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Amplitude, period and phase shift practice problems

Amplitude, period, phase shift and vertical shift of y = A sin(Bx − C) + D. 60 problems with worked solutions; in 60 of them every equation is proved by a computer algebra system.

2 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 13.314

medium

For \( \displaystyle y = 3 \sin{\left(\frac{x}{2} - \frac{1}{2} \right)} - 1 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Write it as A·sin(B(x − h)) + D: A = 3, B = 1/2, h = 1, D = -1.
Reviewed
\[ \frac{2 \pi}{1 \cdot \frac{1}{2}} = 4 \pi \]
Period = 2π/B.✓ Proved
\[ \left(1 - 3 \sin{\left(\frac{x}{2} - 1 \cdot \frac{1}{2} \right)}\right) + \left. 3 \sin{\left(\frac{x}{2} - 1 \cdot \frac{1}{2} \right)} - 1 \right|_{\substack{ x=x + 4 \pi }} = 0 \]
Shifting x by one period gives the same function.✓ Proved
\[ 1 \frac{1}{1 \cdot 2 \cdot \frac{1}{2}} = 1 \]
Phase shift h = C/B.✓ Proved
\[ \left. 3 \sin{\left(\frac{x}{2} - 1 \cdot \frac{1}{2} \right)} - 1 \right|_{\substack{ x=1 + \pi }} = 2 \]
At x = 1 + pi the curve is 3 from its midline y = -1: that is the amplitude.✓ Proved

Answer \( \text{amplitude } 3,\ \text{period } 4 \pi,\ \text{phase shift } 1\text{ to the right},\ \text{vertical shift } -1 \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.315

medium

For \( \displaystyle y = - 4 \sin{\left(\frac{x}{2} - \frac{1}{2} \right)} - 1 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Write it as A·sin(B(x − h)) + D: A = -4, B = 1/2, h = 1, D = -1.
Reviewed
\[ \frac{2 \pi}{1 \cdot \frac{1}{2}} = 4 \pi \]
Period = 2π/B.✓ Proved
\[ \left(4 \sin{\left(\frac{x}{2} - 1 \cdot \frac{1}{2} \right)} + 1\right) + \left. - 4 \sin{\left(\frac{x}{2} - 1 \cdot \frac{1}{2} \right)} - 1 \right|_{\substack{ x=x + 4 \pi }} = 0 \]
Shifting x by one period gives the same function.✓ Proved
\[ 1 \frac{1}{1 \cdot 2 \cdot \frac{1}{2}} = 1 \]
Phase shift h = C/B.✓ Proved
\[ \left. - 4 \sin{\left(\frac{x}{2} - 1 \cdot \frac{1}{2} \right)} - 1 \right|_{\substack{ x=1 + \pi }} = -5 \]
At x = 1 + pi the curve is 4 from its midline y = -1: that is the amplitude.✓ Proved

Answer \( \text{amplitude } 4,\ \text{period } 4 \pi,\ \text{phase shift } 1\text{ to the right},\ \text{vertical shift } -1 \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More amplitude, period and phase shift practice

For \( \displaystyle y = - 2 \sin{\left(2 x \right)} - 1 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.104easy✓ Every equation proved

For \( \displaystyle y = \cos{\left(2 x \right)} - 1 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.269easy✓ Every equation proved

For \( \displaystyle y = 4 \cos{\left(2 x \right)} - 3 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.272easy✓ Every equation proved

For \( \displaystyle y = 2 \sin{\left(x \right)} - 3 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.279easy✓ Every equation proved

For \( \displaystyle y = 4 \cos{\left(4 x \right)} - 2 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.91medium✓ Every equation proved

For \( \displaystyle y = 4 \sin{\left(4 x + 8 \right)} + 3 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.92medium✓ Every equation proved

For \( \displaystyle y = 3 \cos{\left(\frac{x}{2} + \frac{3 \pi}{8} \right)} - 2 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.93medium✓ Every equation proved

For \( \displaystyle y = 3 \cos{\left(2 x + \frac{\pi}{6} \right)} - 1 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.94medium✓ Every equation proved

For \( \displaystyle y = 2 - 2 \sin{\left(\frac{\pi x}{2} \right)} \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.95medium✓ Every equation proved

For \( \displaystyle y = - 3 \cos{\left(\frac{\pi x}{2} \right)} - 1 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.96medium✓ Every equation proved

For \( \displaystyle y = 2 \cos{\left(4 x + 8 \right)} \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.97medium✓ Every equation proved

For \( \displaystyle y = 2 \sin{\left(3 x \right)} + 1 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.98medium✓ Every equation proved

For \( \displaystyle y = 3 \sin{\left(x + \frac{\pi}{4} \right)} + 2 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.99medium✓ Every equation proved

For \( \displaystyle y = - 4 \sin{\left(2 x \right)} \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.100medium✓ Every equation proved

For \( \displaystyle y = 3 \sin{\left(3 x \right)} - 3 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.101medium✓ Every equation proved

For \( \displaystyle y = 3 \sin{\left(2 x + \frac{\pi}{6} \right)} + 1 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.102medium✓ Every equation proved

For \( \displaystyle y = 2 - 2 \sin{\left(x + \frac{\pi}{3} \right)} \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.103medium✓ Every equation proved

For \( \displaystyle y = 4 \sin{\left(x - 1 \right)} \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.105medium✓ Every equation proved

For \( \displaystyle y = 3 \cos{\left(3 x \right)} + 2 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.266medium✓ Every equation proved

For \( \displaystyle y = - \sin{\left(3 x \right)} \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.267medium✓ Every equation proved

For \( \displaystyle y = 4 \cos{\left(\frac{\pi x}{2} \right)} + 3 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.268medium✓ Every equation proved

For \( \displaystyle y = 2 \cos{\left(4 x + \frac{\pi}{3} \right)} - 3 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.270medium✓ Every equation proved

For \( \displaystyle y = - 3 \sin{\left(4 x - 4 \right)} - 2 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.271medium✓ Every equation proved

For \( \displaystyle y = 4 \cos{\left(2 x - 2 \right)} \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.273medium✓ Every equation proved

For \( \displaystyle y = - 2 \sin{\left(\frac{\pi x}{2} \right)} - 2 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.274medium✓ Every equation proved

For \( \displaystyle y = 3 \cos{\left(\frac{\pi x}{2} \right)} + 3 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.275medium✓ Every equation proved

For \( \displaystyle y = \sin{\left(\frac{x}{2} + \frac{5 \pi}{12} \right)} - 2 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.276medium✓ Every equation proved

For \( \displaystyle y = - 3 \cos{\left(3 x \right)} - 1 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.277medium✓ Every equation proved

For \( \displaystyle y = - 4 \sin{\left(\frac{\pi x}{2} + \frac{\pi}{4} \right)} - 1 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.278medium✓ Every equation proved

For \( \displaystyle y = - 3 \sin{\left(2 x \right)} - 2 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.280medium✓ Every equation proved

For \( \displaystyle y = 3 \sin{\left(\frac{x}{2} + \frac{3 \pi}{8} \right)} - 1 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.312medium✓ Every equation proved

For \( \displaystyle y = - 4 \sin{\left(\pi x \right)} \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.313medium✓ Every equation proved

For \( \displaystyle y = - \sin{\left(2 x + 4 \right)} - 2 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.316medium✓ Every equation proved

For \( \displaystyle y = 3 - 3 \sin{\left(2 x - 2 \right)} \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.317medium✓ Every equation proved

For \( \displaystyle y = 2 - \sin{\left(4 x \right)} \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.318medium✓ Every equation proved

For \( \displaystyle y = - 2 \sin{\left(3 x + 6 \right)} \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.319medium✓ Every equation proved

For \( \displaystyle y = - 3 \sin{\left(\pi x \right)} - 3 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.320medium✓ Every equation proved

For \( \displaystyle y = 3 \cos{\left(x + \frac{\pi}{3} \right)} - 2 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.321medium✓ Every equation proved

For \( \displaystyle y = 4 \cos{\left(\pi x \right)} - 2 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.391medium✓ Every equation proved

For \( \displaystyle y = 3 - 4 \cos{\left(3 x - 3 \right)} \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.392medium✓ Every equation proved

For \( \displaystyle y = - 3 \sin{\left(\frac{x}{2} - \frac{1}{2} \right)} - 2 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.393medium✓ Every equation proved

For \( \displaystyle y = 2 \sin{\left(x + \frac{\pi}{3} \right)} - 2 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.394medium✓ Every equation proved

For \( \displaystyle y = 3 \cos{\left(\frac{\pi x}{2} + \frac{\pi}{4} \right)} - 3 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.395medium✓ Every equation proved

For \( \displaystyle y = \cos{\left(4 x \right)} \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.396medium✓ Every equation proved

For \( \displaystyle y = - 4 \cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 2 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.397medium✓ Every equation proved

For \( \displaystyle y = 4 \cos{\left(\pi x \right)} + 2 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.398medium✓ Every equation proved

For \( \displaystyle y = - 3 \sin{\left(3 x \right)} - 1 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.399medium✓ Every equation proved

For \( \displaystyle y = - 3 \sin{\left(\frac{x}{2} \right)} \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.400medium✓ Every equation proved

For \( \displaystyle y = \cos{\left(x \right)} + 2 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.469medium✓ Every equation proved

For \( \displaystyle y = 1 - 4 \sin{\left(3 x \right)} \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.470medium✓ Every equation proved

For \( \displaystyle y = 4 \sin{\left(\pi x \right)} - 2 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.471medium✓ Every equation proved

For \( \displaystyle y = \cos{\left(\frac{\pi x}{2} \right)} + 3 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.472medium✓ Every equation proved

For \( \displaystyle y = - 2 \cos{\left(\frac{x}{2} + \frac{\pi}{6} \right)} - 2 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.473medium✓ Every equation proved

For \( \displaystyle y = 2 - 2 \cos{\left(x + \frac{\pi}{4} \right)} \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.474medium✓ Every equation proved

For \( \displaystyle y = 3 \sin{\left(2 x \right)} + 2 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.475medium✓ Every equation proved

For \( \displaystyle y = \sin{\left(2 x - 2 \right)} + 1 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.476medium✓ Every equation proved

For \( \displaystyle y = - 3 \sin{\left(2 x \right)} \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.477medium✓ Every equation proved

For \( \displaystyle y = - 4 \cos{\left(x + \frac{\pi}{4} \right)} - 2 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Problem 13.478medium✓ Every equation proved