Lesson: Properties of Parabolas

In this lesson, we will learn how to identify the properties of the parabola using graphical representation and the function rule.

Sample Question Videos

  • 00:56

Worksheet: 18 Questions • 1 Video

Q1:

The figure shows the curve ( 𝑦 + 𝑥 3 ) + 4 ( 𝑦 𝑥 ) + 4 = 0 2 , the dashed line 𝑦 + 𝑥 = 3 , and its perpendicular 𝑦 𝑥 = 5 .

Determine the coordinates of the two intersections 𝐴 and 𝐵 .

The vertex of this parabola lies where the line 𝑦 𝑥 = 𝐾 meets the parabola at exactly one point.

What is the value of 𝐾 ? What are the coordinates of the vertex?

The vertex also lies on the line of symmetry 𝑦 + 𝑥 = 3 , which is present in the equation. It turns out that the equation ( 𝑦 + 𝑥 3 ) 8 ( 𝑦 + 𝑥 3 ) + 4 ( 𝑦 𝑥 ) + 2 = 0 2 gives a parabola whose axis of symmetry is parallel to 𝑦 + 𝑥 = 3 .

By completing the square and rewriting this as ( 𝑦 + 𝑥 𝐴 ) + 4 ( 𝑦 𝑥 ) + 𝐵 = 0 2 , determine what the new axis of symmetry is. What is the value of the constant 𝐵 ?

By determining 𝐴 , 𝐵 such that 𝑦 5 𝑥 = 𝐴 ( 𝑦 + 𝑥 ) + 𝐵 ( 𝑦 𝑥 ) , complete the squares and rewrite ( 𝑦 + 𝑥 ) + 𝑦 5 𝑥 + 7 2 as ( 𝑦 + 𝑥 𝑎 ) + 𝑏 ( 𝑦 𝑥 ) + 𝑐 2 . What is this expression? What is the vertex of the parabola?

Q2:

Consider the parabola with Cartesian equation 𝑦 = 2 1 7 𝑥 .

What are the coordinates of its focus?

Write the equation of its directrix.

Q3:

A parabola has the equation 𝑦 = 7 𝑥 2 .

What are the coordinates of its focus?

Write an equation for its directrix.

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