Activity
1. The equation represents the height, as a function of time, of a pumpkin that was catapulted up in the air. Height is measured in meters and time is measured in seconds.
a. Enter the number of seconds after launch.
The pumpkin reached a maximum height of 47 meters. How many seconds after launch did that happen?
3. The pumpkin reached a maximum height of 47 meters 3 seconds after launch.
b. Show your reasoning.
Compare your answer:
c. Suppose someone was unconvinced by your solution. Find another way (besides the steps you already took) to show your solution is correct.
Compare your answer:
- Substitute 3 for in the original expression; evaluating it gives or , which is 47.
- Graphing and shows an intersection is located at .
- Graphing shows a zero at .
2. The equation models the revenue a band expects to collect as a function of the price of one concert ticket. Ticket prices and revenues are in dollars.
A band member says that a ticket price of either $15.50 or $74.50 would generate approximately $1000 in revenue. Do you agree? Show your reasoning.
Compare your answer:
- Partially agree. A ticket price of $15.50 will generate about $1000 in revenue, but a ticket price of $74.50 will generate much less (about $410).
- Using the quadratic formula to solve gives and as the solutions.
- Substituting 15.50 into the expression gives 999.75, which is close to 1000. . Substituting 74.50 into the expression gives 409.75.
- Graphing the equation and shows two intersections at approximately and .
Video: Explaining Your Solutions to Quadratic Situations
Watch the following video to learn more about proving quadratic solutions.
Are you ready for more?
Extending Your Thinking
Function is defined by the equation . Its graph opens downward.
Find the zeros of function without graphing.
Enter the zeros of function .
Compare your answer:
Function has one zero: 3.
Show your reasoning.
Compare your answer:
- is equivalent to . Solving using the quadratic formula gives .
- can be rewritten as and then as . Applying the zero product property to solve gives .
Explain or show how the zeros you found can tell us the vertex of the graph of .
Compare your answer:
Function only has one zero, which means its graph has only one horizontal intercept, occurring when . Because the point is the only intercept, it must also be the vertex of the graph, otherwise there would have been either two horizontal intercepts, or none.
Recall that and . Examine these functions and (which defined the height of the pumpkin). Explain how the maximum of function , once we know it, can tell us the maximum of .
Compare your answer:
The value of the expression for is 47 less than that of , so if we know the maximum of , we know that the maximum of is 47 less than that. We saw that the maximum of is 47, so the maximum of is 47 less than 47, which is 0.
Self Check
Additional Resources
Answering Questions About Situations With the Quadratic Formula
Here is an example of someone solving a quadratic equation that has no solutions:
Step 1 -
Step 2 -
Step 3 -
Study the example. At what point did you realize the equation had no solutions?
- For two numbers to add up to 0, they have to be opposites. Since the number 9 is positive, the other number must be negative, which a square cannot be.
Explain how you know the equation has no solutions.
- A square is always positive, so a square plus a positive number must also be positive and can never equal zero.
Try it
Try It: Answering Questions About Situations With the Quadratic Formula
Function models the height of an object, in meters, seconds after it is launched into the air. It is defined by . How much time will it take the object to reach 15 meters?
Here is how to solve this word problem using the quadratic formula:
To know how much time it would take the object to reach 15 meters, we could solve the equation .
- Rewriting it in standard form gives . The expression on the left side of the equation cannot be written in factored form, however.
- Completing the square isn't convenient because the coefficient of the squared term is not a perfect square and the coefficient for the linear term is an odd number.
- Let's use the quadratic formula, using , , and !
Step 1 -
Step 2 -
Step 3 -
The expression represents the two exact solutions of the equation.
We can get approximate solutions by using a calculator, or by reasoning the .
The solutions tell us that there are two times after the launch when the object is at a height of 15 meters: at about 0.7 seconds (as the object is going up) and 4.3 seconds (as it comes back down).