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IBMathsDPAnalysis & Approaches (AA)HLFlashcards2. FunctionsTransformations of Graphs

Transformations of Graphs (DP IB Analysis & Approaches (AA))

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  • FrontTranslations of Graphs

    What happens to a graph during a translation?

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  • What happens to a graph during a translation?

    During a translation, the graph is moved in the xy plane, but its shape, size, and orientation remain unchanged.

  • What is a translation vector?

    A translation vector describes how far a graph is moved.

    The top value specifies how far left/right and the bottom value how far up/down.

    E.g. the translation vector open parentheses table row 3 row cell negative 7 end cell end table close parentheses describes a translation of 3 units in the positive x-direction and 7 units in the negative y-direction.

  • How is a horizontal translation represented in a function?

    A horizontal translation is represented in a function by the notation f open parentheses x plus a close parentheses.

    This describes a translation of the function f open parentheses x close parentheses by negative a units in the x-direction.

    Note that the sign inside the brackets is the opposite of the direction of movement.

  • What happens to x-coordinates during a horizontal translation?

    During a horizontal translation, described by f open parentheses x plus a close parentheses, the x-coordinates are translated by negative a units in the x-direction.

  • What happens to y-coordinates during a horizontal translation?

    During a horizontal translation, the y-coordinates are unchanged.

  • True or False?

    Vertical asymptotes stay the same during a horizontal translation.

    False.

    Vertical asymptotes do not stay the same during a horizontal translation.

    Graph showing two curves, y=f(x) in red and the vertical translation y=f(x+a) in blue, with vertical asymptotes at x=k and x=k-a.

  • How is a vertical translation represented in a function?

    A vertical translation is represented in a function by the notation f open parentheses x close parentheses plus a.

    This describes a translation of the function f open parentheses x close parentheses by a units in the positive y-direction.

    Note that the sign inside the brackets is the same as the direction of movement.

  • What happens to x-coordinates during a vertical translation?

    During a vertical translation, the x-coordinates are unchanged.

  • What happens to y-coordinates during a vertical translation?

    During a vertical translation, for a translation described by f open parentheses x close parentheses plus a, the y-coordinates are translated by a units in the positive y-direction.

  • True or False?

    Vertical asymptotes stay the same during a vertical translation.

    True.

    Vertical asymptotes stay the same during a vertical translation.

    Graph showing two curves, y = f(x ) in red and the vertical translation y = f(x) + a in blue. Both graphs have the same vertical asymptote at x = k.

  • True or False?

    A reflection changes the size of the graph.

    False.

    A reflection does not change the size of the graph, it only changes its position and orientation.

  • How is a horizontal reflection represented in a function?

    A horizontal reflection is represented in a function by the notation f open parentheses negative x close parentheses.

    This describes a reflection of the function f open parentheses x close parentheses across the y-axis.

    Graph with x and y axes showing two curves: one in black and the other in red. The two curves are reflections of each other in the y-axis.

  • What happens to x-coordinates during a horizontal reflection?

    During a horizontal reflection, the x-coordinates change sign.

    E.g. A point (2, -6) becomes (-2, -6) during a horizontal reflection in the y-axis.

  • True or False?

    Vertical asymptotes stay the same during a horizontal reflection.

    False.

    Vertical asymptotes do not stay the same during a horizontal reflection.

    Graph of two functions y=f(x) in red and its horizontal reflection y=f(-x) in blue. Vertical asymptotes are also reflections of each other with equations x=k and x=-k.

  • How is a vertical reflection represented in a function?

    A vertical reflection is represented in a function by the notation negative f open parentheses x close parentheses.

    This describes a translation of the function f open parentheses x close parentheses across the x-axis.

    Graph with x and y axes showing two curves: one in black and the other in red. The two curves are reflections of each other in the x-axis.

  • What happens to x-coordinates during a vertical reflection?

    During a vertical reflection, the x-coordinates stay the same.

    E.g. A point (2, -6) becomes (2, 6) during a vertical reflection in the x-axis.

  • True or False?

    The equation of a horizontal asymptote will change during a vertical reflection.

    True.

    The equation of a horizontal asymptote will change during a vertical reflection.

    Graph with two curves, y = f(x) in red and its vertical reflection y = -f(x) in blue, The horizontal asymptote for each curve are also reflections of each other in the x-axis and have equations y = k and y = -k .

  • What stays the same during a vertical reflection?

    During a vertical reflection, the x-coordinate of each point and any vertical asymptotes stay the same.

  • What happens to a graph when it undergoes a stretch?

    When a graph undergoes a stretch it is stretched about one of the coordinate axes by a scale factor.

    The size of the graph changes but its orientation stays the same.

  • Define scale factor.

    A scale factor is a constant that determines how much a graph is stretched.

    E.g. a stretch by a scale factor of 2 means that the distance between each point on the graph and the relevant axis is multiplied by 2.

  • True or False?

    A stretch with a scale factor less than 1 is called a compression.

    False.

    A stretch with a scale factor less than 1 is not called a compression.

    In exams, you must use the term "stretch" even for scale factors between 0 and 1.

  • What happens to the points on a graph undergoing a stretch when a scale factor is bigger than 1?

    When a graph undergoes a stretch where the scale factor is bigger than 1, the points on the graph get further away from the relevant axis.

  • What is the equation for a horizontal stretch of a graph f open parentheses x close parentheses by scale factor q?

    The equation for a horizontal stretch of a graph f open parentheses x close parentheses by scale factor q is f open parentheses 1 over q x close parentheses

  • What is the equation for a vertical stretch of a graph f open parentheses x close parentheses by scale factor p?

    The equation for a vertical stretch of a graph f open parentheses x close parentheses by scale factor p is p f open parentheses x close parentheses

  • What happens to y-coordinates during a horizontal stretch?

    During a horizontal stretch, the y-coordinates stay the same.

    E.g. A point (2, -5), undergoing a horizontal stretch with scale factor 3, becomes (6, -5).

  • True or False?

    Vertical asymptotes stay the same during a horizontal stretch.

    False.

    Vertical asymptotes do not stay the same during a horizontal stretch.

    Two graphs, y = f(x) in red and a horizontal stretch of that graph, y = f(qx) in blue. Each graph has a vertical asymptote, one with equation x = k and the other with equation x = (1/q)k.

  • What happens to y-coordinates during a vertical stretch?

    During a vertical stretch, each y-coordinate is multiplied by the scale factor.

    E.g. A point (2, -5), undergoing a vertical stretch with scale factor 3, becomes (2, -15).

  • True or False?

    For a horizontal stretch, the scale factor is the reciprocal of the coefficient of x.

    True.

    For a horizontal stretch, the scale factor is the reciprocal of the coefficient of x.

    E.g. for a function f open parentheses x close parentheses, the notation for a horizontal stretch of scale factor 2 is f open parentheses 1 half x close parentheses.

  • True or False?

    Because horizontal and vertical transformations are dependent on each other, the order in which they are performed makes a difference.

    False.

    Horizontal and vertical transformations are independent of each other.

    Because the transformations work in different directions, they do not affect each other so can be performed in any order.

  • True or False?

    When more than one transformation is applied to a function in the same direction, (i.e. two or more horizontal transformations or two or more vertical ones), the order in which the transformations are applied matters.

    True.

    When more than one transformation is applied to a function in the same direction, the order in which the transformations are applied matters.

    E.g. if a horizontal stretch and a horizontal translation are applied to the same function, the final graph will be different depending on which transformation is applied first.

  • What is the equation for a vertical stretch by factor a followed by a translation open parentheses 0 comma space b close parentheses?

    The equation for a vertical stretch by factor a followed by a translation open parentheses 0 comma space b close parentheses is y equals a f open parentheses x close parentheses plus b.

  • What is the equation for a translation open parentheses 0 comma space b close parentheses followed by a vertical stretch by factor a?

    The equation for a translation open parentheses 0 comma space b close parentheses followed by a vertical stretch by factor a is y equals a open square brackets f open parentheses x close parentheses plus b close square brackets.

  • True or False?

    If a function undergoes multiple vertical transformations, the order of transformations is determined by the order of operations in the equation.

    True.

    If a function undergoes multiple vertical transformations, the order of transformations is determined by the order of operations in the equation.

    E.g. for a function 2 open parentheses f open parentheses x close parentheses minus 8 close parentheses, the brackets comes before multiplication, so a translation of -8 is followed by a stretch of scale factor 2 (both in the y-direction).

  • In the equation y equals f open parentheses a x plus b close parentheses, what is the correct order of transformations?

    In the equation y equals f open parentheses a x plus b close parentheses, the correct order of transformations is:

    • a horizontal translation by negative b,

    • followed by a horizontal stretch with scale factor 1 over a.

  • What is the final equation for a horizontal translation of k units followed by a horizontal stretch by factor a?

    The final equation for a horizontal translation of k units followed by a horizontal stretch by factor a is y equals f open parentheses x over a minus k close parentheses.

  • What is the final equation for a horizontal stretch by factor a followed by a horizontal translation of k units?

    The final equation for a horizontal stretch by factor a followed by a horizontal translation of k units is y equals f open parentheses 1 over a open parentheses x minus k close parentheses close parentheses, or y equals f open parentheses x over a minus k over a close parentheses.

  • True or False?

    When a is negative in y equals f open parentheses a x plus b close parentheses, the reflection and stretch must be done in a specific order.

    False.

    When a is negative in y equals f open parentheses a x plus b close parentheses, the reflection and stretch can be done in any order.

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