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A cosine curve has a maximum of 8 at x = 0, a minimum of 2, and successive maxima 4π apart. With x in radians, which equation fits all three features?

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

TOPIC 01

Trigonometric ratios and function graphs

B confuses amplitude with the full vertical span; C uses the wrong period; D starts at the minimum. Each stated feature helps rule out a different error.

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 / Transfer#Your turn

A cosine curve has a maximum of 8 at x = 0, a minimum of 2, and successive maxima 4π apart. With x in radians, which equation fits all three features?

First model $y=3\cos\left(\frac x2\right)+5$
Second model $y=6\cos\left(\frac x2\right)+2$
Third model $y=3\cos(2x)+5$
Fourth model $y=-3\cos\left(\frac x2\right)+5$

Working and explanation

BUILD THE REASONING

Hint 1

The midline is halfway between the largest and smallest values; amplitude is half their difference.

Hint 2

Use T = 2π/B with positive B, then check whether x = 0 is a maximum or a minimum.

Worked solution

Recover the vertical parameters.
$D=\frac{8+2}{2}=5,\qquad A=\frac{8-2}{2}=3$
Recover the horizontal multiplier from the period.
$B=\frac{2\pi}{4\pi}=\frac12$
A positive cosine coefficient gives the required initial peak.
$y(0)=3\cos0+5=8$

A: y = 3 cos(x/2) + 5.

Checks and common pitfalls: B confuses amplitude with the full vertical span; C uses the wrong period; D starts at the minimum. Each stated feature helps rule out a different error.

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