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6.6.2: Practice Test

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    116132
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    Practice Test

    For the following exercises, sketch the graph of each function for two full periods. Determine the amplitude, the period, and the equation for the midline.

    1.

    f( x )=0.5sinx f( x )=0.5sinx

    2.

    f( x )=5cosx f( x )=5cosx

    3.

    f( x )=5sinx f( x )=5sinx

    4.

    f( x )=sin( 3x ) f( x )=sin( 3x )

    5.

    f( x )=cos( x+ π 3 )+1 f( x )=cos( x+ π 3 )+1

    6.

    f( x )=5sin( 3( x π 6 ) )+4 f( x )=5sin( 3( x π 6 ) )+4

    7.

    f( x )=3cos( 1 3 x 5π 6 ) f( x )=3cos( 1 3 x 5π 6 )

    8.

    f( x )=tan( 4x ) f( x )=tan( 4x )

    9.

    f( x )=2tan( x 7π 6 )+2 f( x )=2tan( x 7π 6 )+2

    10.

    f( x )=πcos( 3x+π ) f( x )=πcos( 3x+π )

    11.

    f( x )=5csc( 3x ) f( x )=5csc( 3x )

    12.

    f( x )=πsec( π 2 x ) f( x )=πsec( π 2 x )

    13.

    f( x )=2csc( x+ π 4 )3 f( x )=2csc( x+ π 4 )3

    For the following exercises, determine the amplitude, period, and midline of the graph, and then find a formula for the function.

    14.

    Give in terms of a sine function.

    A graph of two periods of a sine function, graphed from -2 to 2. Range is [-6,-2], period is 2, and amplitude is 2.
    15.

    Give in terms of a sine function.

    A graph of two periods of a sine function, graphed over -2 to 2. Range is [-2,2], period is 2, and amplitude is 2.
    16.

    Give in terms of a tangent function.

    A graph of two periods of a tangent function, graphed over -3pi/4 to 5pi/4. Vertical asymptotes at x=-pi/4, 3pi/4. Period is pi.

    For the following exercises, find the amplitude, period, phase shift, and midline.

    17.

    y=sin( π 6 x+π )3 y=sin( π 6 x+π )3

    18.

    y=8sin( 7π 6 x+ 7π 2 )+6 y=8sin( 7π 6 x+ 7π 2 )+6

    19.

    The outside temperature over the course of a day can be modeled as a sinusoidal function. Suppose you know the temperature is 68°F at midnight and the high and low temperatures during the day are 80°F and 56°F, respectively. Assuming t t is the number of hours since midnight, find a function for the temperature, D, D, in terms of t. t.

    20.

    Water is pumped into a storage bin and empties according to a periodic rate. The depth of the water is 3 feet at its lowest at 2:00 a.m. and 71 feet at its highest, which occurs every 5 hours. Write a cosine function that models the depth of the water as a function of time, and then graph the function for one period.

    For the following exercises, find the period and horizontal shift of each function.

    21.

    g( x )=3tan( 6x+42 ) g( x )=3tan( 6x+42 )

    22.

    n( x )=4csc( 5π 3 x 20π 3 ) n( x )=4csc( 5π 3 x 20π 3 )

    23.

    Write the equation for the graph in Figure 1 in terms of the secant function and give the period and phase shift.

    A graph of 2 periods of a secant function, graphed over -2 to 2. The period is 2 and there is no phase shift.
    Figure 1
    24.

    If tanx=3, tanx=3, find tan( x ). tan( x ).

    25.

    If secx=4, secx=4, find sec( x ). sec( x ).

    For the following exercises, graph the functions on the specified window and answer the questions.

    26.

    Graph m( x )=sin( 2x )+cos( 3x ) m( x )=sin( 2x )+cos( 3x ) on the viewing window [ 10,10 ] [ 10,10 ] by [ 3,3 ]. [ 3,3 ]. Approximate the graph’s period.

    27.

    Graph n( x )=0.02sin( 50πx ) n( x )=0.02sin( 50πx ) on the following domains in x: x: [ 0,1 ] [ 0,1 ] and [ 0,3 ]. [ 0,3 ]. Suppose this function models sound waves. Why would these views look so different?

    28.

    Graph f( x )= sinx x f( x )= sinx x on [ 0.5,0.5 ] [ 0.5,0.5 ] and explain any observations.

    For the following exercises, let f( x )= 3 5 cos( 6x ). f( x )= 3 5 cos( 6x ).

    29.

    What is the largest possible value for f( x )? f( x )?

    30.

    What is the smallest possible value for f( x )? f( x )?

    31.

    Where is the function increasing on the interval [ 0,2π ]? [ 0,2π ]?

    For the following exercises, find and graph one period of the periodic function with the given amplitude, period, and phase shift.

    32.

    Sine curve with amplitude 3, period π 3 , π 3 , and phase shift ( h,k )=( π 4 ,2 ) ( h,k )=( π 4 ,2 )

    33.

    Cosine curve with amplitude 2, period π 6 , π 6 , and phase shift ( h,k )=( π 4 ,3 ) ( h,k )=( π 4 ,3 )

    For the following exercises, graph the function. Describe the graph and, wherever applicable, any periodic behavior, amplitude, asymptotes, or undefined points.

    34.

    f( x )=5cos( 3x )+4sin( 2x ) f( x )=5cos( 3x )+4sin( 2x )

    35.

    f( x )= e sint f( x )= e sint

    For the following exercises, find the exact value.

    36.

    sin 1 ( 3 2 ) sin 1 ( 3 2 )

    37.

    tan 1 ( 3 ) tan 1 ( 3 )

    38.

    cos 1 ( 3 2 ) cos 1 ( 3 2 )

    39.

    cos 1 ( sin( π ) ) cos 1 ( sin( π ) )

    40.

    cos 1 ( tan( 7π 4 ) ) cos 1 ( tan( 7π 4 ) )

    41.

    cos( sin 1 ( 12x ) ) cos( sin 1 ( 12x ) )

    42.

    cos 1 ( 0.4 ) cos 1 ( 0.4 )

    43.

    cos( tan 1 ( x 2 ) ) cos( tan 1 ( x 2 ) )

    For the following exercises, suppose sint= x x+1 . sint= x x+1 . Evaluate the following expressions.

    44.

    tant tant

    45.

    csct csct

    46.

    Given Figure 2, find the measure of angle θ θ to three decimal places. Answer in radians.

    An illustration of a right triangle with angle theta. Opposite the angle theta is a side with length 12, adjacent to the angle theta is a side with length 19.
    Figure 2

    For the following exercises, determine whether the equation is true or false.

    47.

    arcsin( sin( 5π 6 ) )= 5π 6 arcsin( sin( 5π 6 ) )= 5π 6

    48.

    arccos( cos( 5π 6 ) )= 5π 6 arccos( cos( 5π 6 ) )= 5π 6

    49.

    The grade of a road is 7%. This means that for every horizontal distance of 100 feet on the road, the vertical rise is 7 feet. Find the angle the road makes with the horizontal in radians.


    6.6.2: Practice Test is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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