LEARN · EXPLAIN · REVISE

For x in radians, find the amplitude, period and range. Explain what the minus sign does to the graph.

Read the idea, work independently, then explain what changed.

TOPIC 01

Trigonometric ratios and function graphs

The negative multiplier reflects the cosine curve in its midline. At x = 0 the curve starts at its minimum −1; the amplitude is still positive.

PREDICT → MOVE → EXPLAIN

See amplitude, period and phase together.

Predict the number of cycles and the horizontal shift. Switch angle units without changing the curve.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Standard#Worked example

For x in radians, find the amplitude, period and range. Explain what the minus sign does to the graph.

y=-2\cos(3x)+1

Working and explanation

BUILD THE REASONING

Hint 1

Separate the vertical multiplier −2, horizontal multiplier 3 and vertical shift 1.

Hint 2

Cosine lies between −1 and 1. Its argument must advance by 2π for a full cycle.

Worked solution

Amplitude is the magnitude of the vertical multiplier.
$|A|=|-2|=2,\qquad D=1$
Solve for a change in x that produces a full cycle.
$3T=2\pi\Rightarrow T=\frac{2\pi}{3}$
Add the vertical shift after scaling the range.
$-1\le\cos(3x)\le1\Rightarrow -1\le y\le3$

Amplitude 2; period 2π/3; midline y = 1; range [−1, 3].

Checks and common pitfalls: The negative multiplier reflects the cosine curve in its midline. At x = 0 the curve starts at its minimum −1; the amplitude is still positive.

Think first. Reveal a hint when the class is ready.