Lesson 6 Turn It Around Practice 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 $90$ 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 $x = 4$ and focus $(0, 3)$ .

5.

One more for good measure: Write the equation of the parabola with directrix $x = - 3$ and focus $(- 2, - 5)$ .

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.

a.

Are all parabolas quadratic functions?

b.

Are all parabolas similar?

Ready for More?

Parabolas can have a general form, just like circles. Here’s the equation of a parabola in general form: $y^{2} - 10 x + 6 y - 11 = 0$ . Your challenge is to find the vertex, the focus, and the directrix of the parabola.

Takeaways

Equation of a parabola with vertical directrix, vertex $(h, k)$ , and distance from the focus to the vertex, $p$ .

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 $A B + B C$ .

2.

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

The circle is tangent to the line $y = 4$ and the $x$ -axis.
The line $x = 2$ goes through the center of the circle.