Nate Hansen 01.07.13. & 01.08.13. Blog Post.

Hey guys, so I am blogging for Monday and Tuesday of this week.


Monday: Today we mainly went over the homework that was assigned from Friday {p. 135 #s 1-6, 11, 15, 21, 27-32}. The homework assignment was focused on identifying x-axis symmetry, y-axis symmetry, or symmetry with respect to the origin. Here are some examples:

 

URL: http://tutorial.math.lamar.edu/Classes/Alg/Symmetry_files/image003.gif

Symmetric with respect to the x-axis

As you will remember, an easy visual way to test for x-axis symmetry is to imagine folding the graph along the x-axis. If the parts above and below the x-axis, coincide, the graph is symmetric with respect to the x-axis.

The algebraic test for x-axis symmetry is as follows: replace (y) with (-y). If it comes out to an equivalent equation, then the graph is symmetric to the x-axis.

 

 

URL: http://03.edu-cdn.com/files/static/mcgrawhill-images/9780071439275/f0056-02.jpg

Symmetric with respect to the y-axis

As with the x-axis; in order to find out whether or not a graph represents y-axis symmetry, you can mentally picture folding the graph along the y-axis. If the components of the graph to the right and left of the y-axis match up, the graph is symmetric to the y-axis. 

The algebraic test for y-axis symmetry is almost exactly like the test for x-axis symmetry: replace (x) with (-x). If it comes out to an equivalent equation, then the graph is symmetric to the y-axis.

Remember: If the graph of a function f is symmetric with respect to the y-axis, it is called an even function. Mathematically speaking, for each x in the domain of f; f(x) = f(-x).

 

URL: http://www.uiowa.edu/~examserv/mathmatters/tutorial_quiz/geometry/typesofsymmetry_clip_image029.gif 

 Symmetric with respect to the origin

In order to easily determine whether a graph represents symmetry with respect to the origin, mentally rotate the graph 180˚ about the origin. The resulting figure should coincide with the original graph.

The algebraic method to test for symmetry around the origin is to first replace (x) with (-x) and (y) with (-y). After this, and it comes out to an equivalent equation as your original equation, then the graph is symmetric with respect to the origin.

Remember: If a graph of a function f is symmetric about the origin, we say that it is an odd function. Mathematically speaking, for each x in the domain of f; f(-x) = -f(x).

I'll do problem #17 on page 135 to illustrate the points made above.

Test algebraically whether the graph is symmetric with respect to the x-axis, the y-axis, and the origin. Then check your work graphically, if possible, using a grapher.

17. 3x² – 2y² = 3

This is your baseline equation to which you should compare other equations when you have inserted either (-x), (-y), or both.

Plug in (-y) and simplify: 3x² - 2(-y)²  = 3

3x² – 2y² = 3

This is the same as the baseline equation, so it is symmetric with respect to the x-axis.

Next plug in (-x) and simplify to see if the graph is also symmetric about the y-xis: 3(-x)² – 2y² = 3

3x² – 2y² = 3

This is also equivalent to the original equation, so the graph is also symmetric with respect to the y-axis.

Finally plug in both (-y) and (-x) and simplify to test for symmetry around the origin: 3(-x)² – 2y² = 3

3x² – 2y² = 3

This too is the same as the initial equation so the graph is symmetric to the origin as well as to the x- and y- axes.

All of the material for Monday's class that I blogged about either came from notes that I took from the board, or from pages 122-127 in our Precalculus textbook. Also see the homework assignment mentioned in the first part of this post. If you are still having trouble understanding the concepts of different symmetries and the even vs odd idea, then check out this Khan Academy video, it is pretty helpful: 

https://www.khanacademy.org/math/trigonometry/functions_and_graphs/analyzing_functions/v/recognizing-odd-and-even-functions

That essentially sums up class on Monday: mostly reviewing homework and concepts that we studied during the previous Friday's class. Now we'll move on to talk about Tuesday. 

Tuesday: Today we reviewed homework from the previous night {p.135 # 31-44, and #7 on worksheet: a,b,c,},  and learned more about the different transformations that a function can undergo and how to recognize such a transformation. Go to page 134 in our book to find the following helpful chart:

A

Again, you can find the above chart on page 134 of the textbook.

Using the knowledge from the above table, I will now show problem #63 on page 136.

Write an equation for a function that has a graph with the given characteristics. Check your answer using a grapher. 

63. The shape of y = 1/x but shrunk vertically by a factor of ½ and shifted down 3 units

First put the equation in to function form as follows: f(x) = 1/x

Next multiply the side of the equation with “1/x” by 2 in order to shrink the function. Refer back to the chart above if you need help remembering other transformations.

Your function should now look like this: f(x) = 1/2x

Finally, subtract 3 from the “1/x” side of the equation to vertically shift the function downwards three units.

Your final function will look like: f(x) = 1/2x -3

I will do one more problem, problem #65 on page 136. The instructions are the same as problem # 63

65. The shape of y = x², but upside-down and shifted right 3 units and up 4 units

As with the previous problem, convert the equation into function form: f(x) = x²

Make the x term negative to flip the function across the x-axis

The function will look like: f(x) = -x²

Now, in order to account for shifting the function right 3 units, subtract three from the x term.

The function will now look like: f(x) = -(x – 3)²

Finally, add 4 to the right side of the equation to vertically shift the function up four units

This is the final form of the transformed function: f(x) = -(x – 3)² + 4

That's it for my blog post. Hope it helps clear some stuff up. Thanks. 

Nate