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Quadratic Functions and their Uses
"Gateway Arch", St. Louis, Missouri
We are beginning our study of the concept of "Quadratic Functions". We have learned about linear functions and the properties of their graphs. Now we will study a different type of equation and learn the distinct features of its graph. Complete the Scavenger Hunt below. Answer the questions by referring to the internet links provided. It will be necessary to read through (or view) all of the information provided in these links. After all, it is a Scavenger Hunt, so don't just expect the answers to jump out at you!
(Click Here to download the Student Response Sheet)
I. An Introduction to Quadratics
We'll begin with a short video introducing you to the concept of a quadratic function and teaching you how to recognize one just from the form of its equation. Click on the link below to begin!
The website below is a link to a video which will describe the form of a quadratic equation. You will learn about the 'a' , 'b', and 'c' values in the equation. Pay close attention to where these are located! You'll need to know in order to proceed on in the Scavenger Hunt.
Website Name: virtualnerd.com
Link: http://www.virtualnerd.com/algebra-2/quadratics/function-definition.php
1) In order for a function to be quadratic, the degree of the function must be _____________.
2) When graphed, a quadratic function will create a shape called a ______________.
3) True or False: A quadratic function may have an 'x-to-the-third-power' term in it provided it also has an 'x -squared' term in it as well.
______________
II. The Graph of a Quadratic
Now let's learn a little more about the graphs of Quadratic Functions. Use the website below to explore what happens to a quadratic function's graph when the values of a, b, and c are changed. Then use what you discover to answer the questions below. (Hint: You may want to read over the questions first so you will know what ways you should experiment with the values for a, b, and c.)
The website below is an applet for the graph of a quadratic equation. An applet allows you to manipulate and change the values of the function and study what effect this has upon the function's graph.
Website Name: Ronblond.com
Link: http://www.ronblond.com/M11/QFA.sf.APPLET/index.html
4) The parabola opens upward ("smiley face") when the value of 'a' is _______________ and downward ("frown") when the value of 'a' is
_________________.
5) As the value of 'c' increases, the parabola moves _________________.
6) Which would be thinner - a parabola with an 'a' value of +3 or one with an 'a' value of +5 ? ____________ .
III. Important Parts of a Parabola
Now that you have some idea of what a parabola looks like and how the values of 'a' , 'b' , and 'c' affect the parabola's graph, there are some important vocabulary terms we need to learn in order to be able to talk about the parabola. Use the link below to help with this.
The following website gives an explanation of the different parts of a parabola. For each different part, there are hyperlinks (generally words highlighted in blue) for you to study a certain part in more detail. You will need to click on these hyperlinks if you want to find the hidden answers to the questions below!
Website Name: Mathwords.com
Link: http://www.mathwords.com/p/parabola.htm
7) The Axis of Symmetry of a parabola always passes through which two points of the parabola? The ____________ and the ______________.
8) The Axis of Symmetry of a parabola is always ___________________ to the directrix.
IV. The Vertex
The vertex of a parabola is an especially important point, and we will eventually study specific techniques for how to find a parabola's vertex, both from the graph of the parabola and from its equation.
The website below contains several pictures further describing the parts of a parabola and how the vertex is affected by the direction in which the parabola opens. Look carefully to find answers for the questions below!
Website Name: sites.google.com
Link: https://sites.google.com/a/warrenk12nc.org/integrated-math-iii/all-units/unit-4/find-zeros-vertex-minimum-maximum
9) When a parabola opens upward the vertex is a _______________ point, and when a parabola opens downward the vertex is a
_______________ point.
10) From our previous study of the effects of the value of 'a', we can say that the vertex of a parabola is the highest point of the parabola when 'a' is
_________________ and the lowest pointwhen 'a' is __________________ .
V. The Parabola and the X-Axis
One of the most important processes we will learn for parabolas is how to solve their equations. We will see that solving Quadratic Equations is a very different process from what we have learned in solving linear equations. The reason for this has to do with the ways that parabolas can intersect with the x-axis.
This next website contains a detailed study of Quadratics and Quadratic Equations. The section we will focus on here describes the specific place where a parabola touches the x-axis. Once again, be sure to investigate the hyperlinks! You never know where an answer is lurking!
Website Name: The Biology Project - BioMath
Link: http://www.biology.arizona.edu/biomath/tutorials/quadratic/Roots.html
11) The point(s) where a parabola intersects the x-axis are called the __________________ or may also be called
_______________ or _________________.
12) When we are asked to "solve" a quadratic equation, we are really being asked to find the _____________________ of the parabola.
13) The greatest number of times a parabola can ever intersect the x-axis is ____________.
VI. Parabolas in the Real World
Quadratics are not just theoretical mathematical constructs. They actually can be found in many places in real life. This means that their equations can help us to solve problems involving situations in the real world. Use the links below to investigate several instances where parabolas appear in the world around us, then write a paragraph of at least 7 sentences describing one you find interesting. Remember to use your own words when you write.
i) This website describes real world occurrences of all the conic sections, of which the parabola is one. You will have to scroll
down the page a little to find the section on Parabolas.
Website Name: Occurrences of the Conics
Link: http://britton.disted.camosun.bc.ca/jbconics.htm
ii) This video is very informative about many practical applications of the parabola to technology.
Website Name: Youtube.com
Link: http://www.youtube.com/watch?v=Djnwlj6OG9k
iii) This website studies the parabola as a geometric shape and investigates instances where parabolas are found in nature.
Website Name: Keymath.com
Link: http://www.keymath.com/x6928.xml
14) Response Essay: ___________________________________________________________________________________________________________________________
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