Sinusoidal Graphs Test – Precalculus Math (Printable PDF)

Question 1 Features of a Sinusoid (2 points)

For y = 5 sin(4x − π) + 2: (a) Find the amplitude and the period. (1 point) (b) Find the phase shift and the midline. (1 point)

Question 2 Equation from a Graph (2 points)

The graph shows a sinusoid with a maximum at (2, 2) and minima at (0, −4) and (4, −4).

1. Find the amplitude, midline and period. (1 point)
2. Write an equation of the form y = A sin(B(x − C)) + D. (1 point)

Question 3 Tangent Features (2 points)

Let y = tan(3x). (a) Find the period. (1 point) (b) Find the two vertical asymptotes closest to the y-axis. (1 point)

Question 4 Write a Model from Extremes (3 points)

A sinusoidal function has a maximum of 14 at x = 2 and its next minimum of −6 at x = 8.

1. Find the amplitude and the midline. (1 point)
2. Find the period and B. (1 point)
3. Write an equation using cosine. (1 point)

Question 5 Bobbing Buoy (3 points)

A buoy’s height above the sea floor is h(t) = 2.5 sin(πt/3) + 4 feet, t in seconds.

1. Find the period. (1 point)
2. Find the maximum and minimum heights. (1 point)
3. Find the exact value of h(1) and a decimal approximation. (1 point)

Question 6 Buoy at a Given Height (2 points)

For the buoy with h(t) = 2.5 sin(πt/3) + 4, find every time in the first cycle (0 ≤ t < 6) when the height is 5.25 feet.

Question 7 Monthly Temperatures (3 points)

The average high temperature in a city is 38°F in January (month 1) and 94°F in July (month 7), and the pattern repeats every 12 months. Points for months 1, 4, 7 and 10 are plotted.

1. Write a cosine model T(m) = Acos(B(m − C)) + D. (2 points)
2. Predict the average high in March (month 3). (1 point)

Question 8 Transformation Description (3 points)

The graph of y = cos x is transformed into y = −2cos(x/2) + 3.

1. Describe each transformation in order. (1 point)
2. State the period. (1 point)
3. State the range. (1 point)