Thursday, December 6, 2012

Transformations and the Circle Family

Today in class, we talked about the unit circle.

unit circle: circle with a radius of one








Inside the unit circle is a right triangle. The radius of the circle is one, so you can see that the hypotenuse is one. Because this is a right triangle, you can use the Pythagorean Theorem to derive a formula for the equation. Plug in x and y as a and b, and then use r (radius), in this case 1, as c.
This gives you x^2 + y^2 = 1^2.


  1. Triangle (0,0)(1,0)(0,1) is a right triangle.
  2. segment (0,0)(0,1) is the hypotenuse of the triangle.
  3. hypotenuse=c
  4. a^2 + b^2 = c^2 (Pythagorean Theorem)
  5. a=x, b=y, c=1
  6. x^2 + y^2 = 1^2
  7. the hypotenuse is also the radius of the circle.


Basically, you are looking at distance from the center. If you have the distance from your x-coordinate to the center, and your distance from your y-coordiante to the center, you can construct a stair step from the center of your circle to your point. It's not about the point, it's about how you get there.

Ex. start at the origin (0,0) (this is your center)
      go up 3 and right 2 (this is your point)

      distance in x's = 2
      distance in y's = 3

      the 2 and 3 are the legs of your triangle.
      The hypotenuse will be the radius of the circle.

another example: start at (3,4) (this is your center)
                            go down 2 and right 4 (this is your point)
                         
                            distance in x's = 4
                            distance in y's = 2
                       
                            4 and 2 are the legs of your triangle.
                            the hypotenuse will be the radius of the circle.



Sometimes this x^2+y^2=r^2 (r= radius) is hard to wrap your brain around, because it's not really a nice "equation" for a graph. In other words, there is no neat "input, output" formula. This does not tell you how to take an x-value and spit out a coordinate.
Instead of thinking of this as a formula, another way to look at it is simply as a true statement about the
graph of the unit circle. It doesn't tell you how to get y, but it is a true statement about this graph.

To put the formula into "y=" form.......
x^2 + y^2 = r^2
subtract x^2
y^2 = r^2 - x^2
take the square root
y = sqrt(r^2 - x^2)

*remember...each square has a (+) and (-) square root. If you only have the (+), you will get a semicircle
*also you cannot do the sqrt of r^2 and the sqrt of x^2 separately



Ellipses
to make an ellipse, stretch or shrink a circle into an oval.
*stretch or shrink before you translate

ex. go from a circle with a center at (0,0) to an ellipse with a "center" at (3,0)

  1. x^2 + y^2 = 1
  2. (x/4)^2 + y^2 = 1   stretched along x-axis at a scale factor of 4
  3. [(x-3)/4]^2 + y^2 = 1   stretched along x-axis and translated three to the right

circle at (0,0)



ellipse at (3,0)


------------------------------------->
                                            










Here is a link to an interactive unit circle. This could be helpful in visualizing the right triangles from each of the infinite points on the circle
The page does have information on sin, cosine etc. but you can ignore that and just use the circle. 




3 comments:

  1. Hello, i was wondering if someone could go over the difference between translation, reflection, stretch and shrink. I struggled with determining each one on the homework. Thanks.

    ReplyDelete
  2. Translation:
    A translation moves a graph left or right, up or down, or both

    Example:
    y=x^2+2 moves the graph up 2
    y=(x-3)^2 moves the graph to the right 3
    y=(x+5)^2-1 moves the graph to the left 5 and down one

    Reflection:
    A reflection is a transformation that flips a graph across an axis, creating a mirror image.

    Example:
    Parent function- y=sqrt x

    When you negate the entire equation, the graph reflects along the y axis or across the x axis y= - sqrt x

    When you only make the x negative in an equation, it reflects the graph along the x axis, or across the y axis y= sqrt - x

    Stretching and Shrinking
    A stretch or a shrink is a transformation that expands or compresses a graph either horizontally or vertically

    Examples:
    Parent function= abs value x
    y= 2 IxI stretches the graph along the y axis or shrinks along the x by a scale factor of 2
    y=.5 IxI stretches the graph along the x axis or shrinks along the y by a scale factor of .5
    y= Ix/3I stretches the graph along the x axis or shrinks along the y by a scale factor of 3
    y=Ix/.25I stretches the graph along the y axis or shrinks along the x by a scale factor of .25

    Hope that helps

    ReplyDelete
  3. Im just generally confused about how to make an Ellipse.. What is the general equation for an ellipse? How are ellipses graphed on graphing calculator? Are they translated like normal functions that we've done in the past? And what kind of situations do ellipses represent? Thanks.

    ReplyDelete

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