Cara mudah belajar FUNGSI dan GRAFIK FUNGSI (PART 2)

Main ideas and lessons (Part 2: Functions & Function Graphs)

1) Purpose of “function transformations” (shifts)

The instructor explains how to sketch graphs of transformed functions starting from a basic function.

General idea

• Start with a basic function (written like (y=f(x))).
• Apply modifications inside and/or outside the function.
• Determine how the graph shifts (up/down/left/right) accordingly.

Transformation rules mentioned

• Up/down shift (outside the function)
  • If (y=f(x)+a) → graph shifts up by (a)
  • If (y=f(x)-a) → graph shifts down by (a)
• Left/right shift (inside the function)
  • If (y=f(x+a)) → graph shifts left by (a)
  • If (y=f(x-a)) → graph shifts right by (a)
• Multiple transformations
  • If the expression includes both inside + outside changes, do:
    1. Simplify inside first (figure the left/right shift),
    2. Then apply outside (figure up/down shift).

Why this matters

• Instead of recalculating every point, you can sketch using the transformation from the basic graph.

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2) Basic function types to memorize + their graph shapes

The lesson lists common base functions and their key visual properties.

A) Constant function

• Form: (y=k)
• Graph: horizontal line
• Example: if (k=2), then (y) remains 2 for all (x).

B) Identity function and its opposite

• Identity: (y=x) (a 45° line through the origin)
• Opposite: (y=-x) (reflected across the origin)

C) Linear function (general)

• Form: (y=ax+b)
• Graph: straight line
• Meaning: slope is determined by (a), vertical shift by (b).
• Transforming the basic line uses shifts based on (b).

D) Quadratic function

• Form (general mentioned): (y=ax^2+bx+c)
• Basic: (y=x^2)
  • Opens upward when positive
• Reflected: (y=-x^2)
  • Opens downward

E) Cubic function

• Basic example: (y=x^3)
• Reflected: (y=-x^3)
• Key idea: the sign changes the direction the graph extends.

F) “Polynomial / other functions” note

• Mentions additional polynomial forms (e.g., (ax^n+\dots)).
• Graphing often relies on recognizing base forms and applying transformations.

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3) How to sketch function graphs (method taught through examples)

The instructor provides an approach for sketching graphs without computing every point.

Sketch method (implied by examples)

• Identify the basic function shape (e.g., (x^2), (x^3)).
• Recognize the operations as:
  • shifts up/down,
  • shifts left/right,
  • reflections (negative sign),
  • combinations of these.
• Use transformations to find key features:
  • For quadratics: focus on the vertex shift
• Use a few points to confirm the sketch.

Worked examples (graph sketch tasks)

1. (y=x^2-1)
   • Start from (y=x^2)
   • Shift down 1
2. (y=(x-1)^2) (inside shift)
   • Shift right 1
3. (y=x^2+2)
   • Shift up 2
4. (y=x^3-2)
   • Shift down 2
5. (y=x^2+4x-1) (factor/complete-square style)
   • Complete the square to rewrite as a form like:
     • ((x+2)^2 + (\text{constant}))
   • Then apply shifts
6. A more complex quadratic involving a negative sign
   • Uses reflection plus shifts after simplification (via factoring/completing the square).

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4) More advanced function types beyond algebraic/polynomial

After algebraic graphing basics, the instructor extends to other function families.

A) Trigonometric functions

• Sine: (y=\sin x)
  • Key values at special angles: starts at 0, reaches 1, down to (-1), etc.
• Cosine: (y=\cos x)
  • Similar key values at special angles.

B) Exponential functions

• Mentioned form: (y=e^x) (a transcription issue includes (y=x^x))
• Notes include example evaluations:
  • (e^0=1)
  • (e^{-1}\approx 0.367)
• Also mentioned: reflected exponential behavior depending on sign/inside changes.

C) Natural logarithm

• Natural log: (y=\ln x)
• Key ideas:
  • domain restriction and behavior near asymptotes
  • vertical asymptote at (x=0)
• Relations mentioned/used:
  • (\ln(1)=0)
  • (\ln(e)=1)

(There is also mention of base-10 logs like (\log_{10}), but the focus returns to natural logs.)

D) Even and odd functions

• Odd function (example: sine)
  • Property: (f(-x)=-f(x))
  • Graph symmetry about the origin
• Even function (example: cosine)
  • Property: (f(-x)=f(x))
  • Graph symmetry about the y-axis

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5) Cartesian coordinate system + straight line and circle equations

A) Coordinate system basics

• x-axis: horizontal, y-axis: vertical
• Origin: ((0,0))
• Points: (P(x_1,y_1)), (Q(x_2,y_2))

B) Distance between two points (Pythagoras)

• Distance formula: [ \text{distance }PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} ]

C) Straight line equations (multiple cases)

Forms for a line

• General form: (ax+by+c=0)

• Slope/gradient: [ m=\frac{y_2-y_1}{x_2-x_1} ]

• Point-slope form: (y-y_1=m(x-x_1))

• Slope-intercept form: (y=mx+c)

Parallel lines

• Parallel lines have the same slope.

Perpendicular lines

• If lines are perpendicular: [ m_1m_2=-1 ]

D) Circle equations

• The key idea is center/radius interpretation.

• Circle centered at ((0,0)) with radius (r): [ x^2+y^2=r^2 ]

• Circle centered at ((h,k)) with radius (r): [ (x-h)^2+(y-k)^2=r^2 ]

• Expanded circle equations are also referenced to identify center and radius via completing the square.

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6) Example problems (coordinate geometry)

Example 1: Straight line parallel to a given line

• Given line (example): (7x-14y+8=0)
• Find a line parallel to it through ((-2,5))
• Method:
  • parallel ⇒ same slope
  • use point-slope or slope-intercept form

Example 2: Straight line perpendicular to a given line

• Given a line (example described via gradient derivation)
• Find a line perpendicular to it through ((1,3))
• Method:
  • perpendicular ⇒ (m_1m_2=-1)
  • use point-slope form

Example 3: Determine a circle from an expanded equation

• Given an expanded quadratic-looking equation in (x) and (y)
• Steps:
  • group (x)-terms and (y)-terms
  • complete the square
  • rewrite into standard circle form
• Then determine:
  • center
  • radius

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7) Homework instructions (as stated)

Homework

• Homework 1 (function-related / lines)

  • Find the equation of a line passing through the intersection point of:
    • (3x-4y-5=0) (transcription may vary)
    • (2x+3y=9)
  • The required line is perpendicular to the line (2x+3y=9).
  • Method:
    • find the intersection point first (solve simultaneously),
    • then use the perpendicular-slope rule to get the new line.

• Homework 2 (line through given points / parallel & perpendicular constraints)

  • Determine the equation of a line through ((2,1))
  • Include constraints such as:
    • parallel to (3x-2y=5), and/or
    • perpendicular to (4y+3x-9=0)
  • (Some subtitle details are garbled, but the task is clearly about writing line equations using parallel/perpendicular conditions.)

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Speakers / sources featured

• Mrs. Endang (main instructor/lecturer)