Circles & Parabolas

Circles & Parabolas

Dr. Tom Giles

Multiple Select

Check all of the graphs that are quadratics. (Hint: think of the only directionality types we can have)

Multiple Choice

• a

• b

• c

• d

Multiple Choice

• a. (-6, 1)

• b. (6, 1)

• c. (-6, -1)

• d. (6, -1)

Multiple Choice

• a. (5, 12)

• b. (5, -12)

• c. (-5, 12)

• d. (-5, -12)

From the Equation

• Make the signs on h and k OPPOSITE to get the center

• Take the SQUARE ROOT of r² to get the radius

• If you don't SEE a h or k, then that means the coordinate is ZERO.

Multiple Select

Which equations are in vertex form?

• $$y\ =\ 3x^2-x+10$$

• $$y=\left(x+2\right)^2-4$$

• $$y\ =\ \left(x-1\right)\left(x+2\right)$$

• $$y=\frac{1}{4}\left(x-3\right)^2+15$$

• $$y=12\ +6x^2$$

Multiple Choice

Which equation has a vertex of (4, 2)?

• $$y=\left(x+2\right)^2-4$$

• $$y=\left(x-2\right)^2+4$$

• $$y=\left(x-4\right)^2+2$$

• $$y=\left(x+4\right)^2-2$$

Multiple Choice

Which graph could match the equation below?

$$y=2\left(x+3\right)^2+3$$  

Multiple Choice

Write the equation of the circle with center (-4, 0) and a diameter of 10.

• $$\left(x-4\right)^2+y^2=100$$  

• $$\left(x+4\right)^2+y^2=100$$  
  

• $$\left(x+4\right)^2+y^2=25$$  

• $$x^2+\left(y+4\right)^2=25$$  

• $$\left(x-4\right)^2+y^2=25$$  

Open Ended

What was different about the equations in the last slide that caused the parabolas to look different?

Multiple Choice

Write the equation of the circle with center (2, -9) and a diameter of $$\sqrt{11}$$  .

• $$\left(x+2\right)^2+\left(y-9\right)^2=11$$  

• $$\left(x-2\right)^2+\left(y+9\right)^2=11$$  

• $$\left(x+9\right)^2+\left(y-2\right)^2=11$$  

• $$\left(x-9\right)^2+\left(y+2\right)^2=11$$  

Multiple Choice

$$\left(x+3\right)^2+\left(y+7\right)^2=4$$

What is the radius and center of the circle

• r = 4,  (3, 7)
  

• r = 4, (-3, -7)
  

• r = 2, (3, 7)
  

• r = 2, (-3, -7)
  

Multiple Choice

Find the equation of the circle.

• $$x^2+y^2=12$$

• $$x^2+y^2=144$$

• $$x^2+y^2=36$$

• $$x^2+y^2=576$$

Multiple Choice

Which of the following is the equation of a circle with a center at (-1,3) and a radius of 4?

• (x-1)²+(y+3)²=4

• (x-1)²+(y+3)²=16

• (x+1)²+(y-3)²=4

• (x+1)²+(y-3)²=16

Multiple Choice

Which is the correct equation for the circle?

• (x-3)²-(y+1)²=8

• (x-3)²+(y+1)²=8

• (x+3)²+(y-1)²=16

• (x-3)²-(y+1)²=16

• (x-3)²+(y+1)²=16

Multiple Choice

Which is the correct equation for the circle?

• (x+5)²+(y+2)²=9

• (x-2)²+(y-5)²=9

• (x+2)²+(y+5)²=9

• (x+2)²+(y+5)²=3

• (x-2)²+(y-5)²=6

Multiple Choice

What is the equation of this circle?

• (x - 2)² + (y - 1)² = 4

• (x + 2)²+(y + 1)²=16

• (x + 2)² + (y - 1)² = 4

• (x + 2)²+(y - 1)²=16

Multiple Choice

Write the equation of the circle with center C( -5, 8 ) and radius = 7.

• ( x - 5 )² + ( y + 8 )² = 14

• ( x - 5 )² + ( y - 8 )² = 49

• ( x - 5 )² + ( y + 8 )² = 49

• ( x + 5 )² + ( y - 8 )² = 49

Multiple Choice

Identify the center and radius of a circle with the following equation: $$\left(x+7\right)^2+\left(y-5\right)^2=16$$  

• Center (7, -5) & Radius = 16

• Center (-7, 5) & Radius = 8

• Center (7, -5) & Radius = 4

• Center (-7, 5) & Radius = 4

Multiple Choice

Identify the center and radius of a circle with the following equation: $$\left(x+5\right)^2+\left(y+2\right)^2=48$$  

• Center (5, 2) & Radius = 24

• Center (-5, -2) & Radius  $$\approx$$  6.93

• Center (5, 2) & Radius = 48

• Center (-5, -2) & Radius = 24

Multiple Choice

Which is the correct standard equation of a circle with the given information? Center: (-3, 2) and Radius = 16

• $$\left(x-3\right)^2+\left(y+2\right)^2=256$$

• $$\left(x+3\right)^2+\left(y-2\right)^2=4$$

• $$\left(x+3\right)^2+\left(y-2\right)^2=256$$

• $$\left(x-3\right)^2+\left(y+2\right)^2=16$$

Multiple Choice

Which is the correct standard equation of a circle with the given information? Center (7, -2) and passes through (-1, -6)

• $$\left(x-7\right)^2+\left(y+2\right)^2=6,400$$

• $$\left(x+7\right)^2+\left(y-2\right)^2=80$$

• $$\left(x+1\right)^2+\left(y+6\right)^2=6,400$$

• $$\left(x-7\right)^2+\left(y+2\right)^2=80$$

Multiple Choice

A circle has a center at (1, 1). The diameter is 6. Which of the following points is a point on the circle?

• $$\left(0,\ -1\right)$$

• $$\left(-1,\ 3\right)$$

• $$\left(4,\ 1\right)$$

• $$\left(3,\ 3\right)$$

Multiple Choice

Which point does NOT lie on the following circle? $$\left(x-1\right)^2+\left(y+2\right)^2=10$$  

• (2, 1)

• (4, -3)

• (1, 3)

• (0, -5)

Multiple Select

The equation of a circle is represented by  $$x^2+y^2\ =\ 100$$  
Select all the points below that are on the circle. 

• $$\left(5,\ 8\right)$$  

• $$\left(8,\ 6\right)$$  

• $$\left(10,\ 0\right)$$  

• $$\left(6,\ -8\right)$$  

• $$\left(3,\ -9\right)$$  

Multiple Select

A circle is modeled by the equation $$\left(x-3\right)^2+\left(y\ +3\right)^2\ =\ 25$$  .
Which of the following are also points on the circle. 

• $$\left(7,\ 0\right)$$  

• $$\left(5,\ 2\right)$$  

• $$\left(-1,\ 0\right)$$  

• $$\left(0,\ -7\right)$$  

• $$\left(3,\ -8\right)$$  

Multiple Choice

A circle has center $$\left(4,\ -2\right)$$  and a diameter of 14. Which of the following points is also a point on the circle.

• $$\left(8,\ 4\right)$$  

• $$\left(11,\ -2\right)$$  

• $$\left(7,\ -8\right)$$  

• $$\left(-3,\ -3\right)$$