Lesson 5 Functioning with Parabolas Solidify Understanding

Ready

Find a square piece of paper. (Patty paper works best, but a sticky note would work, too.)

Fold the square 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 different points along the bottom edge. (If you can’t see your fold lines, each time you fold the bottom of your paper up to the dot, put a mark on the fold.)

The fold lines between the focus and the edge should make a parabola.

Experiment with a new paper and move the focus.

Use your experiments to answer the following questions.

1.

How would the parabola change if the focus were moved up, away from the directrix?

2.

How would the parabola change if the focus were moved down, toward the directrix?

3.

How would the parabola change if the focus were moved down, below the directrix?

4.

How would the parabola look if the directrix were a vertical line?

5.

Place a dot representing the focus in an approximate position relative to this curve.

Set

6.

Verify that $(y - 1) = \frac{1}{4} x^{2}$ is the equation of the parabola in figure 1 by substituting in the three points $V (0, 1)$ , $C (4, 5)$ , and $E (2, 2)$ .

Show your work for each point.

7.

If you didn’t know that $(0, 1)$ was the vertex of the parabola, could you have found it by just looking at the equation? Explain.

8.

Use the diagram to derive the equation of a parabola based on the geometric definition of a parabola. Remember that the definition states that $F P = P Q$ .

9.

Recall the equation in problem 6, $(y - 1) = \frac{1}{4} x^{2}$ . What is the value of $p$ ?

10.

In general, what is the value of $p$ for any parabola?

11.

In figure 3, the point $M$ is the same height as the focus and $\overset{―}{F M} ≅ \overset{―}{M R}$ . How do the coordinates of this point compare with the coordinates of the focus?

12.

Fill in the missing coordinates for $M$ and $R$ in the diagram.

Sketch the graph by finding the vertex and the points $M$ and $R$ (the reflection of $M$ ) as defined in the previous diagram. Use the geometric definition of a parabola to find the equation of these parabolas.

13.

Directrix $y = 9$ , Focus $F (- 3, 7)$

Vertex:

Equation:

14.

Directrix $y = - 6$ , Focus $F (2, - 2)$

Vertex:

Equation:

15.

Directrix $y = 5$ , Focus $F (- 4, - 1)$

Vertex:

Equation:

16.

Directrix $y = - 1$ , Focus $F (4, - 3)$

Vertex:

Equation:

Go

Find the maximum or minimum value of each quadratic function by completing the square on the expression that defines it.

17.

$f (x) = x^{2} + 6 x - 5$

18.

$f (x) = 3 x^{2} - 12 x + 8$

19.

$f (x) = - \frac{1}{2} x^{2} + 10 x + 13$

20.

$f (x) = - 5 x^{2} + 20 x - 11$