Lesson 5 Functioning with Parabolas Solidify Understanding

Learning Focus

Write and graph parabolas.

Compare the geometric definition of parabolas with quadratic functions.

How do the parabolas generated with a focus and directrix relate to quadratic functions?

Open Up the Math: Launch, Explore, Discuss

Sketch the graph (accurately enough to have a couple of good points on either side of the line of symmetry). Find the vertex, and use the geometric definition of a parabola to find the equation of these parabolas.

1.

Directrix $y = - 4$ , Focus $A (2, - 2)$

Vertex:

Equation:

2.

Directrix $y = 2$ , Focus $A (- 1, 0)$

Vertex:

Equation:

3.

Directrix $y = 3$ , Focus $A (1, 7)$

Vertex:

Equation:

4.

Directrix $y = 3$ , Focus $A (2, - 1)$

Vertex:

Equation:

5.

Given the focus and directrix, how can you find the vertex of the parabola?

6.

Given the focus and directrix, how can you tell if the parabola opens up or down?

7.

How do you see the distance between the focus and the vertex (or the vertex and the directrix) showing up in the equations that you have written?

8.

Describe a pattern for writing the equation of a parabola given the focus and directrix.

9.

Annika wonders why we are suddenly thinking about parabolas in a completely different way than when we did quadratic functions. She wonders how these different ways of thinking match up. For instance, when we talked about quadratic functions earlier, we started with $y = x^{2}$ . “Hmmmm….I wonder where the focus and directrix would be on this function,” she thought. Help Annika find the focus and directrix for $y = x^{2}$ .

10.

Annika thinks, “OK, I can see that you can find the focus and directrix for a quadratic function, but what about these new parabolas. Are they quadratic functions? When we work with families of functions, they are defined by their rates of change. For instance, we can tell a linear function because it has a constant rate of change.” How would you answer Annika? Are these new parabolas quadratic functions? Justify your answer using several representations and the parabolas in problems 1–4 as examples.

Ready for More?

A parabola has a vertical axis of symmetry with vertex at $(1, 4)$ and focus at $(1, 2)$ . Find the equation of the parabola and the equation of the directrix.

Takeaways

Equation of a parabola with horizontal directrix, vertex $(h, k)$ , and distance $p$ between vertex and focus:

Lesson Summary

In this lesson, we learned that the graph of a quadratic function meets the definition of a parabola. We learned to write equations given the focus and directrix and to find the focus and directrix of the parabola when given the equation of a quadratic function.

Retrieval

1.

This is an activity that you can use to answer questions in the first part of your homework. Try to do it on your own. Your teacher will help you if you have trouble.

Fold a square piece of paper in half vertically, and put a dot anywhere on the fold.
Make several dots along the bottom edge of the paper. Let the bottom edge of the paper be the directrix and the dot be the focus.
Fold the bottom edge of the paper up so each dot on the bottom touches the dot in the vertical fold.
Make a crease each time you match a dot along the bottom to the focus dot.
Do this repeatedly from many different points along the edge. The fold lines between the focus and the edge should make a parabola.

2.

Find the maximum or minimum value of the quadratic equation. Indicate which it is.

$y = - x^{2} + 6 x - 2$