
The one with a quadratic expression in factored form.
Without graphing, determine at what time(s) the height of the object is 0 meters (when the object is on the ground). Show your reasoning.
Enter the times(s) and your reasoning.
Compare your answer:
The object has a height of 0 meters after 6 seconds. Applying the zero product property to solve gives and . Because the object was launched at , the negative solution doesn’t make sense in this context.
Using the function in standard form, determine the height of the object before it started its motion, at seconds. Be prepared to show your reasoning.
Enter the height of the object and your reasoning.
Compare your answer:
The object has a height of 18 meters at 0 seconds. By substituting 0 for into the quadratic form, the starting height can be determined. .
Video: Applying the Zero Product Property
Watch the following video to learn more about applying the zero product property to solve a real-world projectile problem.
Self Check
Additional Resources
Applying the Zero Product Property to Solve a Real-World Projectile Problem
We have previously looked at functions representing the motion of projectiles. We analyzed two equivalent expressions that define the same quadratic function, one in quadratic form and the other in factored form.
Now we’re going to look at a similar problem, but with the additional knowledge of how the function relates to the zero product property.
Let’s look at an example.
The equivalent functions below model the height, in feet, of a baseball seconds after being hit.
Using the zero product property, the quadratic expression in factored form allows you to more easily find out when the height of the baseball is 0 feet.
Without graphing, we can determine at what time the baseball reaches the ground.
Let’s apply the zero product property to solve .
Set each factor equal to 0 and solve.
So, and . These values of represent the roots of the function. Because the object was launched at , the negative solution doesn’t make sense in this context of time. So the baseball reaches the ground in 3 seconds.
There is one benefit of knowing the function in quadratic form. By substituting 0 in for , we can easily determine at what height the baseball was hit, when seconds.
So, the baseball was hit at a height of 3 feet. Note that this point, , is the -intercept of the function when graphed.
Try it
Try It: Applying the Zero Product Property to Solve a Real-World Projectile Problem
The functions below model the height of a projectile in feet, seconds after launch.
1. Without graphing, determine at what time the height of the object is 0 feet.
Here is how to solve this projectile problem using the zero product property:
or
The object has a height of 0 feet at 4.5 seconds.
2. Determine the height of the object before it is launched.
Here is how to solve this projectile problem using the zero product property:
The object has a height of 0 feet at 4.5 seconds.
The object starts at a height of 63 feet.