# Lesson 6Turn It AroundPractice Understanding

## Learning Focus

Write equations for parabolas with vertical directrices.

Determine the direction of opening for any parabola.

Are all parabolas functions? Are all parabolas similar?

## Open Up the Math: Launch, Explore, Discuss

### 1.

Annika is thinking more about the geometric view of parabolas that she has been working on in math class. She thinks, “Now I see how all the parabolas that come from graphing quadratic functions could also come from a given focus and directrix. I notice that all the parabolas have opened up or down when the directrix is horizontal. I wonder what would happen if I rotated the focus and directrix degrees so that the directrix is vertical. How would that look? What would the equation be? Hmmm….” So Annika starts trying to construct a parabola with a vertical directrix. Here’s the beginning of her drawing. Use a ruler to complete Annika’s drawing.

### 2.

Use the definition of a parabola to write the equation of Annika’s parabola.

### 3.

What similarities do you see to the equations of parabolas that open up or down? What differences do you see?

### 4.

Try another one: Write the equation of the parabola with directrix and focus .

### 5.

One more for good measure: Write the equation of the parabola with directrix and focus .

### 6.

How can you predict if a parabola will open left, right, up, or down?

### 7.

How can you tell how wide or narrow a parabola appears?

### 8.

Annika has two big questions left. Write and explain your answers to these questions.

#### b.

Are all parabolas similar?

Parabolas can have a general form, just like circles. Here’s the equation of a parabola in general form: . Your challenge is to find the vertex, the focus, and the directrix of the parabola.

## Takeaways

Equation of a parabola with vertical directrix, vertex , and distance from the focus to the vertex, .

Parabola opens upward if the directrix is and the focus is .

Parabola opens downward if the directrix is and the focus is .

Parabola opens right if the directrix is and the focus is .

Parabola opens left if the directrix is and the focus is .

## Lesson Summary

In this lesson, we learned to work with parabolas that have a vertical directrix. We found how to determine if they opened left or right, and how to write an equation of the parabola given a focus and directrix.

## Retrieval

### 1.

Use the distance formula to find .

### 2.

Use the given information to write the equation of the circle in standard form.

• The circle is tangent to the line and the -axis.

• The line goes through the center of the circle.