Lesson 4 Directing Our Focus Develop Understanding

Ready

Graph each set of functions on the same coordinate axes. Describe in what way the graphs are the same and in what way they are different.

1.

$y = x^{2}$
$y = 2 x^{2}$
$y = 4 x^{2}$

2.

$y = \frac{1}{4} x^{2}$
$y = - x^{2}$
$y = - 4 x^{2}$

3.

$y = \frac{1}{4} x^{2}$
$y = x^{2} - 2$
$y = \frac{1}{4} x^{2} - 2$

4.

$y = x^{2}$
$y = - x^{2}$
$y = {(x + 2)}^{2}$
$y = - {(x + 2)}^{2}$

Set

The points $F (0, 1)$ , $D (0, - 1)$ , $(1, y_{1})$ , $(2, y_{2})$ , $(3, y_{3})$ , $(4, y_{4})$ , and the origin, $(0, 0)$ , have been identified on the diagram. In addition, the dotted lines $x = - 1$ , $x = 1$ , $x = 2$ , $x = 3$ , and $x = 4$ have been graphed to remind you that the points you are seeking lie somewhere on those lines. Let point $F$ be the focus and the line $y = - 1$ be the directrix.

5.

The geometric definition of a parabola states that the distance from the focus $F$ to a point $P$ that lies on the parabola and the distance from point $P$ to the directrix must be equal.

According to this definition, is the point $(0, 0)$ a point on the parabola that is being constructed in the diagram? Justify your answer.

6.

The points $G$ , $H$ , $I$ , and $J$ have not been labeled with ordered pairs in the diagram. Write the coordinates of $G$ , $H$ , $I$ , and $J$ .

7.

Use a straightedge to draw a line segment from $F$ to $(1, y_{1})$ . Then draw a vertical line segment from $(1, y_{1})$ to the line $y = - 1$ . What should the two line segments have in common?

8.

Write an equation that relates the lengths of the two line segments, and calculate the distance from $F$ to $(1, y_{1})$ and the distance from $(1, y_{1})$ to the line $y = - 1$ .

9.

What is the value of $y_{1}$ in the point identified as $(1, y_{1})$ ?

10.

What are the missing values of $y_{2}$ , $y_{3}$ , and $y_{4}$ in the ordered pairs shown on the diagram?

11.

Write the equation of this parabola for any value of using the geometric definition.

Use the point $F (0, 1)$ .

Let $P (x, y_{x})$ be any point of the parabola and $(y_{x} + 1)$ be the distance from $P (x, y_{x})$ to the directrix.

Go

Use completing the square to find the center $(h, k)$ and radius $r$ of the circle.

12.

$x^{2} + y^{2} + 4 y - 12 = 0$

13.

$x^{2} + y^{2} - 6 x - 3 = 0$

14.

$x^{2} + y^{2} + 8 x + 4 y - 5 = 0$

15.

$x^{2} + y^{2} - 6 x - 10 y - 2 = 0$

16.

$x^{2} + y^{2} - 6 y - 7 = 0$

17.

$x^{2} + y^{2} - 4 x + 8 y + 6 = 0$

18.

$x^{2} + y^{2} - 4 x + 6 y - 72 = 0$

19.

$x^{2} + y^{2} + 12 x + 6 y - 59 = 0$