1. Welcome to MS 101
Intermediate Algebra

2. Chapter 1 Linear Equations and Linear Functions
Qualitative Graphs
Graphing Linear Equations
Slope
Finding Linear Equations
Functions

4. 1.1 Qualitative Graphs
Definition: Graph without scaling (tick marks and their numbers) on the axis.
Medication Costs p
Linear Curve
price t
Time

5. Independent/Dependent Variables
Price p is dependent on the year t.
(p is dependent of t) p is a dependent variable
Year t does not depend on the price p.
(t is independent of p) t is an independent variable

6. Examples
You are taking a patient’s pulse counting the beats b per minute m.
The number of people n that can run m miles.
Waiting inline in the cafeteria for lunch. Use n for number of people in front and w for the wait time.
Answers:
Independent - m, m, n
Dependent - b, n, w

7. Independent/Dependent Variable (time)
Independent most the time.
Something happens over time
temperature over time, growth, population, etc.
Dependent
Time it takes for something to happen
wait time, time taken to cook, time taken to move a distance, etc

8. Dependent
Variable
Independent
Variable

9. Intercepts
Average Age of Nurses m A
m-intercept
A-intercept
Average Age
n-intercept
parabola n t
Years Since 1900
An intercept of a curve is the point where the curve intersects an axis (or axes).

10. Graph Characteristics
Increasing Curve
upward from left to right
Decreasing Curve
downward from left to right
Quadrants

11. Examples
Height of a ball thrown straight up
Heart rate of a person on a treadmill as the pace is steadily increased
The temperature in Celsius of boiled water placed in a freezer
The value of a car after a fixed amount of years

12. 1.2 Graphing Linear Equations
Solution, Satisfy, and Solution Set
An ordered pair (a,b) is a solution of an equation in x and y if the equation becomes a true statement when a substituted for x and b is substituted for y. We say (a,b) satisfies the equation.
Solution set is the set of all solutions of the equation.

13. Example y = 4x – 2 Find y if x = 2 y = 4(2) – 2 = 8 – 2 = 6
Thus the ordered pair (2,6) satisfies the equation y = 4x – 2 and (2,6) is a solution of the equation y = 4x – 2

14. Continued Solution set of the equation y = 4x – 2 includes (2,6) x y
Ordered pair (a,b) the independent variable is
First (left) followed by the dependent variable in
the second position (right).

15. Graphing Equations y = 5x + 2 Chose values, ex. 0, 1 Organize in table
= 0 + 2
= 2
Solution: (0,2)
y = 5(1) + 2
= 5 + 2
= 7
Solution: (1,7)
Chose values, ex. 0, 1
Organize in table
x y 7 12 17

16. (3, 17)
(2, 12)
(1, 7)
(0, 2)
(-1,-3)
(-2,-8)

17. (3, 17)
A graph of an equation in two variables is a visual representation of the solutions of an equation.
(2, 12)
(1, 7)
(0, 2)
(-1,-3)
(-2,-8)

18. (1, 9) does not satisfy the equation
(3, 17)
(2, 12)
(1, 9)
(0, 2)
(1, 9) does not satisfy the equation
(-1,-3)
(-2,-8)

19. Linear Equation Forms Slope-Intercept Form Standard Form General Form
y = mx + b
m and b are both constants (m is the slope, b is the y-intercept)
Standard Form
Ax + By = C
A, B, C are integers (A and B are not both equal to zero)
General Form
Ax + By + C = 0
Point-Slope Form
(y – y ) = m (x – x )
Points (x, y) & (x ,y ), slope m 1 1 1 1

20. General to Slope-Intercept Form
3y - 12x + 9 = 0
3y - 12x = 0 -9
3y – 12x = -9
3y - 12x + 12x = x
3y = x
y = x
y = 4x - 3
General to Standard
Try to get y by itself
Divide
Simplify

21. Distributive Law First Distribute 2(2y-3) = 2(3x-4) + 2y
Get y by itself
Divide
Simplify
2(2y-3) = 2(3x-4) + 2y
4y – 6 = 6x – 8 +2y
4y – 6 -2y = 6x – 8 + 2y -2y
2y – 6 +6 = 6x – 8 +6
2y = 6x – 2
y = 3x - 1

22. Graphing Equations with Fraction Slopes
y =¾x - 4
Chose x values that are multiples of the denominator. Ex. 0, 4, 8, 12
Make a chart and graph
x y
0 y = ¾ (0) – 4 = -4
4 y = ¾ (4) – 4 = -1
8 y = ¾ (8) – 4 = 2

23. Intercepts of Graphs
To find the x-intercept, substitute 0 for y and solve for x. It will always be in the form (x,0).
To find the y-intercept, substitute 0 for x and solve for y. It will always be in the form (0,y).
y = - ½x +3
0 = - ½x + 3
0 -3 = - ½x -3
-3 *(-2) = - ½x *(-2)
6 = x
~x-intercept (6,0)
y = - ½ (0) +3
y = 3
~y-intercept (0,3)
y-intercept (0,3)
x-intercept (6,0)

24. Vertical/Horizontal Lines
If a and b are constants,
Vertical lines are in the form x=a
Horizontal lines are in the form y=b
y=6
x=8

25. How do we measure steepness?
1.3
How do we measure steepness?

26. Comparing Steepness
Calculate the ratio of vertical to horizontal distance.
Patient 1 has his foot elevated 2 feet above the bed over a vertical distance of 2.5 feet. Patient 2 has his foot elevated 1.5 feet above the bed over a vertical distance of 2 feet. Which leg is elevated steeper?
Patient 1’s foot is steeper because the ratio is slightly greater.
Vertical Distance
Horizontal Distance 2
1.5
2.5 2 2
1.5
= .8
= .75
2.5 2

27. Slope
Let (x , y ) and (x , y ) be two points on a line. The slope of the line is:
vertical change rise y - y
horizontal change run x - x 1 1 2 2 2 1
m = = = 2 1
Run is positive to the right
Run is negative to the left
Rise is positive going up
Rise is negative going down

28. Increasing vs Decreasing
positive rise
positive run or
negative rise
negative run
positive slope
positive rise
negative run or
negative rise
positive run
negative slope

29. Finding Slope of a Graph
m = rise
run
m = 2 1
m = 2
m = y - y
x - x
m = 3 – 1
2 – 1
m = 2 = 2 1 2 1
(2,3) 2 1 2
(1,1) 1

30. Investigating Slope of Horizontal/Vertical Lines
2 – Slope of a horizontal line
4 – 2 2 is always 0.
3 – 1 2 Slope of a vertical line
1 – 1 0 is always undefined.
m = = = 0
m = = = Undefined
y = 2 (2,2) (4,2)
x = 1 (1,1) (1,3)

31. Parallel/Perpendicular Lines
Parallel lines have the same slope
m = m
Slopes of perpendicular lines are opposite reciprocals
1 1
m 2 l 2
Parallel lines never intersect 2 1 l 1 l 2
m =
2 = 2 l 1 1
Perpendicular lines form 90 degree (right) angles

32. Finding Slope of Linear Equations
1.4
Finding Slope of Linear Equations
y = -3x - 2
Graph
rise
run
If in slope-intercept form:
m = slope, m = -3
b = y-intercept, b = -2 1
m = = = -3 -3 1 -3

33. Using Slope and y-intercept to graph an equation
Slope = 2 = rise 2
run 1
y-intercept (0, -2)
Plot y-intercept (0, b)
Use m = to plot 2nd point.
Sketch line passing through the 2 points.
rise
run
(0, -2)

34. Vertical Change Property
For a line y = mx + b, if the run in 1, then the rise is the slope m. m 1 1 m
Slope Addition Property
For a line y = mx + b, if the independent variable increases by 1, then the dependent variable changes by slope m.
y = -6x + 3 , x increases by 1, y changes by -6

35. Identifying Parallel/Perpendicular Lines
Are the lines 3y = 2x – 6 and 6y – 4x = 24 parallel, perpendicular or neither?
3y = 2x – 6
y = 2/3x - 3
6y – 4x + 4x = 18 +4x
6y = 4x + 24
y = 4/6x + 24/6
y = 2/3x + 4
Get y by itself
Divide
Simplify
Get the equations in slope-intercept form
Slope in both equations is 2/3, therefore the lines are parallel.

36. Indentifying Linear Equations from points
Set 1
Set 2
Set 3
Set 4 X Y 10 24 6 7 33 -2 4 11 20 12 34 -3 16 8 17 35 -4 13 9 23 36 -5 14 27 37 -6

37. Answers Set 1 – slope -4 Set 2 – not a line Set 3 – slope 0

38. Linear Equation Forms Slope-Intercept Form Standard Form General Form
1.5
Slope-Intercept Form
y = mx + b
m and b are both constants (m is the slope, b is the y-intercept)
Standard Form
Ax + By = C
A, B, C are integers (A and B are not both equal to zero)
General Form
Ax + By + C = 0
Point-Slope Form
(y – y ) = m (x – x )
Points (x, y) & (x ,y ), slope m 1 1 1 1

39. Find an equation given slope and a point
Slope 4, point (6,2)
Start with putting the slope into y=mx + b
y = 4x + b
Plug the coordinates of the point in for x and y
(2) = 4(6) + b
Solve for b
2 -24 = 24 + b -24
-22 = b
y = 4x - 22

40. Find an equation of the line using two points
(3, 5) (4, -2)
Find the slope
y - y –
x - x –
Place slope into y = mx + b
y = -7x + b
Then chose either point to plug in for x and y… (3, 5)
(5) = -7(3) + b
Solve for b
5 +21 = b +21
26 = b… y = -7x + 26
m = = = 2 1 2 1

41. Finding the approximate equation of a line
(-4.56,-5.24) (7.72, -4.93)
Find the slope
y - y – (-5.24)
x - x – (-4.56)
Place slope into y = mx + b
y = .03x + b
Then chose either point to plug in for x and y… (7.72, -4.93)
(-4.93) = .03(7.72) + b
Solve for b
= b
-5.16 = b… y = .03x – 5.16
m = = = = = .03 2 1 2 1

42. Finding an equation of a line parallel to a given line
y = 3x – 2 , point (1,6)
m = 3, so slope is 3
Parallel lines have similar slopes, so:
y = 3x + b
Plug in the coordinates for x and y
(6) = 3(1) + b
6 -3 = 3 + b -3
b = 3
y = 3x + 3 is parallel to y = 3x -2

43. Finding an equation of a line perpendicular to a given line
y = ½ x – 6, point (2,-2)
Slopes of perpendicular lines are opposite reciprocals:
flip the numerator and denominator and change the sign
or -2
y = -2x + b
(-2) = -2(2) + b
-2 +4 = -4 + b +4
2 = b, y = -2x + 2 is perpendicular to y = ½ x - 6 12 21

44. Point-slope Form
If a non-vertical line has a slope m and contains the point (x ,y ), then the equation of the line is:
(y - y ) = m (x - x )
Given a point and the slope:
m = 4 point (6,-3)
y – (-3) = 4 (x - 6 )
y = 4x – 24 -3
y = 4x – 27 1 1 1 1

45. Finding an equation of a line using point-slope form
Given two points (2,14) and (-8,-6)
Find slope
-6 –
-8 –
(y - y ) = 2(x - x )
Substitute in one of the points
y – (14) = 2(x – (2))
y – = 2x
y = 2x + 10
m = = = 2 1 1

46. 1.6 Functions Relation – set of ordered pairs
Domain – set of all values of the independent variable (x-values)
Range – set of all values of the dependent variable (y-values)
Function – a relation in which each input leads to exactly one output
Domain = input, Range = output

47. Equations by input/outputx 5 6 7 y 3 1
4 & 3 x 2 4 y 7 8 9
RELATIONS
RELATIONS
input
(domain)
output
(range)
input
(domain)
output
(range)

48. Is it a function? y = ± x y² = x y = x²
x = 5 , y = ± 5 or y = 5 and y = -5
input has 2 outputs, not a function
y² = x
x = 16, y = ± 4 or y = 4 and y = -4
y = x²
x = 3, y = 3² = 9
input has 1 output, function

49. Is it a function? Set 1 Set 2 Set 3 Set 4 x y 4 1 5 6 10 9 -1 7 11 2
input y
output 4 1 5 6 10 9 -1 7 11 2 12 8 13 -2 14

50. Answers Set 1 – not function, slope undefined, vertical line
Set 2 – not a function
Set 3 – function, slope 0, horizontal line
Set 4 – function, slope -3, decreasing line

51. Is it a function?
When x =2
y = 3 and
y = -3, not a
function

52. Vertical line test
A relation is a function if and only if every vertical line intersects the graph of the relation at no more than one point.

53. Functions from equations
Is y =-3x -2 a function?
Graph it!
Any vertical line
would intersect
only once….Yes.

54. Linear Function All non-vertical lines are functions. Linear function:
a relation whose equation can be put in
y=mx + b form (m and b are constants)

55. Rule of Four Four ways to describe functions x y -2 -8 -1 -3 0 2 7 12
y = 5x + 2
Four ways to describe functions
Equation
Graph
Table
Verbally (input-output)
x y 7 12 17
For each input-output pair, the output is two more
than 5 times the input.

56. Finding Domain and Range
Domain = width = input = independent variable = x-values
Range = height = output = dependent variable = y-values
Leftmost point (-3,1)
Rightmost point (4, 1)
Domain -3 ≤ x ≤ 4
Highest point (2, 3)
Lowest point (-1,-2)
Range -2 ≤ y ≤ 3

57. Domain/Range Continued
D = all real numbers
R = y ≥ -1
D = x ≥ 0
R = y ≥ -1
D = all real numbers
R = all real
numbers