Transcript x - Math KSU

A graph is a diagram of a relationship of
(at least) two variables with changing
values.
The current value of each variable is
represented as a distance from the origin.
The coordination of the variables is
represented as a point on the graph
All points on the graph are equally
important.
A graph can just be a set of discrete points
{(0,1), (1.5,3), (2.3,π), (5,0.7), (5+π,6), (9,2)}
Or a shape…
The equation of a graph
• The algebraic relationship between variables
on a graph
• The equation is test: If you pick a pair of
values that makes the equation true, that
point is on the graph. If you pick a pair that is
not true, the point is not on the graph.
• Graphing is finding all the points that make
the equation true, and changing their color.
How would we find the equation of this graph?
Tool: The distance between two points
Tool: The distance between two points
Horizontal distance between points is x-a
Vertical distance between points is y-b
We have a right triangle.
Pythagorean Theorem
(x - a) + (y - b) = ?
2
2
2
(x - a) + (y - b) = ?
2
2
Distance Formula
• The distance between (a,b) and (x,y) is
(x - a) + (y - b)
2
2
Back to a circle problem
Circle
• A circle is the set of all points a given distance
(the radius) from a given point (the center).
Our circle is all the points 4 units away from (1,2)
All points 4 units away from (1,2)
• For any point (x,y)
• The distance between (1,2) and (x,y) is 4
• Using the Pythagorean Theorem…
(x -1) + ( y - 2) = 4
2
• Is the equation of the circle
2
2
Equation of a circle
• For radius r and center (a,b)
(x - a) + (y - b) = r
2
2
2
Consider a circle with equation
x2 + (y+4)2 = 289. The center and radius are given by:
A)
B)
C)
D)
E)
(0,4), radius=289
(0,-4), radius=289
(4,0), radius=289
(-4,0), radius=289
None of the above.
Consider a circle with equation
x2 + (y+4)2 = 289. The center and radius are given by:
(x - 0) + (y - -4) = (17)
2
2
2
center (0, -4)
radius 17
E
Midpoint formula
• The point halfway between (a,b) and (x,y) is
æ x +a y+bö
,
ç
÷
è 2
2 ø
Functions
A function is a relationship between
two changing variables
• An “input” variable
• An “output” variable
– The result of “doing” the function to the output
variable
• Both variables change so that the “input”
variable always tells you exactly what the
“output” variable is.
– You never get two outputs for the same input.
Function
Output
Any time I know the input, that’s enough
information to tell me the output.
Example: when input is 2, output is 20.
input
Not a Function
Input
Just knowing the input is not enough to tell
me the output
Example: when input is 20, the output could
be 2, 3.6, or 6.9.
Output
Function
output
input
Not a Function
output
input
Function notation
•
•
•
•
For a function named ƒ
And an input variable named x
The output variable is named ƒ(x).
ƒ(x) is the number that is the result of doing
the action ƒ to the number x
WARNING
• ƒ(x) DOES NOT MEAN ƒ*x
– You can only multiply numbers. f is NOT a number.
f is the name of a relationship.
– x and ƒ(x) are the numbers.
• Brangelina is not a person.
– Brangelina is the name of a relationship
– Brad and Angelina are the people
How to do a function to the input
number
• Algebra: Substitute
f ( x) = x - 3
Function definition
x=2
Input variable x has value 2
2
f ( 2) = 2 - 3
2
Write 2 anywhere you see an x
f (2) = 2 - 3 = 1 Simplify
2
f (2) = 1
Output variable has value 1
How to do a function to the input
number
ƒ(x)
ƒ
x
Find your input value on the x axis. Here the input value is 5.5
How to do a function to the input
number
ƒ(x)
ƒ
x
Go up and over to Find your output value on the ƒ(x) axis.
Here the output value is 25
How to do a function to the input
number
ƒ(x)
ƒ
x
Input 5.5, output 25, name of function ƒ. ƒ(5.5)=25.
Given the function f(x)=x2 -2Mx, where M is
some parameter, find f(3).
A)
B)
C)
D)
E)
3
3M
9-6M
6-9M
None of the above.
Given the function f(x)=x2 -2Mx, where M is
some parameter, find f(3).
f ( x ) = x - 2Mx
Function definition
x=3
Input variable x has value 3
2
f ( 3) = 3 - 2M 3 Write 3 anywhere you see an x
2
f (3) = 3 - 2M 3 = 9 - 6M Simplify
2
f (3) = 9 - 6M
Output variable has value 9-6M
C
Domain and Range
• Function is a relationship between a changing
input variable and a changing output variable.
• The domain is a description of all the values
that the changing input variable takes on.
• The range is a description of all the value that
the changing output variable takes on.
Example
ƒ(x)
Domain: [0,9]
Range: [0,30]
ƒ
x
State the range of the function whose graph is
pictured here. Select the best answer!
A) [3,1]  [1,3]
B)
[3,1)  [1,3]
C)
[3,1]  [2,3]
D) [3,1)  [2,3]
E) None of the above
State the range of the function whose graph is
pictured here. Select the best answer!
First, the output changes from -1
to -3 (excluding -1).
The output can be any value
between -3 and -1, but can’t be 1. Range: [-3,-1)
Later, the output changes from 2
to 3.
The output can be any value
between 2 and 3, including 2
and 3. Additional Range: [2,3]
[3,1)  [2,3]
D