1. Chapter 8
Pre Algebra

2. Relations and Functions 2/18
A set of ordered pairs
Relation
(2, 4), (5, 6), (-1, 5) X Y 2 4 5 6 -1

3. The y values of a relation
Domain
The x values of a relation
Range
The y values of a relation
Domain
Range 2 4 5 6 -1

4. Function
Each domain value is paired with only ONE range value
Domain
Range 2 4 5 6 -1

5. X Y 2 8 3 9 5 X Y 2 5 -4 9 3 1

6. Plot the points (ordered pairs) on a coordinate plain, if any points lay on the same vertical line, then the relation is not a function.
Vertical Line Test
Look at the grid lines that cross the x-axis or use your pencil to make a vertical line.

8. You Try
Workbook
Page 129 and 130

9. Equations with two Variables 2/19
y = mx + b
Where m and b are integers (numbers)
Linear equations
Y = 2x + 4
An ordered pair that makes the equation true
solution
A solution of y = 2x + 4 is ( 1, 6)
6 = 2(1) +4
6 = 2 + 4
6 = 6 YES!!!

10. Find a solution to each equation when x = 15 y = 4x -10 y = x/5 +7
Examples
Find a solution to each equation when x = 15
y = 4x -10
y = x/5 +7
y = -3x - 13

11. Linear equations have infinitely many solutions
Graphing
Linear equations have infinitely many solutions
For every x value (including fractions, decimals, and negative numbers) there is a y that makes the equation true.
Keep your work organized by making a chart.

12. Corresponding y-value
Charts
There are many different ways to chart information. You need at minimum a “t chart” x y
X-value
Corresponding y-value
X-value
Corresponding y-value
X-value
Corresponding y-value

13. I like a chart with more information
Charts
I like a chart with more information
equation
(x, y) x y
X-value
work
y-value
y-value
X-value
work
X-value
work
y-value

14. Find 3 solutions and then graph y = 4x -10 y = x/5 +7 y = -3x - 13
Examples
Find 3 solutions and then graph
y = 4x -10
y = x/5 +7
y = -3x - 13

15. Writing Rules for Linear Functions 2/23
y = mx + b
Where m is slope and b is the y-intercept
Linear equations
y = 2x + 4
Function Notation
Replace y with f(x)
f(x) is read f of x
f(x) = 2x + 4
Helps keep you organized when you are plugging and chugging without making a table
WHY????

16. Write using function notation y = 4x -10 y = x/5 +7 y = -3x - 13
Examples
Write using function notation
y = 4x -10
y = x/5 +7
y = -3x - 13

17. An equation that describes a function
Function Rules
An equation that describes a function
y = 1/2x -4 f(x) = 1/2x-4
Both are function rules
Commit to a notation
Remember x is the input and y is the output so f(x) is also the output
If the problem you are given is in function notation f(x) = mx + b
Then your table needs to have f(x) as the output (replace y with f(x)).

18. y = 1/2x -4 f(x) = 1/2x-4 Example Make a table with 3 values x f(x) -4-4 2 -3 4 -2 x y -4 2 -3 4 -2

19. what is the value when x = 3?
Organization
Function notation keeps you organized because it lets you see that you plugged in for x
f(x) = 3x -7
what is the value when x = 3?
Replace all the x’s
f(3) = 3(3) -7
f(3) = 9 – 7
f(3) = 2

20. Finding function rules
Two ways
Graph and find m and b and use y=mx+b
Look at table to find m and b
Slope
m = rise / run
m = change in y / change in x
m = (y1-y2)/(x1-x2)

21. Example
Find the fuction rules x y -5 1 -1 2 3 x
f(x) -4 4 -2 2

22. Solve by Graphing 2/25 Many points are graphed
Scatter Plots
Many points are graphed
Look at the graph for the trends.
Find a specific trend line
Write a function rule (equation) for the trend line
Use the trend line and function rule to make predictions

30. Examples

31. Solving Systems of Linear Equations 2/26
Two or more linear equations
Systems
An ORDERED PAIR that satisfies both equations
Solution
y = -x + 1 and y = 2x + 4 are a system
(-1, 2) is a solution

33. Plug and chug ordered pair into both equations
Checking a solution
Plug and chug ordered pair into both equations
y = -x + 1 and y = 2x + 4 are a system
(-1, 2) is a solution
y = 2x + 4
2 = 2(-1) +4
2 =
2 = 2
y = -x + 1
2 = -(-1) + 1
2 = 1 + 1
2 = 2

34. You Try
Is (-3, -10) a solution to
y = x – 7
y = 4x +2

35. Parallel lines have no solution
Parallel lines will have the same slope but different y-intercepts
y = 3x + 4
y = 3x - 5

37. Every point on the line is a solution
Infinitely many solutions
The same line
Every point on the line is a solution
Same slope same y -intercept
y = 3x + 4

39. Where lines intersect is the solution
Solve by graphing
Graph both lines on same coordinate plane
Where lines intersect is the solution
Write the point as an ordered pair
x – y = -4
x + y = 6

42. Homework Check if (-1, 5) is a solution of each system. Show work
x + y = 4 2. y = -2x x = y – 7
x – y = y = x – y = -x + 9
Solve by graphing. Check each solution.
4. y = x y = 2x y = x + 1
y = 3x – x + y = y = -x - 3

43. Graphing Linear Inequalities 3/1
Replace the equal sign of a linear equation with an inequality sign:
≤ ≥ < >
Examples
y ≤ 2x – 4
y ≥ -3x + 3
y < ⅔x
y > -4x -5

44. Plot b on the coordinate plane Use slope to find a 2nd point
Graphing
Solve for y
Find m and b
Plot b on the coordinate plane
Use slope to find a 2nd point
Look at the inequality sign
If > or < draw a dashed line
If ≤ or ≥ draw a solid line
Find 2 test points, one above and one below the line
Plug each test point into the original inequality
Shade in the area of the graph where the test point was a solution

45. Will the line be solid or dashed?
You try
Will the line be solid or dashed?
dashed
1. y > x - 6
2. y ≤ -x + 8
solid
3. y < -⅓ x + 1
dashed
4. y ≥ 2x - 1
solid

46. Graph y > x + 2 m = 1 and b = 2 Test points (0, 0) y > x + 2
Graphing Example
Test points
Graph y > x + 2
(0, 0)
y > x + 2
0 > 0 + 2
0 > 2 NOPE
m = 1 and b = 2
(-3, 0)
y > x + 2
0 >
0 > -1 YEP

47. You try
y ≤ -x + 8
2x + 3y ≤ 7
-y ≤ 2x
4x + y < -3

48. Graphing Linear Inequalities 3/2
Solving
Solve a system of linear inequalities (more than one at a time) by graphing both on the same coordinate plane.
Solution
The solution is where both shaded areas overlap

52. Examples
y > -x
y > 2x + 3
2. 3x + y > 5
y ≥ - 2
3. Medium Drinks cost $2 and large drinks cost $3. Let x be the number of medium drinks sold and y be the number of large drinks sold. How many drinks must a vender sell to have at least $60 in sales? Show all possible solutions by graphing.

53. Patterns and Sequences 3/3
A set of numbers that follow a pattern
Four types
Arithmetic
Geometric
Both
Neither

54. Sequence changes by adding a fixed number
Arithmetic Sequence
Sequence changes by adding a fixed number
Called a common difference
The common difference can be positive or negative.
Find the common difference, rule, and next 3 terms
Examples
2, 5, 8, 11, . . .
7, 7, 7, 7, 7, . . .
63, ,

56. Sequence changes by multiplying a fixed number
Geometric Sequence
Sequence changes by multiplying a fixed number
Called a common ratio
Remember that ratios can be whole numbers or fractions
Examples
Find the common ratio, rule, and next 3 terms
3, 3, 3, 3, . . .
3, -6, 12, -24, . . .
100, 50, 25, 12.5,
1/10, 1/30, 1/90, 1/270, . . .

58. Both Uncommon Examples 0, 0, 0, 0, . . . . -3, -3, -3, -3, . . .
Sequence changes by multiplying a fixed number or adding a common difference
Uncommon
Find the rule and next 3 terms
Examples
0, 0, 0, 0,
-3, -3, -3, -3,
5, 5, 5, 5, . . .
1/2, 1/2, 1/2, 1/2, . . .

59. Sequence does not have a common difference or ratio
Neither
Sequence does not have a common difference or ratio
Have to be creative to find a pattern
Find the rule and next 3 terms
Examples
1, 1, 2, 3, 5, 8, 13, 21,
5, 6, 8, 11, 15, . . .
3, 6, 18, 62,
10, 100, 10000, ,