The first application of quadratic transformations you will investigate is dropping an object from a specific height.

Let's begin with the height function:

h(t) = -16t² + c

This quadratic function can be used to approximate the height h in feet above the ground of a falling object t seconds after it is dropped from a height of c feet. This model is only used to approximate the height of the falling object because it does not take into account air resistance, wind, or other factors that would affect the speed of the object.

How is this function related to the quadratic equation in vertex form?

f(x) = a(x − h)²+ k

Answer the following questions. Click on the button to check your answer.

What is “a” in the height function?

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a = -16

What is “h” in the height function?

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h = 0
h(t) = -16(t – 0)² + c
h(t) = -16(t)² + c

What is “k” in the height function?

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k = c

Let’s continue and look at a specific falling object problem.

Example 1

Jack was climbing up his beanstalk. When he climbed 20 feet from the base of the beanstalk at ground level, he stopped and dropped one of his golden eggs. He continued to climb the bean stalk and when he reached 80 feet above the ground, he stopped and dropped another golden egg. Click on the golden eggs to watch them fall. To reset the activity, click on the eggs again.

Answer the following on your own paper or in your notes.

1. Write the height functions for the golden eggs that Jack dropped.

a. First egg Jack drops: h₁(x) =

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h₁(x) = -16t² + 20
Falls from 20 feet.


b. Second egg Jack drops:  h₂(x) =

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h₂(x) = -16t² + 80
Falls from 80 feet.

2. What kind of translation do you think occurred between the graphs of the two functions? How do you know this?
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There is a vertical translation between h₁(x) and h₂(x) because the k value changed from 20 to 80.

3. Use your own graphing paper or click on the image below to use an online graph plotter to graph both functions on the same coordinate (x,y) grid.
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Follow this link for directions on how to enter the equations and set a good window for the graph.
To enter in your equations you must use the asterisk “*” for multiplication and “^” for an exponent. Here is how to enter the first graph. -16*x^2+20. To create a good viewing window change the range of the y-axis values from -5 and 5 to -100 and 100. Click on Draw or hit Enter to create the graphs.



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4. What do the x values represent?
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The x values represent time in seconds.

5. What do the y values represent?
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The y values represent height in feet.

6. Describe the translation between the two parabolas? What does this mean in terms of the golden eggs that where dropped?
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There is a vertical translation from a height of 20 feet to a height of 80 feet, which means the second egg was dropped 60 feet higher than the first egg.

7. What other information can you gather from the graph? How long does it take for the eggs to hit the ground?
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We can also determine approximately how long it takes the eggs to hit the ground. Looking at the graph, you can see that for the egg that drops from 20 feet, it hits the ground in a little over 1 second. You can also see that for the egg that drops from 80 feet, it hits the ground in about 2.25 seconds.

In this problem, you discovered that dropping an object at different heights created a graph with a vertical translation. In the next section, you will investigate what happens when gravitational forces are changed.