The solutions of a quadratic function are the zeros, or where the graph of the quadratic function crosses the x–axis.

Example: The Kemah Boardwalk has a new game that offers a $1000 bonus for the person with the highest score. Jose and Mirza decide to join the competition. A ball is put on the end of a lever that is on a table. The buckets are located at different distances from the lever. Each time the ball lands in a bucket the player earns points depending on the distance the bucket is from the original starting point.

The object of the game is to earn the most points. How a player earns points:

1. If the ball does not go into a bucket, 0 points are earned.
2. If the ball goes into the first bucket, 5 points are earned.
3. If the ball goes into the second bucket, 10 points are earned.
4. If the ball goes into the third bucket, 15 points are earned.
5. If the ball goes into the fourth bucket, 25 points are earned.

How to play the game:

1. Press down on the lever to release the ball.
2. The ball goes (hopefully) into a bucket.
3. Each player tosses the ball three times for his or her total score.

Toss the ball by clicking on the left end of the lever. First, toss the ball three times for Mirza and then three times for Jose. Make sure to write down the points each one earned for each toss.

• When the ball is tossed into the air, what type of graph best describes the height of the ball versus the horizontal distance from the starting point?

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  What path did the ball trace out as it flew through the air?

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  Parabola, or quadratic function

• The horizontal distance represents the number of points scored on the toss. Write the coordinates for a ball landing in each of the 4 buckets shown.

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  The coordinates will be of the form (x, 0).

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  (5, 0), (10, 0), (15, 0), and (25, 0)

• If there were an unlimited number of buckets, what coordinates would represent that a player had earned 45 points on one toss?

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  What does the x-axis represent in this situation?

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  (45, 0)

When Mirza and Jose were playing the bucket game, the number of points earned was based on where the parabola crossed the x-axis. There are actually two points of intersection; the first one is located at the origin, and the second one is located at the bucket.

Practice

Bernadette throws the javelin for her school’s track and field team. The graph below shows the height of a javelin and the horizontal distance traveled by the javelin for three of Bernadette’s practice throws.

Use the graph to answer the questions below.

1. Which javelin throw landed closest to 45 yards from where Bernadette started?

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   The x-intercept on the graph represents where the javelin landed.

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   Throw #2

2. What is the approximate distance of the javelin throw that went the farthest?

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   The x-intercept on the graph represents where the javelin landed.

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   Approximately 56 yards

3. What is the total horizontal distance where the javelin was higher than 12 feet on throw #2?

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   Locate the points along the path of throw #2 where the height was 12 feet, and count the horizontal distance between these two points.

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   Approximately 25 yards

4. What is the total horizontal distance where the javelin was higher than 6 feet on throw #1?

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   Locate the points along the path of throw #1 where the height was 6 feet, and count the horizontal distance between these two points.

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   Approximately 50 yards