**Chapter 6 – Section 1: Graphing Parabolas **

When graphing a parabola, fol low the following steps :

A. Find the vertex, focus, directrix, and the direction of the parabola.

B. Find the endpoints of the focal chord by adding and subtracting 2c to either
the x- value or the y -

value of the focus depending on the direction of the parabola.

C. Plot and label the vertex, focus, and the endpoints of the focal chord in
point form, draw the curve of

the parabola with a solid line , draw the directrix with a dashed line, and label
the directrix with its

equation.

**Example**: Graph

A. The vertex is (0, 0) and the standard form is x^{2} = 4cy.
Therefore,1/2=4c and .c =1/8 Because x is the

squared term and c is positive, the parabola opens up. The focus must therefore
be on the y-axis

above (0, 0). Since c=1/8 the focus must beSince
the parabola opens up, the directrix must

be parallel to the x-axis and below (0, 0). Therefore the equation of the
directrix is y = -1/8

B. If c = 1/8 then Since the parabola
opens up,1/4 is added and subtracted from the x value

of the focus while the y value is unchanged. Therefore, the endpoints of the
focal chord are

**Example**: Graph y^{2} = -5x

A. The vertex is (0, 0), and the standard form is y^{2}
= 4cx. Therefore, -5 = 4c and . c = -5/4

Because y is the squared term and c is negative , the parabola opens left. The
focus must therefore be on the x-axis

left of (0, 0). Since c =-5/4 the focus must beSince
the parabola opens left, the directrix

must be parallel to the y-axis and right of (0, 0). Therefore the equation of
the directrix is x = 5/4

B. If c =-5/4 then Since the parabola opens
left, -5/2 is added and subtracted from the y

value of the focus while the x value is unchanged. Therefore, the endpoints of
the focal chord are

**Exercises for Graphing Parabolas**

Graph the following parabolas. Label the vertex, focus, and endpoints of the
focal chord. Draw the directrix

and label with its equation.

**Chapter 6 – Section 1**: Additional Application
Problems

**5. Translate into a mathematical model and solve:**

The diameter of a parabolic flashlight is 4 centimeters. The focal length
(distance from the focus to the

parabolic surface) is 0.5 centimeters. De termine the maximum depth of the
flashlight. Round the answer

to the nearest centimeter. **Include correct units.**

a. Equation of parabola:_____________________

b. Depth:_____________

**6. Translate into a mathematical model and solve:**

A parabolic satellite TV receiving dish will have a receiver placed at the
focus. Where should the

receiver be placed if the dish is 3 feet in diameter and has a depth of 1 foot?
Round the answer to 2

decimal places .

a. Equation of parabola:___________________

b. The receiver should be placed ____ feet from the
vertex.

**Chapter 6, Section 2** – Additional Application
Problem

**7. Translate into a mathematical model and solve:**

An arch of a bridge has the shape of the top half of an ellipse . The arch is 40
feet wide and 12 feet high

at the center. Find the equation of the complete ellipse. Find the height of the
arch 10 feet from the

center at the bottom. **Round to the nearest hundredth.**

a. Equation of ellipse: ____________________

b. Height of arch:________________________** Include correct units.**

**Chapter 6, Section 2** – Finding the Equation of a
Hyperbola given vertices and asymptotes

Find the equation of the hyperbola with vertices (6, 0) and (–6, 0) and
asymptotes and.

Solution :

Since the vertices are (±6,0) , the x-axis must be the major axis, the center
must be (0, 0), the general

equation to be used must be , and a must be
6. Therefore, the equation
can be written

with confidence. Only b still needs to be found. The slope of the equation for
the asymptote of a hyperbola

is the change in y divided by the change in x. 6 is the change in x and b is the
change in y, so

Solving this proportion gives b = 4 , and the equation of the hyperbola is
or

Chapter 6, Section 2 – Additional Exercise for Finding Hyperbola Equations

8. Find the equation of the hyperbola with vertices (–4, 0) and (4, 0) and
asymptotesand

**Chapter 6, Section 2 **– Graphing Hyperbolas

When graphing a hyperbola, follow the following steps:

A. Determine the major axis, the center, and the vertices. Plot and label the
vertices.

B. Determine helpful points on the other axis and use these points and the
vertices to draw a box around

the center of the hyperbola.

C. Draw the asymptotes through the corners of the box using dashed lines.

D. Find the foci by using c^{2} =a^{2} +b^{2} to find c.
Add and subtract c to either the x or the y value of the

center of the hyperbola depending on the direction the hyperbola opens.

**Example**:

A. Because the x^{2} is positive, the major axis is the x-axis, the

standard form of the equation is
, and the

center is (0, 0). Since Since the major axis

is the x-axis, the vertices are found by adding and

subtracting a to the x value of the center giving

( , 0) and (-
, 0).

Plot and label the vertices.

B. Helpful points on the y-axis are found by adding and

subtracting b to the y value of the center. Since b^{2} = 9,

b = 3, and the helpful points are (0, 3) and (0, –3).

Plot the helpful points and draw a box around the center

using the vertices and these points. These helpful points

do not have to be labeled but certainly can be.

C. Draw the asymptotes through the corners of the box
using

dashed lines.

D. Since c^{2} = a^{2} + b^{2} , c^{2}
= 2 + 9 = 11, and c =
. Because

the major axis is the x-axis, c is added and subtracted to

the x value of the center. Therefore the foci are (
, 0)

and (- , 0).

Plot and label the foci. Then draw in the hyperbola

making sure that it approaches the asymptotes but does

not touch the asymptotes or look as if it will cross the

asymptote.

**Example**:

A. Because the y^{2} is positive, the major axis is the y-axis, the
standard form of the equation is

, and the center is
(0, 0). Since a^{2} = 9, a = 3. Since the major axis is the y-axis, the

vertices are found by adding and subtracting a to the y value of the center
giving (0, 3) and (0, –3).

Plot and label the vertices.

B. Helpful points on the x-axis are found by adding and subtracting b to the x
value of the center. Since

b^{2} = 16, b = 4, and the helpful points are (4, 0) and (–4, 0).

Plot the helpful points and draw a box around the center using the vertices and
these points. These

helpful points do not have to be labeled on the graph but certainly can be.

C. Draw the asymptotes through the corners of the box using dashed lines.

D. Since c^{2} = a^{2} + b^{2} , c^{2}
= 9 +16 = 25, and c = 5. Because

the major axis is the y-axis, c is added and subtracted to

the y value of the center. Therefore the foci are (0, 5)

and (0, –5).

Plot and label the foci. Then draw in the hyperbola

making sure that it approaches the asymptotes but does

not touch the asymptotes or look as if it will cross the

asymptote.

**Chapter 6, Section 2** – Exercises for Graphing
Hyperbolas

Graph each of the following. Label the vertices, and the foci. Graph must
approach directly drawn

asymptotes, if applicable.

**Answers for Chapter 6, Section 1 & 2 Exercises**

5. a. Equation: x^{2} = 2 y or y^{2} = 2x;
either is correct b. 2 centimeters

6. a. Equation: or ;
either is correct. b. 0.56

7. a. Equation:or
either is correct. b. 10.39 feet

8.or