Algebra 2 Question of the Day

1. Amplitude $2$; midline $y=0$; period $\pi$
2. Amplitude $2$; midline $y=-2$; period $2\pi$
3. Amplitude $1$; midline $y=2$; period $\pi$
4. Amplitude $1$; midline $y=-2$; period $2\pi$ (correct answer)

Explanation: This question tests your ability to graph trigonometric functions by identifying their characteristic features like amplitude, midline, and period. Trigonometric functions like f(x) = a·sin(bx) + d have periodic (repeating) graphs with three key features: amplitude |a| is the vertical distance from the midline to a peak, period 2π/|b| is the horizontal length of one complete cycle, and midline y = d is the horizontal center line the graph oscillates around. The graph oscillates between y = d - |a| (minimum) and y = d + |a| (maximum), repeating this wave pattern every 2π/|b| units. For f(x) = 3sin(2x) + 1: amplitude 3, period π, midline y = 1, oscillating between -2 and 4. For r(x) = sin(x) - 2, amplitude |1|=1, period 2π/|1|=2π, midline y=-2, oscillating between -3 and -1. Choice A correctly identifies amplitude 1, midline y=-2, and period 2π. A distractor like choice D might double the amplitude to 2, perhaps confusing the vertical shift with stretching. For trigonometric functions (sine and cosine): (1) Amplitude = |a| tells you how far from midline to peak (vertical stretch), (2) Period = 2π/|b| tells you how long one complete wave takes (horizontal compression if |b| > 1), (3) Midline y = d tells you the horizontal center (vertical shift). To sketch: draw the midline as a dashed horizontal line at y = d, mark one period length, sketch wave oscillating ±a from the midline. Sine starts at midline going up, cosine starts at maximum. The wave repeats every period!