Amplitude and Midline
A free Trigonometry lesson from the “Graphs of Trig Functions” unit, with a worked example and practice problems including step-by-step solutions.
For a sinusoid y = a*sin(x) + d or y = a*cos(x) + d, the amplitude is |a| (the vertical distance from the midline to a peak) and the midline is the horizontal line y = d about which the graph oscillates. When given a maximum and minimum value instead, amplitude = (max - min)/2 and midline = (max + min)/2. The coefficient a stretches or reflects the wave vertically, while d shifts the whole graph up or down.
What you'll learn
- Read amplitude |a| and midline y = d directly from y = a*sin(x) + d or a*cos(x) + d
- Compute amplitude as (max - min)/2 and midline as (max + min)/2
- Describe how a and d stretch, reflect, and shift the graph
Worked example
Problem. Find the amplitude and midline of y = -5cos(x) + 3.
- Amplitude is |a|, the absolute value of the outside coefficient: |-5| = 5.
- The constant added is d = 3, so the midline is y = 3.
- The negative sign reflects the graph but does not change the amplitude.
Answer: amplitude = 5, midline y = 3
Practice problems
1. Find the amplitude of y = 6sin(x) + 2.
Show solution
- Warm-up: First identify exactly what the question is asking: Find the amplitude of y = 6sin(x) + 2.
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Amplitude is the absolute value of the outside coefficient a.
- Here a = 6.
- So the amplitude is 6.
- Check the result by substituting or estimating: the response should match 6 and make sense in the original problem.
Answer: 6
2. Find the midline of y = 4cos(x) - 7. Write it as y = d.
Show solution
- Warm-up: First identify exactly what the question is asking: Find the midline of y = 4cos(x) - 7. Write it as y = d.
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- The constant added at the end is d.
- Here d = -7.
- The midline is y = -7.
- Check the result by substituting or estimating: the response should match y = -7 and make sense in the original problem.
Answer: y = -7
3. Find the amplitude of y = -8cos(x) + 1.
Show solution
- Warm-up: First identify exactly what the question is asking: Find the amplitude of y = -8cos(x) + 1.
- For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
- Amplitude is |a|, so drop the negative sign.
- |-8| = 8.
- The amplitude is 8.
- Check the result by substituting or estimating: the response should match 8 and make sense in the original problem.
Answer: 8
4. Find the amplitude and midline of y = 0.5sin(x) + 9. Write your answer as 'amplitude, y = d'.
Show solution
- Core Practice: First identify exactly what the question is asking: Find the amplitude and midline of y = 0.5sin(x) + 9. Write your answer as 'amplitude, y = d'.
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Amplitude is |0.5| = 0.5.
- The constant d = 9 gives midline y = 9.
- So amplitude = 0.5 and midline y = 9.
- Check the result by substituting or estimating: the response should match 0.5, y = 9 and make sense in the original problem.
Answer: 0.5, y = 9
5. A function reaches a maximum of 24 and a minimum of -6. Find its amplitude.
Show solution
- Core Practice: First identify exactly what the question is asking: A function reaches a maximum of 24 and a minimum of -6. Find its amplitude.
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Amplitude = (max - min)/2.
- (24 - (-6))/2 = 30/2.
- The amplitude is 15.
- Check the result by substituting or estimating: the response should match 15 and make sense in the original problem.
Answer: 15
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