Modeling Real-World Situations
Is this situation best modeled by an exponential or linear function?
A population doubles every year
Exponential — the population is multiplied by the same factor each year.
In A(t)=800(1.05)t , identify the growth factor.
1.05
Is the function f(x)=3(2)x showing growth or decay?
Growth, because the growth factor (2) is greater than 1.
What happens to the graph of an exponential function when a constant is added to the x?
The graph shifts horizontally to left
This value is the common ratio of the geometric sequence
3, 6, 12, 24,
2
Describe what the initial value represents in an exponential model.
The initial value represents the starting amount before any growth or decay occurs.
Explain what the number 800 represents in
A(t)=800(1.05)t
800 represents the initial amount at time t=0t = 0t=0.
Describe what the y-intercept of an exponential graph represents in context.
The y-intercept represents the initial value of the situation.
Describe how the graph of f(x) changes when it becomes f(x)+k
The graph shifts up if k>0 and down if k<0
The sequence 100, 50, 25, 12.5,
Exponential Decay
A population starts at 1,200 and grows by 6% each year.
Write an exponential function to model the situation.
P(t)=1200(1.06)t
Interpret the meaning of 0.92 in
P(t)=500(0.92)t
The quantity decreases by 8% each time period (decay).
or
92% of the quantity remains each time period (decay)
Sketch the graph of
f(x)=2(0.5)x
and identify the asymptote.
The graph shows decay, crosses the y-axis at 2, and has a horizontal asymptote at y=0y = 0y=0.
Compare
f(x)=2x g(x)=2x+4
The shape and growth rate stay the same. The graph shifts up 4 units, and the asymptote moves from y=0 to y=4
Which grows faster as nnn increases:
Sequence A: adds 5 each term
Sequence B: multiplies by 1.2 each term
Sequence B
Prompt
Each year, a population increases by 4% of the current population.
Three students make the following claims about how this situation should be modeled:
Alex: “This situation should be modeled with a linear function because the population increases every year.”
Jordan: “This situation should be modeled with an exponential function because the population increases by a percent.”
Taylor: “A linear model is better because exponential growth always increases too fast to be realistic.”
Question:
Which student is correct? Explain why the other two claims are incorrect.
Jordan is correct because an increase by a constant percent (4%) each year indicates exponential growth.
Alex is incorrect because simply increasing every year does not mean the change is linear. Linear growth requires a constant number added each year, not a constant percent.
Taylor is incorrect because exponential growth does not always increase “too fast.” The rate of growth depends on the growth factor and time frame, and exponential models are commonly used for realistic situations like population growth.
Explain the difference in meaning between
500(1.04)t and 500(1.04)t+200
In the first model, all 500 grows by 4% each period.
In the second model, 500 grows by 4% each period, and 200 is added once and does not grow.
Two exponential functions are graphed. One starts lower but grows faster.
Which function will eventually be larger?
The function with the larger growth factor will eventually be larger, even if it starts lower.
Explain the difference between multiplying an exponential function by a constant and adding a constant.
Multiplying scales the output and affects the steepness, while adding shifts the graph vertically without changing the growth rate.
This explicit formula represents a geometric sequence with a first term of 4 and a common ratio of 3.
What is an=4(3)n-1
Create a real-life situation that would be modeled by
f(x)=500(1.04)x
Explain how each part of the equation matches your situation.
A savings account starts with $500 and grows by 4% each year.
500 is the starting amount, 1.04 is the growth factor, and x represents time.
A student says the bonus in a word problem should always be added to the initial value.
Evaluate this claim.
(Tell me if you think this is true or not, and why)
The claim is not always true. A bonus should only be added inside the exponential part if it also grows. One-time changes should be added outside the function.
Explain how you can determine key features of an exponential function without graphing it.
By examining the equation: the initial value gives the y-intercept, the growth factor shows growth or decay, and the asymptote is usually y=0 unless shifted.
Create a real-life situation that would result in the transformation
f(x)→f(x)-k
Example: A population shrinks at a constant percent, but an extra fixed number of individuals are lost at the start.
A geometric sequence starts at 10 and decreases by 20% each step. This is the explicit formula.
What is an=10(0.8)n−1