- Post a relationship from a real-world situation or concept that illustrates the idea of a mathematical function. For example, we use the function F =9/5(C + 32) to convert from degrees Celsius to degrees Fahrenheit.
Function relates input values (independent values) to exactly one possible output value (dependent value).
Usually, it follows f(x) =y where x = input and y = output.
One common application is projectile motion: f(t) = `at^2 +bt+c`
where a <0 and f(t) as height of the object at time "t".
The graph of parabola opens downward (projectile path) when a <0.
Example: `f(t) = -4t^2 +14t+6 ` where f(t) is height in feet and t is time in seconds.
a) Finding initial height:
Let t=0 (starting time) then
f(0) =` -4(0)^2+14(0)+6` = 6 ft
Interpretation: An object is launched from the height of 6 feet.
b) Finding maximum height.
Hint: Vertex is at the peak of the parabola in projectile trajectory.
`Vertex t = -b/(2a) = -(14/(2*(-4))) =1.75 seconds`
maximum height: `f(3.5) = -4(3.5)^2 +14(3.5) +6 =18.25 feet.`
Interpretation: The object will reach 18.25 feet after 1.75 seconds before it falls down along its projectile trajectory.
c) Time it hits the ground.
At the ground level, the height is 0 or f(t)=0
f(t) = -4t^2 +14t +6
0 = -4t^2 +14t +6
0 = -(t-2)(4t+3) Factoring
t =2 seconds
4t +3 =0
4t = -3
t= -0.75 seconds (negative time value does not exists).
Interpretation: The object hits the ground after 2 seconds.
Another application of mathematical function from a real-world situation is profit, revenue and cost functions.
Note that profit = revenue – cost.
P(x) = R(x) – C(x) where x= number of items sold.
Note: P(x) = positive value means it earns a profit and R(x) > C(x)
P(x) = negative value means loss and R(x) < C(x)
P(x) =0 or break-even where there is no loss or gain of profit. R(x)=C(x).
Suppose a store franchised a toy for $5 each with a fixed cost of $230.
Cost function = total variable cost + total fixed cost.
C(x) = 5x + 230.
The toys are sold for a price $8.
Revenue function = price * items
R(x) = 8*x or R(x) = 8x.
P(x) = 8x – (5x+230)
= 8x – 5x -230
= 3x -230
*Break-even when R(x) = C(x).
8x = 5x +230
3x = 0 +230
3x = 230
x = 76.7
Round off to 77 toys since 0.70 toy is not possible.
The number of items can be counted as 0,1,2,3,…
The store should at least had sold 77 toys to earn a profit.
When the sold items is less than 77 toys, there will be a loss.
Let x = 77 toys then profit is:
P(x) = 3(x) -230
P(77) = 3(77) -230
= 231 -230
The store earns $1 profit for 77 toys sold.
As a student, one function that I have seen a lot in school being tied to real-life examples is y = mx + b. Here are a few of the most common ones in my memory.
- Simple phone plans: The question typically states that a phone plan costs b amount of money for basic functions (such as domestic calls and possibly texts) and then m amount of money per minute for international calling (as an example).
- Production: The equipment needed to produce widgets costs b amount of money to buy. From there, the cost per widget in terms of keeping the machines running is m dollars.
- Car rentals: You pay an upfront fee of b dollars for the car, and m dollars per mile.
In all the examples above, I went ahead and used the same variables m and b as they would be used in the equation.