Properties and Shifts of Parabolas

The video tutorial explains the properties of parabolas, focusing on their vertex form and how to shift them. It covers the concept of focal length and how to determine the focus and directrix. The tutorial also demonstrates how to shift the focus and directrix when the parabola is moved from the origin, using examples to illustrate these transformations.

What is a common property of all parabolas in the basic form discussed?

• They all open to the left.

• They all have a vertex at the origin.

• They all pass through the point (1,1).

• They all have a focal length of 10.

How can you move the vertex of a parabola from the origin?

• By introducing parameters h and k.

• By changing the focal length.

• By rotating the parabola.

• By changing the coefficients of x and y.

What remains constant when a parabola is shifted?

• The focal length.

• The directrix position.

• The vertex position.

• The direction it opens.

In the example x^2 = 32y, what is the focal length?

• 8

• 4

• 16

• 32

When shifting the parabola x^2 = 32y to x-3)^2 = 32(y+1), how is the focus affected?

• It moves 3 units left and 1 unit up.

• It moves 3 units right and 1 unit up.

• It moves 3 units right and 1 unit down.

• It remains at the origin.

What happens to the directrix when the parabola is shifted?

• It moves vertically only.

• It moves horizontally only.

• It remains unchanged.

• It moves both horizontally and vertically.

What is the new directrix of the parabola after shifting x^2 = 32y to (x-3)^2 = 32(y+1)?

• y = 0

• y = -9

• y = -8

• y = 1

How do you determine the new focus after shifting a parabola?

• By rotating the parabola.

• By using the original focus and adding the shift values.

• By finding the midpoint between the vertex and the directrix.

• By calculating the new vertex and adding the focal length.

What is the importance of the focal length in a parabola?

• It changes with the vertex position.

• It remains constant and helps locate the focus.

• It affects the direction the parabola opens.

• It determines the width of the parabola.

What should you do first when shifting a parabola?

• Identify the new vertex coordinates.

• Change the focal length.

• Change the coefficients of x and y.

• Rotate the parabola.