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    General Mathematics - M04 - L04 - WEEK 1

    Uploaded by

    Ji Pao
    This document provides examples and explanations of how to graph transformations of exponential functions based on a base function. It covers reflections about the x-axis and y-axis, vertical stretching and shrinking by multiplying the y-values by constants, vertical shifting by adding or subtracting constants, and horizontal shifting by adding or subtracting constants inside the function. Several examples are worked through step-by-step and then generalized. The document concludes with evaluation problems asking to identify the transformations, sketch the graphs, and provide key features like domain, range, y-intercepts and horizontal asymptotes.

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    © All Rights Reserved
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    General Mathematics - M04 - L04 - WEEK 1

    Uploaded by

    Ji Pao
    52 views4 pages
    This document provides examples and explanations of how to graph transformations of exponential functions based on a base function. It covers reflections about the x-axis and y-axis, vertical stretching and shrinking by multiplying the y-values by constants, vertical shifting by adding or subtracting constants, and horizontal shifting by adding or subtracting constants inside the function. Several examples are worked through step-by-step and then generalized. The document concludes with evaluation problems asking to identify the transformations, sketch the graphs, and provide key features like domain, range, y-intercepts and horizontal asymptotes.

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    General Mathematics_M04_L04 - WEEK 1

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    This document provides examples and explanations of how to graph transformations of exponential functions based on a base function. It covers reflections about the x-axis and y-axis, vertical stretching and shrinking by multiplying the y-values by constants, vertical shifting by adding or subtracting constants, and horizontal shifting by adding or subtracting constants inside the function. Several examples are worked through step-by-step and then generalized. The document concludes with evaluation problems asking to identify the transformations, sketch the graphs, and provide key features like domain, range, y-intercepts and horizontal asymptotes.

    Copyright:

    © All Rights Reserved
    Download as DOCX, PDF, TXT or read online from Scribd
    Download as docx, pdf, or txt
    52 views4 pages

    General Mathematics - M04 - L04 - WEEK 1

    Uploaded by

    Ji Pao
    This document provides examples and explanations of how to graph transformations of exponential functions based on a base function. It covers reflections about the x-axis and y-axis, vertical stretching and shrinking by multiplying the y-values by constants, vertical shifting by adding or subtracting constants, and horizontal shifting by adding or subtracting constants inside the function. Several examples are worked through step-by-step and then generalized. The document concludes with evaluation problems asking to identify the transformations, sketch the graphs, and provide key features like domain, range, y-intercepts and horizontal asymptotes.

    Copyright:

    © All Rights Reserved
    Download as DOCX, PDF, TXT or read online from Scribd
    Download as docx, pdf, or txt
    of 4

    JC Excellente Christian Academy Inc.

    ACADEMIC EXCELLENCE.LEADERSHIP POTENTIAL. CHRISTIAN VALUES


    Blk. 40 Lot 73 Road 1 Minuyan II, CSJDM Bulacan

    General Mathematics
    2nd GRADING - WEEK 1
    Module 4
    Lesson 3

    TOPIC: Graphing Transformations of Exponential Functions

    INSTRUCTION:

    Reflecting Graphs

    Example #1: Use the graph of y=2x to graph the functions y=−2x and y=2−x .

    Solution:
    Some y-values are shown on the following table.

    The y-coordinate of each point on the graph of y=– 2x the negative of the y-coordinate of the graph of y=2x .
    Thus, the graph of y=– 2x is the reflection of the graph of y=2x about the x-axis.
    The value of y=2−x at x is the same as the value of y=2x at – x. Thus, the graph of y=2−x is the reflection of the
    graph of y=2x about the y-axis.

    The corresponding graphs are shown below:

    The results in Example #1 can be generalized as follows:


    Reflection
     The graph of y=– f (x) is the reflection about the x-axis of the graph of y=f ( x ).
     The graph of y=f (– x) is the reflection about the y-axis of the graph of y=f ¿).

    Example #2:
    Use the graph of y=2x to graph the functions y=3 (2x ) and y=0.4(2 x ).

    Solution:
    Some y-values are shown on the following table.
    The y-coordinate of each point on the graph of y=3 (2x ) is 3 times the y-coordinate of each point on y=2x .
    Similarly, the y-coordinate of each point on the graph of y=0.4 ( 2 x ) is 0.4 times the y-coordinate of each point
    on y=2x .

    The graphs of these functions are shown below.

    Observations:
    1. The domain for all three graphs is the set of all real numbers.
    2. The y-intercepts were also multiplied correspondingly. The y-intercept of y=3 (2x ) is 3, and the y-
    intercept of y=0.4 ( 2 x ) is 0.4.
    3. All three graphs have the same horizontal asymptote: y=0.
    4. The range of all three graphs is the set of all y >0.

    The results of Example #2 can be generalized as follows:


    Vertical Stretching or Shrinking
    Let c be a positive constant. The graph of y=cf ( x) can be obtained from the graph of y=f (x ) by multiplying
    each y-coordinate by c. The effect is a vertical stretching (if c >1) or shrinking (if c <1) of the graph of y=f ( x ).

    Example #3:
    Use the graph of y=2x to graph y=2x – 3 and y=2x +1.

    Solution:
    Some y-values are shown on the following table:

    The graphs of these functions are shown below:


    Observations:
     The domain for all three graphs is the set of all real numbers.
     The y-intercepts and horizontal asymptotes were also vertically translated from the y-intercept and
    horizontal asymptote of y=2x .
     The horizontal asymptote of y=2x is y=0. Shift this 1 unit up to get the horizontal asymptote of
    y=2x +1 which is y=1, and 3 units down to get the horizontal asymptote of y=2x – 3 which is
    y=– 3.
     The range of y=2x +1 is all y >1, and the range of y=2x – 3 is all y > – 3.

    The results of Example #3 can be generalized as follows:


    Vertical Shifts
    Let k be a real number. The graph of y=f ( x )+ k is a vertical shift of k units up (if k > 0) or k units down (if k < 0
    ) of the graph of y=f (x ).

    Example #4:
    Use the graph of y=2x to graph y=2x−2and y=2x+ 4.

    Solution.
    Some y-values are shown on the following table.

    The graphs of these functions are show below:

    Observations:
     The domain for all three graphs is the set of all real numbers.
     The y-intercepts changed. To find them, substitute x=0 in the function. Thus, the y-intercept of
    y=2x+ 4 is 24 =16 and the y-intercept of y=2x−2 is 2 – 2=.25.
     The horizontal asymptotes of all three graphs are the same ( y=0). Translating a graph horizontally does not
    change the horizontal asymptote.
     The range of all three graphs is the set of all y >0 .

    The results of Example #4 can be generalized as follows:


    Horizontal Shifts
     Let k be a real number. The graph of y=f ( x – k ) is a horizontal shift of k units to the right (if k > 0) or k
    units to the left (if k < 0) of the graph of y=f ( x ).
    SOLVED EXAMPLES:

    1. Sketch the graph of F (x)=3 x+1 – 2, then state the domain, range, y-intercept, and horizontal asymptote.

    Solution:
    Transformation:
    The base function f (x)=3x will be shifted 1 unit to the left and 2 units down

    Steps in Graph Sketching:


    Step 1: Base function: f (x)=3x ; y-intercept: (0,1) ; horizontal asymptote: y=0
    Step 2: The graph of F (x) is found by shifting the graph of the function f left one unit and down two units.
    Step 3: The y-intercept of f (x) (0,1) will shift to the left by one unit and down two units towards ( – 1 , – 1).
    This is not the y-intercept of F (x).
    Step 4: The horizontal asymptote will be shifted down two units, which is y=– 2.
    Step 5: Find additional points on the graph; F ( 0 )=30 +1 – 2 = 1 and F ( 1 )=3 1+1 – 2=7 .
    Step 6: Connect the points using a smooth curve.

    Domain: All real numbers


    Range: (−2 , ∞)
    y-intercept: (0,1)
    Horizontal Asymptote: y=−2

    EVALUATION:

    Find the following for each of the given.


    (a) use transformations to describe how the graph is related to its base exponential function y=b x,
    (b) sketch the graph;
    (c) identify its domain, range, y-intercept, and horizontal asymptote.

    1. F ( x )=2 ( 3 x )

    x+1
    1
    2. G ( x ) = ()4
    −4

    3. H ( x ) =−2(3 x−1 )

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