Question

example 4
a pendulum swings to 90% of
its height on each swing and
starts at a height of 80 cm.
the height of the pendulum in
centimeters, y, is recorded
after x number of swings.
determine the key features
of this function.
number of swings (x) height in cm (y)
0 80
1 72
2 64.8
3 58.32
5 47.24
10 27.89
20 9.73
40 1.18
60 0.14
80 0.02

Explanation:

Step1: Identify function type

This is an exponential decay function, since each swing's height is 90% (0.9) of the prior height, starting at 80 cm. The general form is $y = a(b)^x$, where $a$ is the initial value, $b$ is the decay factor.

Step2: Define function parameters

Initial height $a = 80$, decay factor $b = 0.9$. So the function is $y = 80(0.9)^x$.

Step3: Find initial value

The initial value is the height at $x=0$, which is $y=80$ cm.

Step4: Identify decay factor/rates

Decay factor $b=0.9$, so decay rate is $1 - 0.9 = 0.1$ or 10% per swing.

Step5: Determine asymptote

For exponential decay functions with positive initial value, the horizontal asymptote is $y=0$ (height approaches 0 but never reaches it).

Step6: Analyze domain/range

Domain: All non-negative integers $x \geq 0$ (number of swings can't be negative).
Range: All positive real numbers $0 < y \leq 80$ (height is positive, maximum at initial swing).

Answer:

1. Function Equation: $y = 80(0.9)^x$
2. Initial Value: 80 cm (height at 0 swings)
3. Decay Factor: 0.9 (90% of previous height per swing)
4. Decay Rate: 10% of height lost per swing
5. Horizontal Asymptote: $y=0$ (height approaches 0 as swings increase)
6. Domain: All non-negative integers ($x \geq 0$)
7. Range: All positive real numbers where $0 < y \leq 80$