Download Algebra One Notes for Chapter Four

Algebra One Notes for Chapter Four
Section 4-­‐1 – Graphing Basics Rectangular Coordinates and Lines
Label the following
1. quadrant 1
2. quadrant 2
3. quadrant 3
4. quadrant 4
5. origin
6. x-axis
7. y-axis
8. Ordered Pair (x, y) at (2, 1)
Graphing Ordered Pairs
Start at the origin
The first number is the x value – it tells how Left or Right to move
Positive moves Right
Negative moves Left
The second number is the y value – it tells how far Up or Down to move
Positive moves Up
Negative moves Down
Section 4-­‐3 – Domain, Range, Function Characteristics
Domain and Range
Relation: a set of ordered pairs
Function: a relation in which each x value maps to exactly 1 y value.
Domain: the x values of the relation (without repeat)
Range: the y values of the relation (without repeat)
Functions
Function: a relation in which each x value maps to exactly 1 y value.
Vertical Line Test: if a vertical line can hit the graph of a relation more than once it fails the vertical line
test
and is not a function.
Examples Graphically – Function? State Domain and Range.
Ordered Pairs, Maps, and Tables
Example of a Function: Focus on how no x value does two things.
Ordered Pairs
Map
(-8, 4), (-1, 1),
(0, 6), and (5, 4)
Table
x
-8
-1
0
5
Graph
y
4
1
6
4
Example of a Non-Function: Focus on how some of the x values do more than one thing.
Ordered Pairs
(-7, 7), (2, 0), (2, 4),
(3, 7), (5, 0), and (5,
8)
Map
Table
x
-7
2
2
3
5
5
Graph
y
7
0
4
7
0
8
Section 4-­‐4 – Domain, Range, Function Characteristics
Independent vs. Dependent Variables
x is the independent variable
y is the dependent variable
Example: The equation y = 40x will tell y, the distance travelled, for travelling for x hours and 40 mph.
So the distance travelled depends on the time.
This shows that y is the dependent variable and x is the independent variable.
Solutions to equations of 2 variables.
If an ordered pair makes a true statement, then it is part of the solution.
To test this – plug in the ordered pair and see if the right is equal to the left.
Example: Given the equation y = 4x – 2, are (3, 3) and (0, -2) part of the solution?
Plug (3, 3) into y = 4x – 2
3 = 4(3) – 2
3 = 10 NO
so (3, 3) is NOT part of the solution
plug (0, -2) into y = 4x – 2
-2 = 4(0) – 2
-2 = -2 YES
so (0, -2) IS part of the solution
Example: create a table to show solutions to the equation y = 2x + 1
Plug -2 in for x
Plug 1 in for x
Plug 0 in for x
Plug 3 in for x
y = 2(-2) + 1 and y = -3
y = 2(1) + 1 and y = 3
y = 2(0) + 1 and y = 1
y = 2(3) + 1 and y = 7
for x = -2, 1, 0, 3
x
y
-2
-3
1
3
0
1
3
7
Section 4-­‐5 – Graphing Linear Equations
Linear vs. Non-linear
•
•
•
•
Linear equations can be written as Ax + By = C or y = mx + b
The power of x is 1 and the power of y is 1. X and y cannot be multiplied together. X and y not in
the bottom of a fraction.
There is a consistent change for y as x changes. This is called the slope.
The graph of a linear equation is a linear equation.
Standard Form: Linear equations can be written as Ax + By = C
Y = mx + b form: Linear equations can be written as y = mx + b
Examples: Are these linear?
yx = 8
y = 4x + 7
y = x2 + 1
3x + 4y = 12
y=4
if yes, write in
Standard form:
Graphing Constant Functions ( x =
•
•
•
•
or y =
)
When there is only 1 variable, the graph will be vertical or horizontal.
X = creates a vertical line (not a function) and y = creates a horizontal line (function).
Solve for the variable.
The variable is the axis that is crossed and the number is where it crosses.
Examples: graph
y=4
x = -1
3x + 1 = 7
x=2
If you can make a t-table, you can graph anything!
•
•
•
•
•
Solve for y.
Create a table using -2, -1, 0, 1, 2 as the x values (unless x is multiplied by a fraction).
If x is multiplied by a fraction, then use the denominator to make the table from the multiples of the
denominator.
o If y = ½ x
table is -4, -2, 0, 2, 4
o If y = 1/3 x table is -6, -3, 0, 3, 6
Find the y value for each x value.
Plot the order pairs and connect to make a line.
Examples: graph y = 4x + 1
x
y
-2
-7
-1
-3
0
1
1
5
2
9