Quadratic Functions Word Problems

Analyzing QuadraticsObjective: In each of these problems, students calculate and interpret key features of quadratic functions (axis of symmetry, vertex, y-intercept, x-intercepts) based on a real world context. They can factor for the first problem, but will need the quadratic formula for the second two.

How it works: This includes three, multi-part word problems that focus on using a quadratic model to analyze questions in context. The data for these problems is all genuine!

Problem 1: Happiness vs. the Cheeseburger. Students explore how much happier people get when they eat cheeseburgers… and how too much of a good thing can bring us down.

Problem 2: The Hail Mary. Students take on the role of a quarterback and filmographer, trying to figure out how much time they need to wait before throwing the pass and how to catch it all on camera. (There’s a link to a great, quick video for this one!)

Problem 3: Pokémon Go. Students explore the rise and fall of the game Pokémon Go. They take on the role of a game designer, determining exactly how to make it more popular.

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Happiness vs. The Cheeseburger Name: __________________________________

Your friend can eat a lot. Cheeseburgers are his favorite, so he gets happy when he eats them… until he has too many. If you measured his happiness, h(x ), based on how many cheeseburgers he eats, x, you’d get a curve that looked like this: h(x )=−x2+6 x+16.

1. Graph h(x ). Don’t forget to label your axes!

2. If your friend wants to be as happy as possible, how many cheeseburgers should he eat? How do you know?

3. What does the y-intercept of this graph represent?

4. How many cheeseburgers can your friend eat before he becomes unhappy?

5. Calculate your friend’s level of happiness if he ate 10 cheeseburgers. What does this mean?

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The Hail Mary Name: __________________________________

In football, some of the most important plays happen in the final seconds of the game. If a team needs to score, sometimes they throw a Hail Mary--a long pass that goes almost the whole length of the field. If the team catches it, they score. If they don't, they lose.

The University of Washington Huskies tried throwing one of the longest Hail Marys in history—about 73 yards! For this pass, the ball's height over time was modeled by the function h( t)=−16 t 2+64 t+6, where h( t) was the height in feet and t was the time in seconds. Let's break down whether the Huskies ever had a chance.

1. 73 yards isn't just a long way to throw, it's a long way to run. To give his team the best chance of running all the way down the field and catching the ball before it hits the ground, the quarterback (the passer) needs to keep the football in the air for a long time. How long was the ball in the air?

2. It takes a good college football player about 8 seconds to run 73 yards. How long should the quarterback have waited before he threw the ball?

3. After the quarterback threw the ball, the cameras lost sight of it. It went so high that it was out of sight. If the camera can only see 40 feet high, how long was the ball out of sight?

4. How high did the ball go?

Even when the pass is accurate, the ball goes far enough, and the team has time to catch it... Hail Marys don't usually work. If you need to know whether it worked for the Huskies, here's your answer: bit.ly/2PAno93.

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The Rise and Fall of Pokémon Go Name: __________________________________

Back in the day, people played a little game called Pokémon Go. Over 50 million people played it, which made it one of the most popular games of all time. But like most games, people got tired of it. Today, you’re a game designer and investor. Your goal is to figure out how you could have made the game better… and how you could have made a lot of money on it. Go time.

Section 1: Taking it in1. The graph to the right

shows the popularity of Pokémon Go over time. To start, describe any trends you see in the data.

2. Why do you think the data looks like this?

Section 2: Amping it upThe popularity of Pokémon Go can be modeled a quadratic function. This is true for lots of things that become popular quickly, then become old news. The function that best models this data is f ( x )=−0.06 x2+3.75 x−4.87. Use this equation to answer the following questions.

3. If you can track how many people use the game each day, you can predict when people will stop playing. That way you can fix the game while it’s still popular—before it’s too late. What key feature of our model tells us when the game was most popular?

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4. Let’s go back in time, before Pokémon Go started losing popularity. As the game designer, it’s your job to figure out how to keep the game popular—how to stop people from quitting. To keep people playing, you decide to launch an amazing new feature to the game exactly when it reaches the peak of popularity. Based on your model, when should you launch the new feature? (Bonus: What will the feature be?)

5. You wanted to launch the amazing new feature when the game was most popular, but it turned out to be harder to build than you thought. You won’t be ready to launch until day 50. It’s a very expensive new feature, so it will only be worth it to build it if it helps the number of users increase to at least 40 million. The new feature should get about 9 million more people playing. Will it be worth it to build? How do you know?

6. If you have a lot of users, it doesn’t take much to make a lot of money from an ultra-popular game. Let’s say you make just $0.04 per user each day. Let’s also pretend that you never released that new feature. Based on the model, f ( x )=−0.06 x2+3.75 x−4.87, will you ever earn at least $1 million per day? If so, for how long? Explain how you know!

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Happiness vs. The Cheeseburger: Answer Key

Your friend can eat a lot. Cheeseburgers are his favorite, so he gets happy when he eats them… until he has too many. If you measured his happiness, h(x ), based on how many cheeseburgers he eats, x, you’d get a curve that looked like this: h(x )=−x2+6 x+16.

1. Graph h(x ). Don’t forget to label your axes!

2. If your friend wants to be as happy as possible, how many cheeseburgers should he eat? How do you know?

If he wants to be as happy as possible, he should eat 3 cheeseburgers. (3, 25) is the vertex, which is the point at which h(x) is highest.

3. What does the y-intercept of this graph represent?

The y-intercept of the graph, h(0)=16, represents how happy he is when he has eaten zero cheeseburgers.

4. How many cheeseburgers can your friend eat before he becomes unhappy?

To avoid being unhappy, he shouldn't eat more than 8 cheeseburgers.

5. Calculate your friend’s level of happiness if he ate 10 cheeseburgers. What does this mean?

If he ate 10 cheeseburgers, his level of happiness would be -24. This means he'd be very unhappy, because his happiness level is negative.

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Happiness

Cheeseburgers Eaten

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The Hail Mary: Answer Key

In football, some of the most important plays happen in the final seconds of the game. If a team needs to score, sometimes they throw a Hail Mary--a long pass that goes almost the whole length of the field. If the team catches it, they score. If they don't, they lose.

The University of Washington Huskies tried throwing one of the longest Hail Marys in history—about 73 yards! For this pass, the ball's height over time was modeled by the function h( t)=−16 x2+64 x+6, where h( t) was the height in feet and t was the time in seconds. Let's break down whether the Huskies ever had a chance.

1. 73 yards isn't just a long way to throw, it's a long way to run. To give his team the best chance of running all the way down the field and catching the ball before it hits the ground, the quarterback (the passer) needs to keep the football in the air for a long time. How long was the ball in the air?

The ball was in the air for 4.09 seconds.

2. It takes a good college football player about 8 seconds to run 73 yards. How long should the quarterback have waited before he threw the ball?

The quarterback should have waited for at least 3.91 seconds before throwing the ball.

3. After the quarterback threw the ball, the cameras lost sight of it. It went so high that it was out of sight. If the camera can only see 40 feet high, how long was the ball out of sight?

The ball was out of sight for 2.74 seconds--from 0.63 seconds until 3.37 seconds after it was thrown.

4. How high did the ball go?

At its highest point, the ball was 70 feet high.

Even when the pass is accurate, the ball goes far enough, and the team has time to catch it... Hail Marys don't usually work. If you need to know whether it worked for the Huskies, here's your answer: bit.ly/2PAno93.

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The Rise and Fall of Pokémon Go: Answer KeyBack in the day, people played a little game called Pokémon Go. Over 50 million played it, which made it one of the most popular games of all time. But like most games, people got tired of it. Today, you’re a game designer and investor. Your goal is to figure out how you could have made the game better… and how you could have made a lot of money on it. Go time.

Section 1: Taking it in1. The graph to the right

shows the popularity of Pokémon Go over time. To start, describe any trends you see in the data.

The game got popular quickly, then started to level off. Eventually it got less popular because the line starts to go down after around 33 weeks.

2. Why do you think the data looks like this?

Answers may vary.

Example answer: Lots of things are popular at first, but then get less interesting. It looks like this game was popular for a while, but then people found more interesting things to do with their time instead so they stopped playing.

Section 2: Amping it upThe popularity of Pokémon Go can be modeled a quadratic function. This is true for lots of things that become popular quickly, then become old news. The function that best models this data is f ( x )=−0.06 x2+3.75 x−4.87. Use this equation to answer the following questions.

3. If you can track how many people use the game each day, you can predict when people will stop playing. That way you can fix the game while it’s still popular—before it’s too late. What key feature of our model tells us when the game was most popular?

The vertex tells us when the game was most popular.

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4. Let’s go back in time, before Pokémon Go started losing popularity. As the game designer, it’s your job to figure out how to keep the game popular—how to stop people from quitting. To keep people playing, you decide to launch an amazing new feature to the game exactly when it reaches the peak of popularity. Based on your model, when should you launch the new feature? (Bonus: What will the feature be?)

The model predicted that Pokémon Go would be most popular about 31 days after it launched. That's when I should launch the new feature.

Bonus: While answers may vary, we all know the new feature would be to let Pikachu evolve into a train, so it can be a Pika-choo-choo.

5. You wanted to launch the amazing new feature when the game was most popular, but it turned out to be harder to build than you thought. You won’t be ready to launch until day 50. It’s a very expensive new feature, so it will only be worth it to build it if it helps the number of users increase to at least 40 million. The new feature should get about 9 million more people playing. Will it be worth it to build? How do you know?

Yes, it should be worth it to build the new feature. After 50 days, about 32.6 million people should still be playing. If the new feature gets another 9 million people to play, a total of 41.6 million will end up playing—just higher than the 40 million we need.

6. If you have a lot of users, it doesn’t take much to make a lot of money from an ultra-popular game. Let’s say you make just $0.04 per user each day. Let’s also pretend that you never released that new feature. Based on the model, f ( x )=−0.06 x2+3.75 x−4.87, will you ever earn at least $1 million per day? If so, for how long? Explain how you know!

If I earn $0.04 per user per day, I need 25 million users to earn $1 million each day. Based on the model, I'll have 25 million users between 9.4 and 53.1 days. Therefore, I will make at least $1 million each day for over 43 full days.

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