Algebra III
Lesson 42
Conic Sections – Circles – Constants in
Exponential Functions
Conic Sections
Imagine two cones, one sitting on a table point up, the other placed
such that the points meet and it is perfectly ‘straight up’ from the other
one.
Now picture a thin sheet of metal.
Hold it perfectly parallel to the
floor and imagine it passing
through the cones.
Now tilt one end a little.
Keep tilting the same end until
it comes out of the top of the
cone.
Keep tilting it until it is vertical and
cutting through the top and bottom
of the cone.
Circles
Definition: All the points in a plane that are equidistant from a
point called the center of the circle.
From the Pythagorean theorem:
x2 + y2 = r2
This is the standard form.
x2 + y2 - r2 = 0 This is the general form.
r
y
x
y
r
x
General form is all on one side and
multiplied out.
This is for a circle centered at the origin.
How about for a circle not centered at the origin.
Use translating graph skills
(x – 5)2 + (y + 5)2 = 72
Standard form
7
y
(5,-5)
x
y
7
x
Example 42.1
Find the standard form of the equation whose center is at (h,k) and
whose radius is r.
Basic equation for a circle is:
r
y
(h,k)
x
x2 + y2 = r2
Translate
y
r
x
(x – h)2 + (y – k)2 = r2
Example 42.2
Find the standard form of the equation of a circle of radius 4
whose center is at (3,-3). Then write the equation in general
form.
Start with the basic equation.
(x – h)2 + (y – k)2 = r2
Plug in the values.
(x – 3)2 + (y + 3)2 = 16
Standard form.
Multiply.
x2 – 6x + 9 + y2 + 6y + 9 = 16
Rearrange.
x2 + y2 – 6x + 6y + 2 = 0
General form.
Constants in Exponential Functions
Review: Exponential Functions
y = 3x
y = (⅓)x
New twist.
y = 32x
Technique for solving:
y = 4-x
Repackage
y = 32x
y = 4-x
y = (32)x
y = (4-1)x
y = 9x
y = (¼)x
y = (⅓)3x+2
Example 42.3
Sketch the graph of the function y = 3-x
Repackage
y = 3-x
y = (3-1)x
y = (⅓)x
Points for graphing
(0,1)
(-1,3)
Example 42.4
Sketch the graph of the function y = (½)2x
Repackage
y = (½)2x
y = (½2)x
y = (¼)x
Points for graphing
(0,1)
(-1,4)
Example 42.5
Sketch the graph of the function y = (½)-x-2
Repackage
y = (½)-x-2
y = (½)-x(½)-2
y = 4(2)x
Points for graphing
(0,4)
(1,8)
Practice
a) Sketch the graph of the function y = (½)x-2
Repackage
y = (½)x-2
y = (½)x(½)-2
y = 4(½)x
Points for graphing
(0,4)
(-1,8)
b) Find the standard form of the equation of a circle of radius 3
whose center is at (3,4).
Start with the basic equation.
(x – h)2 + (y – k)2 = r2
Plug in the values.
(x – 3)2 + (y - 4)2 = 9
Standard form.
c) Given the function f(x) = |x|, write the equation of the function g
whose graph is the graph of f translated two units to the right and three
units up.
Rewrite the function.
(y) = |(x)|
Put in the translations.
(y – 3) = |(x – 2)|
Rearrange.
y = |(x – 2)| + 3
Rewrite
g(x) = |x – 2| + 3
d) A runner can run 100 yards in 9.8 seconds and 440 yards in 49
seconds. what is the ratio of her average speed in the 440 to her
average speed in the 100?
Ratio – means fraction
r440 / r100
r = d/t
r440 = 440/49
= 8.98
r100 = 100/9.8
= 10.2
r440 / r100
= 8.98 / 10.2
= .88