Understanding Parabolas: Vertex, Focus, and Directrix

The video tutorial explains how to find the vertex, focus, and directrix of a parabola given by the equation y^2 - 6y + 8x + 41 = 0. It involves rewriting the equation in standard form, identifying the vertex and focus, and determining the directrix. The tutorial also includes sketching the parabola based on these components.

What is the first step in finding the vertex, focus, and directrix of a parabola?

• Determine the direction of opening

• Find the x-intercepts

• Identify the axis of symmetry

• Convert the equation to vertex form

Which form is used for a parabola that contains y^2 and not x^2?

• y^2 = 4px

• x^2 = 4py

• x = ay^2 + by + c

• y = ax^2 + bx + c

What is the purpose of completing the square in the context of parabolas?

• To find the x-intercepts

• To calculate the y-intercepts

• To determine the axis of symmetry

• To rewrite the equation in standard form

What is the vertex of the parabola given by the equation y^2 - 6y + 8x + 41 = 0?

• (-4, 3)

• (4, -3)

• (3, -4)

• (-3, 4)

How do you determine the direction in which a parabola opens?

• By the value of the vertex

• By the sign of p

• By the sign of the coefficient of y

• By the sign of the coefficient of x

What is the distance from the vertex to the focus of the parabola?

• Equal to half the value of p

• Equal to twice the value of p

• Equal to the value of p

• Equal to the absolute value of p

If the vertex of a parabola is at (-4, 3) and p = -2, where is the focus located?

• (-4, 1)

• (-6, 3)

• (-2, 3)

• (-4, 5)

What is the equation of the directrix for the parabola with vertex (-4, 3) and p = -2?

• y = -3

• x = -2

• x = -6

• y = 3

How is the directrix related to the vertex of a parabola?

• It is parallel to the axis of symmetry

• It is perpendicular to the axis of symmetry

• It is a horizontal line

• It is a diagonal line

What does the standard form of a parabola help to identify?

• The vertex, focus, and directrix

• The x-intercept

• The slope of the parabola

• The y-intercept