17.1 Equation of a Circle Exit Ticket

Essential Question: What is another way to rewrite quadratic equations that are given to us in the form [math]x^2+ax=b[/math]?[br][br]Slide the [math]a[/math] and [math]b[/math] sliders to change your original quadratic to whatever you like and carefully take notice of how the equations change.

Geometric Visualization of the Method for Completing the Square on a Quadratic Equation

1.

What's the general expression for the area of the larger blue region in the above applet?

[math]ax[/math] [math]x^2[/math] [math]b[/math]

2.

What was done to the larger blue area on the left to get the two smaller blue areas on the right?

It was randomly split and arranged in some other fashion It was split into two equal pieces but with one dimension now half of what it used to be or [math]\frac{a}{2}[/math]

3.

What is the area of the yellow region and how is it related to the larger blue region?

Depends on how you split the original larger blue region [math]\left(\frac{a}{2}\right)^2[/math], which makes it the square of half the numerical part of the larger blue region. Oh, it's not related

4.

So uh, why is this process called COMPLETING THE SQUARE?

It just is! There doesn't always have to be reasons for stuff man! Well, adding in the little yellow area COMPLETES THE SQUARE. That is, we started with a rectangle and now we have a square. Bam!!

5. Complete the square on the following quadratic equation

[math]x^2+4x=23[/math]

6. Complete the square on the following quadratic equation

[math]y^2-6y=36[/math]

7. Complete the square on the following quadratic equation

[math]x^2-8x=45[/math]

8. Complete the square on the following quadratic equation

[math]y^2+12y=100[/math]

9. Complete the square on the following "double" quadratic equation - the equation of a circle. Then write the coordinates of its center and its radius.

[math]x^2+2x+y^2-8y+13=0[/math]

10. Type the original equation from (9) into the input bar to graph the circle and see if the center and radius you found are correct. Use the given tools to determine the center and radius of the circle.

11. Complete the square on the following "double" quadratic equation - the equation of a circle. Then write the coordinates of its center and its radius.

[math]x^2-6x+y^2-12y+20=0[/math]

12. Type the original equation from (11) into the input bar to graph the circle and see if the center and radius you found are correct. Use the given tools to determine the center and radius of the circle.