As you learned in the previous sections, the quadratic function f(x) = a(x − h)2+ k, can be used to describe and predict the effects of changes in a, h, and k on the graph of the function given applied and purely mathematical situations.

No matter if you are dropping an object, throwing an object, finding maximum area, or maximum profit, just to name a few, how you interpret the quadratic function in vertex form and use it to describe and predict the effect of changes in a, h, and k on the graph is the same.

Answer the following problems on your own paper or in your notes before moving on to Test Your Understanding.

  1. Two coconuts fall from a coconut tree, the first coconut falls from 35 feet and second coconut falls from 48 feet.
    1. Using the height function from section 2, write the functions for the first and second coconuts that drops in terms of t seconds.
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      h1(t) = -16t2 + 35 and h2(t) = -16t2 + 48.

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    2. Use this online graph plotter or your graphing calculator to graph the two functions that represent the falling coconuts.
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      This is a graph that represents the height of two falling coconuts from 35 and 48 feet after t seconds
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    3. Describe the translation between the graph of the coconut that falls from 35 feet to the graph of the coconut that falls from 48 feet.
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      The graph is vertically translated 13 units up.

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    4. What does this translation mean?
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      This means that the second coconut that fell was 13 feet higher than the first coconut that fell.

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  2. Now consider that one of the coconut trees grows on the moon. The height function for the coconut falling 35 feet on the moon is represented by hM(t) = -2.66x2 + 35. The height function for a coconut falling 35 feet on Earth is represented by hE(t) = -16x2 + 35.
    1. What type of translations occurs between the graph of the coconut falling on the Earth to the graph of the coconut falling on the moon. If you need to graph the functions to help you see the translation, click here.
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      There is a vertical compression of the moon's function as compared to the Earth's function.

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    2. Which coconut will hit the ground faster?
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      The coconut falling on Earth will hit the ground faster than the coconut falling on the moon. Close Pop Up
  3. Two arches of a bridge are modeled by the functions
    h1(f) = -0.0015(x − 120)2 + 25 and
    h2(f) = -0.0015(x − 360)2 + 25
    1. Describe the translation of the graph of second arch as compared to the graph of the first arch?
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      The graph of the second arch is translated 240 units to the right of the graph of the first arch. Close Pop Up

Now go to Test Your Understanding to complete the assessment.