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Trigonometric Graphs Practice Quiz

Explore key features of trigonometric graphs, such as amplitude, period, midline, key coordinate configurations, and asymptotic behavior. Select your answers below to verify your logic.

Question 1: Amplitude of Sine Waves

Identifying Peak Displacement

Determine the amplitude of the following periodic function:
y = -3 sin(2θ) + 1

Step-by-Step Worked Solution:
  • Represent the equation in general transformational form:
    y = a sin(b(θ - c)) + d
  • Identify the parameters from the given function: a = -3, b = 2, and d = 1.
  • Recall the definition of amplitude: Amplitude represents half the distance between the maximum and minimum values of the wave and is mathematically defined as the absolute value of coefficient a:
    Amplitude = |a|
  • Substitute the value of a:
    Amplitude = |-3| = 3
  • Therefore, even though the graph is reflected vertically across its midline, the positive amplitude remains 3.
Question 2: Period of Cosine Waves

Calculating Cycle Length

Determine the exact horizontal period of the graph representing:
y = cos(4θ)

Step-by-Step Worked Solution:
  • Locate the horizontal frequency coefficient b from the standard form equation y = cos(bθ). Here, b = 4.
  • Recall the formula used to calculate the period of sine or cosine functions:
    Period =
    360° |b|
    (or
    2π |b|
    radians)
  • Substitute the frequency value into the equation:
    Period =
    360° 4
    = 90°
  • In radians, this corresponds to:
    2π 4
    =
    π 2
  • This indicates that the function repeats its full wave pattern every 90°.
Question 3: Key Points on parent graphs

Tracking Curve Extremes

On a standard graph of the parent sine function y = sin(θ) over the interval 0° ≤ θ ≤ 360°, at what exact value of θ does the curve reach its absolute minimum point?

Step-by-Step Worked Solution:
  • Recall the coordinates of the key boundary points within the standard period of the sine function:
    • At θ = 0°: sin(0°) = 0 (Starting midline intercept)
    • At θ = 90°: sin(90°) = 1 (Maximum peak point)
    • At θ = 180°: sin(180°) = 0 (Midline intercept)
    • At θ = 270°: sin(270°) = -1 (Minimum trough point)
    • At θ = 360°: sin(360°) = 0 (Ending midline intercept)
  • Identify the lowest coordinate value yielded by the function, which is -1.
  • This absolute minimum occurs exactly at θ = 270° (or 3π/2 radians).
Question 4: Midlines and Vertical Shifts

Identifying the Horizontal Axis of Symmetry

What is the mathematical equation representing the midline (average equilibrium line) of the following trigonometric graph?
y = 2 cos(θ) - 5

Step-by-Step Worked Solution:
  • Represent the equation in standard form: y = a cos(θ) + d, where a controls vertical stretching (amplitude) and d controls vertical shift.
  • Identify parameters: a = 2, and d = -5.
  • Recall that the midline is always defined as the horizontal line representing the vertical shift of the function:
    y = d
  • Substitute the vertical shift parameter to find the midline equation:
    y = -5
  • This means the graph oscillates 2 units above and 2 units below this equilibrium boundary, between y = -3 and y = -7.
Question 5: Asymptotes of Tangent Graphs

Identifying Domain Constraints

At which of the following values of θ does the graph of the tangent function y = tan(θ) exhibit a vertical asymptote?

Step-by-Step Worked Solution:
  • Express the tangent function in terms of its ratio identities:
    tan(θ) =
    sin(θ) cos(θ)
  • Recall that rational expressions are mathematically undefined wherever their denominators evaluate to zero. This occurs where:
    cos(θ) = 0
  • Identify the angle inputs within standard bounds where the cosine function equals zero:
    θ = 90° (π/2), 270° (3π/2), etc.
  • Since the quotient cannot be calculated at these points, the values approach infinity or negative infinity, producing vertical asymptotes.
  • Among the choices provided, the asymptote is situated at θ = 90°.