Grade 9 MATH Term 1 Week 4: Slopes and Zeros of Linear Functions | MATATAG First Term/Q1 Tagalog

Main ideas / lessons

• The video introduces linear functions, focusing on:
  • Slope (rate/steepness of the line)
  • Zeros (the input value(s) that make the output equal to zero)
• It also uses a real-life scenario to motivate linear change and the idea of a function’s zero.

Key concepts

1) Linear function (what it means)

A linear function is defined by these characteristics:

• The graph is a straight line
• You can recognize linearity from a table of values when:
  • The change in output is constant as input changes by the same amount
• It corresponds to an equation where the degree/exponent of the variable is 1 or less
  • Example: (f(x)=150-30x) (degree 1)

Input/output notation

• The function is written as (f(x)), meaning:
  • input: (x)
  • output: (f(x))

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2) Distinguishing linear vs non-linear (from tables/graphs/equations)

From tables

• Linear: output changes by a constant difference when input increases by 1 each time
• Non-linear: output change is not constant

From graphs

• Linear: line is straight
• Non-linear:
  • Graph is curved
  • Graph is not one straight line (e.g., changes direction)

From equations

To be linear, the “degree 1” idea must be satisfied.

• Linear examples:
  • (y = 2x + 3) (degree 1 in (x))
  • Equations where (x) appears only to the first power
• Non-linear examples:
  • Squared term: (y = x^2 + \dots)
  • (x) in the denominator (not a simple degree-1 form)
  • Variables multiplied (e.g., (xy = 6))
  • Cubic: (x^3)

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3) Slopes (steepness / rate of change)

Definition of slope

• Slope describes how steep a line is and equals:
  • vertical change ÷ horizontal change
• Written as: [ m = \dfrac{\Delta y}{\Delta x} ]

Types of slope

• Positive (rises left to right)
• Zero (horizontal line)

• Negative (falls left to right)

• Vertical lines have undefined slope

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4) Zero of a linear function

Definition

• The zero of a function is the x-value that makes the output equal to 0:
  • Find (x) such that (f(x)=0)

How to find it

• From a table: locate where output becomes 0, then take the corresponding input (x)
• From a graph: the line crosses the x-axis
  • The x-intercept’s x-coordinate is the zero
  • Emphasis: the “zero” refers to the x-value, not the full coordinate point

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Methodologies / step-by-step procedures (explicitly taught)

A) Finding slope and zero from a graph (3 main steps)

1. Choose two points with integer coordinates (between grid lines if needed).
2. Compute the rise and run:
   • rise = change in (y)
   • run = change in (x)

3. Compute slope: [ m = \dfrac{\text{rise}}{\text{run}} ]

4. Find the zero:

   • Identify where the line touches/crosses the x-axis
   • Write just the x-value of the x-intercept

Alternative slope method (two-point formula):

• [ m = \dfrac{y_2 - y_1}{x_2 - x_1} ]

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B) Finding slope and zero from an equation

1. Recall slope-intercept form:
   • (f(x) = mx + b)
   • where:
     • (m) = slope
     • (b) = y-intercept
2. To find the zero, set the output to zero:
   • set (f(x)=0)
   • solve for (x)
3. If not in slope-intercept form:
   • rearrange/rewrite into (y = mx + b) first (using algebraic rearrangement and dividing coefficients)

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C) Finding slope and zero from a table

1. Find the slope using change patterns:

   • When (x) increases by 1, compute how much (f(x)) changes
   • Use: [ m = \dfrac{\Delta y}{\Delta x} ]

2. Find the zero:

   • locate where the output (f(x)=0)
   • the corresponding input (x) is the zero
3. The video also shows using the slope formula by selecting two table points:
   • choose two ((x, y)) pairs and apply: [ m = \dfrac{y_2-y_1}{x_2-x_1} ]

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Activity/lesson wrap-up instructions

• The teacher assigns an activity divided into three parts:

  1. Activity 1: Determine slope and zero from a graph
     • Students answer, take a screenshot, and write answers on paper.
  2. Activity 2: Determine slope and zero from an equation
  3. Activity 3: Determine slope and zero from a table

• Students can check answers with teachers, especially if teachers have answer keys from PowerPoint presentations.

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Speakers / sources featured

• Math Easy YouTube Channel (host/teacher speaking on-screen)
• The video’s presenters/teachers referenced as audience support (no individual names given)
• MATATAG First Term/Q1 Tagalog (referenced in the video title; no individual credited)