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 – 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

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

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

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.

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.

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.

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

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

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

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

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

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

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.

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

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

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

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

3.4 – Linear Inequalities in Two Variables

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

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

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.

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.

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

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

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

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.

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

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

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

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

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

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).

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)

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 ●

3.6 – Function Notation

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

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

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

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

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

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

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

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

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 402.12 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 = 402.12 cubic inches, h = 8 inches and r = 4 inches. Find V when h = 10 and r = 2. 𝑉=𝑘 ℎ𝑟 2 402.12=𝑘 8 4 2 3.142=𝑘 𝑉=3.142ℎ 𝑟 2 𝑉=3.142 10 2 2 𝑉=125.68 𝑖𝑛 3