Lesson 2.2 Linear Relations and Functions By Nate Derrick Period 6

Vocabulary Overview Linear Relation: Relations that have straight line graphs Nonlinear Relation: Relations that are not linear Linear Equation: An equation that has no operations other than addition, subtraction, multiplication, of a variable by a constant Linear Function: is a function written with ordered pairs that satisfy a linear equation Standard Form: A form used to write linear equations (Ax+By=C) y-intercept: y-coordinate of the point which the graph crosses the y-axis x-intercept: x-coordinate of the point which the graph crosses the x-axis These are all vocabulary terms used in the lesson

Major Topics Identifying a Linear Functions and Equations Evaluate a Linear equation Putting equations in standard form Using intercepts to graph a line

Identifying Linear Equations and Functions Has no operations other than addition, subtraction, multiplication, and division of a variable by a constant Any function that can be written in y=mx+b or f(x)=mx+b is linear No variable in denominator when Dividing, and No exponents greater than 1 Linear Equations and Functions: Non Linear Equations and Functions: 6y-x=7 9x= 18/y f(x) = 2 - x/11 f(x)= 4-x³

Practice Problems Are the following equations and functions linear? 1.6x -2.4y = 4 2yx - 3y + 2x = 0 Yes No, variables can’t be multiplied by each other in a linear equation f(x)= 4/x 6x - 4y= 16 No, because there is a variable in the denominator Yes

Evaluating a linear Equation The Growth rate of a sample of bermuda grass is given by the function f(x) = 5.9x + 3.25 where f(x) is the total height in inches after x days after an initial measurement How tall is the sample after 3 days f(x) = 5.9x + 3.25 Original Function f(3) = 5.9(3) + 3.25 Substitute 3 for x 20.95 inches Height of the sample after 3 days

Practice Problems You want to make sure you have enough music for a car trip. If each CD is an average of 45 minutes long, the linear function m(x)= 0.75x could be used to find out how many CDs you need to bring. How many hours of music are there on 4 CD’s m(x) = 0.75x Original function m(4) = 0.75(4) Substitute 4 for x 3 If you have 4 CD’s you have 3 hours of music

Using Standard form Always written in Ax + By = C A, B, C are integers (positive or negative whole numbers) No fractions or decimals in standard form However the "Ax" term is positive Write the following equations in standard form Identify A, B, and C y = -4x - 7 Original Equation 3x = -2y - 1 Original Equation 4x + y= -7 Add 4x to both sides 3x + 2y = -1 Add 2y to both sides A = 4, b = 1, C = -7 A = 3, B = 2, C = -1

Practice Problems Write the Equation in Standard Form y = 6x + 5 Original Equation -6x + y = 5 Subtract 6x 6x - y = -5 Multiply the Equation by -1

Practice Problems Write the Equation in Standard Form -0.08x = 1.24y - 3.12 Original Equation -0.08 - 1.24y = -3.12 Subtract 1.24y -8x - 124y = -312 Multiply the Equation by 100 8x + 124y = 312 Multiply the Equation by -1

Using Intercepts to Graph a line Find the x-intercept and the y-intercept of the graph 2x - 3y + 8 = 0 Plug in the coordinate (0,0) then solve 2x - 3(0) + 8 = 0 2(0) - 3y + 8 = 0 2x = -8 -3y= -8 x= -4 y= 8/3

Practice Problem Find the x and y intercepts of the graph 2x + 5y - 10 = 0 2x + 5(0) - 10 = 0 2(0)+ 5y - 10 = 0 2x - 10 = 0 5y - 10 = 0 2x = 10 5y = 10 x= 5 y = 2

Practice Problem Graph