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Trigonometric Functions & Graphs (DP IB Analysis & Approaches (AA))

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  • FrontGraphs of Trigonometric Functions

    Which function is this the graph of?

    Periodic graph ranging from -180 to 360 on the x-axis. Vertical asymptotes at -90, 90, 270 degrees. The curve repeats every 180 degrees, transitioning from negative to positive infinity.

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Cards in this collection (33)

  • Which function is this the graph of?

    Periodic graph ranging from -180 to 360 on the x-axis. Vertical asymptotes at -90, 90, 270 degrees. The curve repeats every 180 degrees, transitioning from negative to positive infinity.

    The graph is of the tangent function.

    It is a periodic graph with the following features:

    • x-intercepts at -180º, 0º, 180º, 360º (and every 180º in either direction),

    • a y-intercept at (0, 0),

    • asymptotes at -90º, 90º, 270º (and every 180º in either direction),

    • and a range of y element of straight real numbers.

  • Which function is this the graph of?

    Periodic graph ranging from -180 to 360 on the x-axis. The curve intersects the x-axis at 90 and every 180 from there in either direction. The graph intercepts the y-axis at (0, 1) and cycles between a maximum of 1 and a minimum of -1.

    The graph is of the cosine function.

    It is a periodic graph with the following features:

    • x-intercepts at -90º, 90º, 270º (and every 180º in either direction),

    • a y-intercept at (0, 1),

    • and a range of -1 ≤ y ≤ 1.

  • Which function is this the graph of?

    Periodic graph ranging from -180 to 360 on the x-axis. The curve intersects the x-axis at 0 and every 180 from there in either direction. The graph intercepts the y-axis at (0, 0) and cycles between a maximum of 1 and a minimum of -1.

    The graph is of the sine function.

    It is a periodic graph with the following features:

    • x-intercepts at -180º, 0º, 180º, 360º (and every 180º in either direction),

    • a y-intercept at (0, 0),

    • and a range of -1 ≤ y ≤ 1.

  • True or False?

    The graph of sin x passes through the origin.

    True.

    The graph of sin x passes through the origin.

  • What is the period of tan x?

    The period of tan x is 180º (or pi radians).

  • What is the period of sin x and cos x?

    The period of both sin x and cos x is 360º (or 2pi radians).

  • Define the range of sin x and cos x.

    The range of both sin x and cos x is -1 ≤ y ≤ 1.

  • Define the range of tan x.

    The range of tan x is y element of straight real numbers. I.e., tan x can take any real number value (positive, negative or zero).

  • What are the equations of the asymptotes in the graph of tan x, negative pi less or equal than x less or equal than 2 pi.

    The equations of the asymptotes in the graph of tan x, negative pi less or equal than x less or equal than 2 pi are

    x equals negative pi over 2, x equals pi over 2 and x equals fraction numerator 3 pi over denominator 2 end fraction.

  • What is the relationship between sin(-x) and sin(x)?

    The relationship between sin(-x) and sin(x) is sin(-x) = -sin(x).

  • What is the relationship between cos(-x) and cos(x)?

    The relationship between cos(-x) and cos(x) is cos(-x) = cos(x).

  • True or False?

    tan(x) = tan(x ± 180°)

    True.

    tan(x) = tan(x ± 180°) or tan(x ± pi)

  • What is the relationship between sin(x) and sin(pi - x)?

    The relationship between sin(x) and sin(pi - x), is sin(x) = sin(pi - x), (or sin(x) = sin(180° - x) ).

  • What does y equals sin open parentheses x close parentheses plus a represent?

    y equals sin open parentheses x close parentheses plus a represents a vertical translation of y equals sin open parentheses x close parentheses by a units in the positive y-direction.

  • How is a horizontal translation to the left by a units represented for sin open parentheses x close parentheses?

    For sin open parentheses x close parentheses, a horizontal translation to the left by a units is represented by sin open parentheses x plus a close parentheses.

  • What transformation does y equals a sin open parentheses x close parentheses represent?

    y equals a sin open parentheses x close parentheses represents a transformation of y equals sin open parentheses x close parentheses by a vertical stretch with scale factor a.

  • How is a horizontal stretch with scale factor a represented for sin open parentheses x close parentheses?

    For sin open parentheses x close parentheses, a horizontal stretch with scale factor a is represented by y equals sin open parentheses x over a close parentheses.

  • What transformation does y equals negative sin open parentheses x close parentheses represent?

    y equals negative sin open parentheses x close parentheses represents a reflection of sin open parentheses x close parentheses about the x-axis.

  • In the function a space sin open parentheses b x close parentheses, what does the absolute value of a represent?

    In the function a space sin open parentheses b x close parentheses, the absolute value of a represents the amplitude of the graph.

  • What is the period of a space sin open parentheses b x close parentheses in degrees?

    The period of a space sin open parentheses b x close parentheses in degrees is fraction numerator 360 º over denominator b end fraction.

  • In a space sin open parentheses b open parentheses x minus c close parentheses close parentheses plus d, what does c represent?

    In a space sin open parentheses b open parentheses x minus c close parentheses close parentheses plus d, the c represents a horizontal translation.

  • What is the equation of the principal axis for the function a space sin open parentheses b open parentheses x minus c close parentheses close parentheses plus d?

    The equation of the principal axis for the function a space sin open parentheses b open parentheses x minus c close parentheses close parentheses plus d is y equals d.

  • What is the period of a space tan open parentheses b open parentheses x minus c close parentheses close parentheses plus d in radians?

    The period of a space tan open parentheses b open parentheses x minus c close parentheses close parentheses plus d is pi over b radians.

  • True or False?

    The order of applying transformations doesn't matter.

    False.

    The order in which transformations are applied is important.

  • In what order should transformations be applied?

    Any stretches should be applied first, followed by any translations. Reflections should be applied last.

  • What line will the maximum points for a space sin open parentheses b open parentheses x minus c close parentheses close parentheses plus d lie on?

    The maximum points for a space sin open parentheses b open parentheses x minus c close parentheses close parentheses plus d will lie on the line y equals a plus d.

  • What type of phenomena can be modelled using trigonometric functions?

    Any phenomena that fluctuates periodically can be modelled using a trigonometric function.

    E.g. the vertical height above the ground of a person on a Ferris wheel over time, or the water depth of a tidal river over time, could be modelled using a trig function.

  • If a trigonometric function y equals a space sin open parentheses b open parentheses x minus c close parentheses close parentheses plus d is used to model the water depth of a tidal river, what would a represent?

    In a trigonometric model y equals a space sin open parentheses b open parentheses x minus c close parentheses close parentheses plus d, the absolute value of a represents the amplitude of the function.

    If the function is used to model the water depth of a tidal river, a would represent the maximum distance that the water depth would be above and below the principal axis.

  • In the trigonometric function y equals a space sin open parentheses b open parentheses x minus c close parentheses close parentheses plus d, how does the value of b affect the model?

    In the trigonometric function y equals a space sin open parentheses b open parentheses x minus c close parentheses close parentheses plus d, changing the value of b changes the period of the function. The bigger the value of b, the quicker the function repeats a cycle.

  • In the trigonometric function y equals a space sin open parentheses b open parentheses x minus c close parentheses close parentheses plus d, how does the value of d affect the model?

    In the trigonometric function y equals a space sin open parentheses b open parentheses x minus c close parentheses close parentheses plus d, y equals d is the principal axis.

    As the value of d increases, the graph is shifted up, as the value of d decreases, the graph is shifted down.

  • In the trigonometric function y equals a space sin open parentheses b open parentheses x minus c close parentheses close parentheses plus d, what does c represent?

    In the trigonometric function y equals a space sin open parentheses b open parentheses x minus c close parentheses close parentheses plus d, the horizontal shift of the graph is represented by c.

  • True or False?

    In real-life scenarios modelled by trigonometric functions, the time to complete a cycle always remains constant.

    False.

    In real-life scenarios modelled by trigonometric functions, the time to complete a cycle does not always remain constant.

    The time taken to complete a cycle may change over time.

    This is one of the limitations of a trigonometric model.

  • True or False?

    One limitation of a trigonometric model is that the the amplitude is constant.

    True.

    One limitation of a trigonometric model is that the the amplitude is constant.

    In real life, the amplitude is not always constant.

    For example, the function may get closer to the principal axis over time.

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