Linear and Nonlinear Relationships

Learning Target I can determine whether a table, equation, or graph represents a linear or nonlinear relationship.

Length of a Side of a Square Yard Compare and Contrast Yards to Feet Number of Feet Number of Yards Yards to Square Yards Length of a Side of a Square Yard Area of Square (yd2)

Definition of Linear Relationship When graphed, a linear relationship will create a straight line. There will be no curve or bend in the line. Linear relationships have a constant rate of change (slope).

Definition of Nonlinear Relationship When graphed, a nonlinear relationship will NOT result in a straight line. The graph may be curved, U-shaped, or V-shaped. Nonlinear relationships do NOT have a constant rate of change.

How to Identify Linear/Nonlinear Relationships You can look at an equation, a graph, or a table of x and y values and determine whether the relationship is linear or nonlinear. The following slides will explain how to tell the difference between linear/nonlinear.

Equations- Linear or Nonlinear? Linear Equations: All linear equations will either be written in the form or y=mx+b, or they can be rearranged into that form. The “x” value can only have an exponent of 1 or 0 in linear equations. If there is no exponent written above the “x,” the exponent is assumed to be 1. If there is no “x” present in the equation, the exponent for “x” is 0. “X” cannot be in the denominator. “X” cannot be an exponent. “X” cannot be in a radical sign (Square Root). “X” and “y” cannot be multiplied together. “Y” must also only have an exponent of 1 or 0.

Equations- Linear or Nonlinear? Nonlinear Equations: In most nonlinear equations, “x” will have an exponent of 2 or higher. If “x” is in the denominator, the equation is nonlinear. If the equation cannot be written in the form of y=mx+b, it is nonlinear.

Equations: Linear or Nonlinear REMEMBER: x1 = x and x0 = 1 In “y = ” form, is x raised to a power of 1 or 0? Does x appear in the numerator? y = 6/x y = x3 + 1 y = x + 4 nonlinear y = ½ x nonlinear y = 4 linear linear linear

Identify:Linear or Nonlinear Equation? 3x 2 y = .6x1 y = x2 + 8 y = 2/x + 5 linear nonlinear linear nonlinear

Graphs: Linear or Nonlinear? Linear graphs will ALWAYS be a STRAIGHT LINE. The straight line can be: Vertical Horizontal Diagonal

Graphs: Linear or Nonlinear? Nonlinear graphs will NOT be a straight line. Nonlinear graphs may be: Curved U-Shaped V-Shaped

Graphs: Linear or Nonlinear Is the graph a straight line? linear nonlinear linear

Identify: Linear or Nonlinear Graph?

Tables: Linear or Nonlinear? Linear Tables: A table is LINEAR if there is a constant rate of change. What does that mean? If the “x” values in the table increase or decrease by the same amount each time AND the “y” values in the table increase or decrease by the same amount each time, the table is linear.

Tables: Linear or Nonlinear? Nonlinear Tables: A table is NONLINEAR if there is NOT a constant rate of change. What does that mean? If the “x” values in the table DO NOT increase or decrease by the same amount each time and/OR the “y” values in the table DO NOT increase or decrease by the same amount each time, the table is NONLINEAR.

Tables: Linear or Nonlinear Is the rate of change constant (the same)? x y 2 50 4 35 6 20 8 5 x y 1 4 16 7 49 10 100 +3 +15 +2 -15 +2 -15 +3 +33 +2 +3 +51 -15 linear nonlinear

Identify: Linear or Nonlinear Table? x y 20 5 16 10 12 15 8 x 2 4 6 y 8 18 nonlinear linear

Exit Slip Identify if linear or nonlinear. Table A Equation y = + 5 3 6 9 12 y 10 8 y = + 5 8 x Graph a b c