Equations Lineares e Slope

Early Bird Question

• A question posed at the start of a class session.
• Example given: "What is your favorite subject in school?"

Learning Targets

• Students are given learning objectives for a particular lesson.

• Example: "I will use slope to graph linear equations in two variables".

Linear Equations

• Includes various forms of linear equations.

• General form: Ax + By + C = 0

• Vertical line: x = a

• Horizontal line: y = b

• Slope-intercept form: y = mx + b

• Point-slope form: y - y₁ = m(x - x₁)

• Two-point form: .

Using Slope

• Discussion points about slope.
• What is slope?
• What kinds of things have slope?
• Slope-intercept form of an equation: y = mx + b, where m is the slope and b is the y-intercept.

Finding Slope

• Calculate slope from two points (x₁, y₁) and (x₂, y₂).
• The formula: m = (y₂ - y₁)/(x₂ - x₁).

Finding the Slope (multiple points)

• Calculation of slope using specific points, with instructions to maintain consistency in the order of subtraction.

Writing Linear Equations

• Describing the point-slope form as a helpful method for finding equations of lines.

Parallel and Perpendicular Lines

• Definition of parallel lines (equal slopes).
• Definition of perpendicular lines (negative reciprocal slopes).
• How slope relates to parallel and perpendicular lines.

Application

• Discussion on how slope is used in real situations.

• Explaining the difference between slope as a ratio and a rate.

Application Examples

• Slope used in ramp steepness.
• Linear sales for a company over time
• Discussing the slope as a ratio or a rate in real-world situations

Summary of Equations of a Line

• General form: Ax + By + C = 0
• Vertical line: x = a
• Horizontal line: y = b
• Slope-intercept form: y = mx + b
• Point-slope form: y - y₁ = m(x - x₁)
• Two-point form: (y-y₁)/(x-x₁) = (y₂-y₁)/(x₂-x₁)

Function and Function Notation

• Relation: A rule of correspondence that relates two variables.

• Function: A special relation that matches each item from one set with exactly one item from a different set.

• Definition of Function: A definition of Function (relation, assigns to each element of set A, exactly one element of set B. The set A, is the domain, the set B contains the range.)

Function and Function Notation - Discrete

• Representing functions in discrete mathematics.

Function and Function Notation - Continuous

• Representing functions using equations and formulas in algebra.

Function and Function Notation - Examples

• Examples comparing relations and functions.

• (1,9), (2,13), (3, 15), (4,15), (5,12), (6,10)

• Graphs showing increasing, decreasing, and constant behavior.

Graphing Functions or relations

• Four ways of representing a function.

• Verbally by an equation describing relationship of input/output variables.

• Numerically with table/list of ordered pairs.

• Graphically as points on coordinates.

• Algebraically by an equasion with 2 variables.

Function and Function Notation - Input/output/relationship/equation examples

• Graph of function values(input/output).

Function and Function Notation - Discrete vs Continuous

Function and Function Notation - Name each part

• y= dependent variable
• x= independent variable
• Relationship of inputs and outputs (domain and range)

Function and Function Notation - Determining if an equation represents a function of x

Function and Function Notation - Input / Output/ equation example

• Example of input/output (f(x)).

Piecewise-Defined Function

• Definition of Piecewise function
• Example of a piecewise function.

Finding Zeros

• Zeros of a function f, these are the x values for which f(x) = 0
• x-intercept of the graph of the function

Finding Intercepts

• Calculate x-intercepts (x_0) or y-intercepts (y_0).

Domain

• Definition of the domain of a function ( all real numbers for which an expression is defined).

• Implied domain examples ( avoid even roots of negative numbers in functions).

• Examples of functions with given domains.

Difference Quotient

• one of the basic definitions in calculus employing the ratio (f(x + h) - f(x)) / h

Difference Quotient, Examples (Function)

• Calculation of the difference quotient for given functions.

Application – Volume of a can

• Calculating the volume of the can as a function of radius.
• Calculating the volume as a function of height.

Summary

• Summary of Key ideas (bullet-point list)

Homework

• Homework assigned (assignment details).

Early Bird Question (example)

• Question posed at the start of the lesson
• Example given: "What is your favorite sport to play?"

Homework

• Instructions for turning in homework via the Teams assignment system.

Quick Check

• Quick check for the lesson (instructions and examples).

Learning Targets (example)

• I will find zeros of functions.
• I will identify even and odd functions.
• I will apply the Greatest Integer Function.

Increasing and Decreasing

• Definitions of increasing, decreasing and constant behaviour of a function.

Average Rate of Change

• Finding the average rate of change for a function (given input and output values and specific points).

Even and Odd Functions

• Identifying even and odd functions.

• Determining whether a function qualifies as even or odd.

Calculators

• Using calculators for tasks.
• Specific tasks like graphing, finding zeros, relative minimums and maximums.

Library of Parent Functions

• The parent functions (various mathematical relationships).

Step Functions (Greatest Integer Function)

• Definition of Greatest Integer function.
• Examples of Greatest Integer function.

Este quiz aborda a compreensão e a aplicação de equações lineares, incluindo a forma de interseção de slope e a forma ponto-slope. Os estudantes aprenderão a calcular o slope entre dois pontos e aplicar diferentes formas de equações lineares. Ideal para aqueles que desejam melhorar suas habilidades matemáticas.