Think of situations you have observed that could be represented by a graph that increases and then decreases, or decreases and then increases.
See solution.
Practice makes perfect
There are plenty of real-life situations that can be modeled by quadratic functions. Let's see some of them!
Example 1
When an object is tossed, it goes up into the air and then comes down again. The height h of the object in meters can be expressed as a function of the time t in seconds since it was thrown.
h(t)=- 4/9t^2+8/3t+1
Let's now graph the function. Note that neither h(t) nor t can be negative, since they represent height and time, respectively.
Let's now interpret the three points on the parabola above.
It is possible that the population of a city increases until, affected by certain factors, it starts to decrease. The population p, in thousands, could be expressed with a quadratic function in terms of the number of years, t, after the city was founded.
p(t)=- 11/1800t^2+11/30t+1/2
Let's see the graph of the function. Note that since t and p(t) represent time and population, respectively, they cannot take negative values.
Let's now interpret the three points on the above curve.
The intersection with the vertical axis represents the initial population.
The vertex represents after how many years the population reached its maximum, and the value of this maximum.
The intersection with the horizontal axis represents after how many years the city has no inhabitants.
Example 3
The profit p in hundreds of a small company can be expressed in terms of the number of years t it has been in the business.
p(t)=5/3t^2-10t-10
We can conclude certain things by considering the graph of the function.
The points above give us information about the company.
The intersection with the vertical axis represents the initial investment of the company. Note that a negative profit represents loss.
The vertex represents after how many years the company will start having an income greater than the expenses.
The intersection with the horizontal axis represents after how many years the company will start making a profit.
