MATH 1001 (Quantitative Skills and Reasoning)
Quadratic Functions and Modeling
In this unit, we will study quadratic functions and the relationships for which they provide suitable models. An important application of such functions is to describe the trajectory, or path, of an object near the surface of the earth when the only force acting on the object is gravitational attraction. What happens when you toss a ball straight up into the air? What about an outfielder on a baseball team throwing a ball into the infield? If air resistance and outside forces are negligible, what is the mathematical model for the relationship between time and height of the ball?
Definition: Quadratic
Functions
A quadratic function is one of the form
where a, b, and c are real numbers with a ≠ 0.
The graph of a quadratic function is called a parabola and its shape resembles that of the graph in each of the following two examples.
Example 1
Figure 1 shows the graph of the quadratic function
.
figure
1
Observe that there is a lowest point V(2, −3) on the graph in figure 1. The point V is called the vertex of the parabola.
Example 2
Figure 2 shows the graph of the quadratic function
.
figure
2
Again, observe that there is a highest point V(1, 5) on the graph in figure 2. This point V is also the vertex of the parabola.
By completing the square,
the quadratic function (example 1) 
can be written as
;
the quadratic function (example 2) 
can be written as
.
Note that the parabola in example 1 opens upward, with vertex V(2, −3) and a vertical axis of symmetry x = 2. The parabola in example 2 opens downward with vertex V(1, 5) and a vertical axis of symmetry x = 1.
Standard Form of a
Quadratic Function Whose Graph (a Parabola) has a Vertical Axis
The graph of 
is a parabola that has vertex V(h, k) and a vertical axis.
The parabola opens upward if a > 0 or downward if a < 0.
If a > 0, the vertex V(h, k) is the lowest point on the parabola, and the function f has a minimum value f (h) = k.
If a < 0, the vertex V(h, k) is the highest point on the parabola, and the function f has a maximum value f (h) = k.
Quadratic Models and Equations
Quadratic Population Models:
.
Here, we denote the independent variable by t (time) instead of x, and the constant c by 
because substitution
of t = 0 yields 
.
We refer to as the initial
population.
Example 3
Suppose that the future population
of
is described (in thousands) by the quadratic model
.
(a)
What is the population of
(b)
In what month of what calendar year will the population
of
Solution
(a) We only need to substitute t = 7 in the population function P(t) and calculate
thousand.
(b) We need to find the value of t when P(t) = 180. That is we need to solve the
equation,
Equation (i) is an example of a quadratic equation which we can solve in a
variety of ways.
First, by graphing both sides of the equation (figure 3):
figure 3
The line 
and the
parabola 
are shown in the
calculator window −100 ≤ x ≤ 80, −100 ≤ y ≤ 300.
To solve equation (i), we find the x-coordinate of the intersection point in the first quadrant. The negative solution of the intersection point in the second quadrant would be in the past. Figure 3 indicates that we have already used the intersection-finding feature to locate the point (14.047, 180).
Hence, t = 14.047 years
= 14 years + (0.047 x 12) months
= 14 years + 0.56 month.
Thus
Hence, sometime during January 2014.
Alternatively, by using the Quadratic
Formula:
The
quadratic equation 
has
solutions 
.
To use the quadratic formula, we first write equation (i) in the form
Here, a = 0.07, b = 4, c = −70, giving
The negative solution would be in the past. So, we only accept the positive
solution, t = 14.047 years = 14 years, 0.56 month.
The Position
Function (Model) of a Particle Moving Vertically:
If an object is projected straight upward at time t = 0 from a point 
feet above ground, with an initial velocity 
ft/sec, then its height above ground after t seconds is given
by
.
Example 4
A projectile is fired vertically upward from a height of 600 feet above the ground, with an initial velocity of 803 ft/sec.
(a) Write a quadratic model for its height y(t) in feet above the ground after t seconds.
(b) During what time interval will the projectile be more than 5000 feet above the ground?
(c) How long will the projectile be in flight?
Solution
(a) Given 
ft and 
ft/sec,
(b) We need to find the values of t for which the height y(t) > 5000 feet.
That is
We solve the inequality (i) by graphing (figure 4) both the parabola
and the line 
, in the calculator
window
−20 ≤ x ≤ 70 and
−2000 ≤ y ≤ 12000.
figure 4
The parabola in figure 4 shows the height of the projectile at time t and the horizontal line is the graph of a height of 5000 feet.
The projectile is more than 5000 feet above the ground when the graph of the parabola is above the horizontal line.
Using the intersection-finding feature of the calculator, we find the approximate intersection points to be t = 6.3 sec and t = 43.9 sec.
Hence, the projectile is above 5000 feet when
6.3 sec < t < 43.9 sec.
(c) The projectile will be in flight until its height h(t) = 0. This corresponds to the x-intercept (not the origin) in figure 4.
Using the root or zero feature of the calculator, we obtain
t = 50.9 sec.
EXERCISES
1. The population (in thousands) for Alpha
City, t years after January 1, 2004 is modeled
by the quadratic function 
In what month of what
year does Alpha City’s population reach twice its initial (1/1/2004)
population?
2. The population (in thousands) for Beta
City, t years after January 1, 2005 is modeled by the quadratic function 
How long will it take
Beta City’s population to reach 350 thousand?
3. The population (in thousands) for Gamma
City, t years after January 1, 2002 is modeled by the quadratic function 
How long will it take
Gamma City’s population to reach 500 thousand?
4. The population (in thousands) for Delta
City, t years after January 1, 2003 is modeled by the quadratic function 
In what month of what
year does Alpha City’s population reach twice its initial (1/1/2003)
population?
5. The population (in thousands) for 
In what month of what
year does
6. A ball is thrown straight up, from ground zero, with an initial velocity of 48 feet per second. Find the maximum height attained by the ball and the time it takes for the ball to return to ground zero.
7. From the top of a 48 feet tall building, a ball is thrown straight up with an initial velocity of 32 feet per second. Find the maximum height attained by the ball and the time it takes for the ball to hit the ground.
8. A ball is thrown straight up from the top of a 160 feet tall building with an initial velocity of 48 feet per second. The ball soon falls to the ground at the base of the building. How long does the ball remain in the air?
9. A ball is dropped from the top of a 960 feet tall building. How long does it take the ball to hit the ground?
10. Joshua drops a rock into a well in which the water surface is 300 feet below ground level. How long does it take the rock to hit the water surface?
Fitting Quadratic Models to Data:
Find the
quadratic model (with t = 0 for the earliest year given in the
data) that best fits the population census data in Problems 11 – 16.
In each case, calculate the average error of this optimal model, and use the model to predict the population in the year 2007.
11.
|
t (years) |
1970 |
1980 |
1990 |
2000 |
|
P (people) |
46,850 |
50,508 |
59,735 |
62,220 |
12.
|
t (years) |
2000 |
2001 |
2002 |
2003 |
2004 |
|
P (thous) |
5,131 |
5,320 |
5,473 |
5,581 |
5,744 |
13.
|
t (years) |
1970 |
1980 |
1990 |
2000 |
2004 |
|
P (thous) |
6,789 |
9,746 |
12,938 |
15,982 |
17,346 |
14.
|
t (years) |
1970 |
1980 |
1990 |
2000 |
2004 |
|
P (thous) |
4,590 |
5,463 |
6,478 |
8,186 |
8,825 |
15.
|
t (years) |
1970 |
1980 |
1990 |
2000 |
2004 |
|
P (thous) |
489 |
801 |
1,202 |
1,998 |
2,315 |
16.
|
t (years) |
1970 |
1980 |
1990 |
2000 |
2004 |
|
P (millions) |
203 |
227 |
249 |
281 |
294 |
Answers to Exercises
1. February, 2013.
2. 8 years, 143 days.
3. 6 years, 183 days.
4. February, 2011.
5. May, 2014.
6. 36 feet, 3 sec.
7. 64 feet, 3 sec.
8. 5 sec.
9. 7.75 sec.
10. 4.33 sec.
11. 
; 1377 people; 66,000 people
12. 
; 14 thousand; 6,030
people
13. 
; 44 thousand; 18,300 thousand
14. 
; 59 thousand; 9,400 thousand
15. 
; 31 thousand; 2,600 thousand
16. 
; 1.4 thousand; 300 thousand
Note: Answers
to questions 11-16 are approximate values.
References
1. Elementary Mathematical Modeling: Functions and Graphs.
Mary Ellen Davis & C. Henry Edwards. Prentice Hall, 2001.
2. Intermediate Algebra.
R. David Gustafson & Peter D. Frisk. Thomson (Brooks/Cole), 2005.
4. www.gpec.org