2. Three Forms of an Equation of a Line
3.3 – The Equation of a Line
Three Forms of an Equation of a Line
Slope-Intercept Form:
This form is useful for graphing, as the slope and the y-intercept are readily visible.
Point-Slope Form:
The point-slope form allows you to use ANY point, together with the slope, to form the equation of the line.
Standard Form:
Note: A, B, and C cannot be fractions or decimals.

3. 3.3 – The Equation of a Line
Slope-Intercept Form– requires the y-intercept and the slope of the line.
m = slope of line
b = y-intercept

4. 3.3 – The Equation of a Line
Slope-Intercept Form– requires the y-intercept and the slope of the line.
m = slope of line
b = y-intercept

5. 3.3 – The Equation of a Line
Write an equation of a line given the slope and the y-intercept.

6. 3.3 – The Equation of a Line
Point-Slope Form – requires the coordinates of a point on the line and the slope of the line.

7. 3.3 – The Equation of a Line
Point-Slope Form – requires the coordinates of a point on the line and the slope of the line.

8. Writing an Equation Given Two Points
3.3 – The Equation of a Line
Writing an Equation Given Two Points
1. Calculate the slope of the line.
2. Select the form of the equation.
a. Standard form
b. Slope-intercept form
c. Point-slope form
3. Substitute and/or solve for the selected form.

9. 3.3 – The Equation of a Line
Writing an Equation Given Two Points
Given the two ordered pairs, write the equation of the line using all three forms.
Calculate the slope. or

10. 3.3 – The Equation of a Line
Writing an Equation Given Two Points
Point-slope form

11. Writing an Equation Given Two Points
3.3 – The Equation of a Line
Writing an Equation Given Two Points
Slope-intercept form

12. Writing an Equation Given Two Points
3.3 – The Equation of a Line
Writing an Equation Given Two Points
Standard form
LCD: 4

13. 3.3 – The Equation of a Line Solving Problems 𝑦−3=−2 𝑥−2
𝑦−3=−2 𝑥−2
𝑦= −2 −4 𝑥+ 5 −4
𝑦−3=−2𝑥+4
𝑦= 1 2 𝑥− 5 4
𝑦=−2𝑥+7
2𝑥+𝑦=7
𝑚= 1 2
𝑚 ⊥ =− 2 1 =−2

14. 3.4 – Linear Inequalities in Two Variables
Are the ordered pairs a solution to the problem?

15. 3.4 – Linear Inequalities in Two Variables
Are the ordered pairs a solution to the problem?
true
true .
false
true
The shaded region represents the solution set for the inequality.

16. 3.4 – Linear Inequalities in Two Variables
Graph the solution set of the linear inequality
false
true

17. 3.4 – Linear Inequalities in Two Variables
Graph the solution set of the linear inequality
false
true

18. 3.4 – Linear Inequalities in Two Variables
Graph the solution set of the linear inequality
false
true

19. 3.4 – Linear Inequalities in Two Variables
Graph the solution set of the linear inequality
true
false

20. 3.4 – Linear Inequalities in Two Variables

22. 3.3 – The Equation of a Line
Slope-Intercept Form– requires the y-intercept and the slope of the line.
m = slope of line
b = y-intercept

23. 3.3 – The Equation of a Line
Slope-Intercept Form– requires the y-intercept and the slope of the line.
m = slope of line
b = y-intercept

24. 3.3 – The Equation of a Line
Point-Slope Form – requires the coordinates of a point on the line and the slope of the line.

25. 3.5 – Introduction to Functions
Defn: A relation is a set of ordered pairs.
Domain: The values of the 1st component of the ordered pair.
Range: The values of the 2nd component of the ordered pair.

26. 3.5 – Introduction to Functions
State the domain and range of each relation. x y 1 3 2 5 -4 6 4 x y 4 2 -3 8 6 1 -1 9 5 x y 2 3 5 7 8 -2 -5

27. 3.5 – Introduction to Functions
Defn: A function is a relation where every x value has one and only one value of y assigned to it.
State whether or not the following relations could be a function or not. x y 4 2 -3 8 6 1 -1 9 5 x y 1 3 2 5 -4 6 4 x y 2 3 5 7 8 -2 -5
function
not a function
function

28. 3.5 – Introduction to Functions
Functions and Equations.
State whether or not the following equations are functions or not. x y -3 5 7 -2 -7 4 3 x y 2 4 -2 -4 16 3 9 -3 x y 1 -1 4 2 -2
function
function
not a function

29. 3.5 – Introduction to Functions
Graphs can be used to determine if a relation is a function.
Vertical Line Test
If a vertical line can be drawn so that it intersects a graph of an equation more than once, then the equation is not a function.

30. 3.5 – Introduction to Functions
The Vertical Line Test y
function x y -3 5 7 -2 -7 4 3 x

31. 3.5 – Introduction to Functions
The Vertical Line Test y
function x y 2 4 -2 -4 16 3 9 -3 x

32. 3.5 – Introduction to Functions
The Vertical Line Test y
not a function x y 1 -1 4 2 -2 x

33. 3.5 – Introduction to Functions Domain and Range from Graphsx y
Find the domain and range of the function graphed to the right. Use interval notation.
Domain
Range
Domain:
[–3, 4]
Range:
[–4, 2]

34. 3.5 – Introduction to Functions Domain and Range from Graphsx y
Find the domain and range of the function graphed to the right. Use interval notation.
Range
Domain:
(– , )
Range:
[– 2, )
Domain

35. 3.6 – Function Notation
Function Notation
Shorthand for stating that an equation is a function.
Defines the independent variable (usually x) and the dependent variable (usually y).

36. 3.6 – Function Notation
Function notation also defines the value of x that is to be use to calculate the corresponding value of y.
f(x) = 4x – 1
find f(2).
g(x) = x2 – 2x
find g(–3).
find f(3).
f(2) = 4(2) – 1
g(–3) = (-3)2 – 2(-3)
f(2) = 8 – 1
g(–3) = 9 + 6
f(2) = 7
g(–3) = 15
(2, 7)
(–3, 15)

37. 3.6 – Function Notation f(5) = 7 f(4) = 3 f(5) = 1 f(6) =
Given the graph of the following function, find each function value by inspecting the graph. x y ●
f(x) ●
f(5) = 7
f(4) = 3 ●
f(5) = 1
f(6) = 6 ●

38. 3.6 – Function Notation

39. 3.7 – Variation
Direct Variation: y varies directly as x (y is directly proportional to x), if there is a nonzero constant k such that
The number k is called the constant of variation or the constant of proportionality
Verbal Phrase
Expression
𝑦 𝑣𝑎𝑟𝑖𝑒𝑠 𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 𝑤𝑖𝑡ℎ 𝑥
𝑦=𝑘𝑥
𝑠 𝑣𝑎𝑟𝑖𝑒𝑠 𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑜𝑓 𝑡
𝑠=𝑘 𝑡 2
𝑦 𝑖𝑠 𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 𝑝𝑟𝑜𝑝. 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑐𝑢𝑏𝑒 𝑜𝑓 𝑧
𝑦=𝑘 𝑧 3
𝑢 𝑖𝑠 𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 𝑝𝑟𝑜𝑝. 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑠𝑞. 𝑟𝑡. 𝑜𝑓 𝑣
𝑢=𝑘 𝑣

40. 3.7 – Variation Direct Variation
Suppose y varies directly as x. If y is 24 when x is 8, find the constant of variation (k) and the direct variation equation.
direct variation equation
constant of variation x y 3 5 9 13 9 15 27 39

41. 3.7 – Variation
Hooke’s law states that the distance a spring stretches is directly proportional to the weight attached to the spring. If a 56-pound weight stretches a spring 7 inches, find the distance that an 85-pound weight stretches the spring. Round to tenths.
direct variation equation
constant of variation

42. 3.7 – Variation
Inverse Variation: y varies inversely as x (y is inversely proportional to x), if there is a nonzero constant k such that
The number k is called the constant of variation or the constant of proportionality.
Verbal Phrase
Expression
𝑦 𝑖𝑠 𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝑙𝑦 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝑤𝑖𝑡ℎ 𝑥
𝑦= 𝑘 𝑥
𝑠 𝑣𝑎𝑟𝑖𝑒𝑠 𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝑙𝑦 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑜𝑓 𝑡
𝑠= 𝑘 𝑡 2
𝑦 𝑖𝑠 𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝑙𝑦 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝑡𝑜 𝑧 4
𝑦= 𝑘 𝑧 4
𝑢 𝑣𝑎𝑟𝑖𝑒𝑠 𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝑙𝑦 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑐𝑢𝑏𝑒. 𝑟𝑡. 𝑜𝑓 𝑣
𝑢= 𝑘 3 𝑣

43. 3.7 – Variation Inverse Variation
Suppose y varies inversely as x. If y is 6 when x is 3, find the constant of variation (k) and the inverse variation equation.
inverse variation equation
constant of variation x y 3 9 10 18 6 2
1.8 1

44. 3.7 – Variation
The speed r at which one needs to drive in order to travel a constant distance is inversely proportional to the time t. A fixed distance can be driven in 4 hours at a rate of 30 mph. Find the rate needed to drive the same distance in 5 hours.
inverse variation equation
constant of variation

45. 3.7 – Variation Joint Variation
If the ratio of a variable y to the product of two or more variables is constant, then y varies jointly as, or is jointly proportional, to the other variables.
Verbal Phrase
Expression
𝑦 𝑣𝑎𝑟𝑖𝑒𝑠 𝑗𝑜𝑖𝑛𝑡𝑙𝑦 𝑤𝑖𝑡ℎ 𝑥 𝑎𝑛𝑑 𝑧
𝑦=𝑘𝑥𝑧
𝑧 𝑣𝑎𝑟𝑖𝑒𝑠 𝑗𝑜𝑖𝑛𝑡𝑙𝑦 𝑤𝑖𝑡ℎ 𝑟 𝑎𝑛𝑑
𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑜𝑓 𝑡
𝑧= 𝑘𝑟 𝑡 2
𝑉 𝑖𝑠 𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝑡𝑜 𝑇 𝑎𝑛𝑑
𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝑙𝑦 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝑡𝑜 𝑃
𝑉= 𝑘𝑇 𝑃
𝐹 𝑣𝑎𝑟𝑖𝑒𝑠 𝑗𝑜𝑖𝑛𝑡𝑙𝑦 𝑤𝑖𝑡ℎ 𝑚 𝑎𝑛𝑑 𝑛 𝑎𝑛𝑑
𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝑙𝑦 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑜𝑓 𝑟
𝐹= 𝑘𝑚𝑛 𝑟 2

46. z varies jointly as x and y.
3.7 – Variation
Joint Variation
z varies jointly as x and y.
x = 3 and y = 2 when z = 12.
Find z when x = 4 and y = 5.
𝑧=𝑘𝑥𝑦
12=𝑘 3 2
2=𝑘
𝑧=2𝑥𝑦
𝑧=2 4 5
𝑧=40

47. V varies jointly as h and 𝑟 2 .
3.7 – Variation
Joint Variation
The volume of a can varies jointly as the height of the can and the square of its radius. A can with an 8 inch height and 4 inch radius has a volume of cubic inches. What is the volume of a can that has a 2 inch radius and a 10 inch
height?
V varies jointly as h and 𝑟 2 .
V = cubic inches, h = 8 inches and r = 4 inches.
Find V when h = 10 and r = 2.
𝑉=𝑘 ℎ𝑟 2
402.12=𝑘
3.142=𝑘
𝑉=3.142ℎ 𝑟 2 𝑉=
𝑉=125.68
𝑖𝑛 3