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MATH 1001 (Quantitative Skills and Reasoning) Gordon College, Barnesville, GA. 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 f ( x ) ax^2 bx c , 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 y f ( x ) x^2 4 x (^1). 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.
Quadratic Functions and Modeling Example 2 Figure 2 shows the graph of the quadratic function y g ( x ) 2 x^2 4 x 3. 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) f^ (^ x ) ^ x^2 ^4 x ^1 can be written as f ( x ) ( x 2 )^2 (^3) ; the quadratic function (example 2) (^ )^243 g x x^2 x can be written as g ( x ) 2 ( x 1 )^2 (^5). 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.
Quadratic Functions and Modeling Solution (a) We only need to substitute t = 7 in the population function P ( t ) and calculate P ( 7 ) 110 4 ( 7 ) 0. 07 ( 7 )^2 141. (^43) thousand. (b) We need to find the value of t when P ( t ) = 180. That is we need to solve the equation, 110 4 t 0. 07 t^2 180 (i ) 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): 180 110 4 0. 07 2 2 1 Y Y t t figure 3 The line Y 2^ ^180 and the parabola 2 Y 1 (^) 110 4 x 0. 07 x 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 Stockton City should reach a population of 180 thousand 14 years, 0.56 month after January 1, 2000. Hence, sometime during January 2014.
Quadratic Functions and Modeling Alternatively , by using the Quadratic Formula: The quadratic equation ax^2 ^ bx c ^0 , a ^0 has solutions a b b ac x 2 ^2 4 . To use the quadratic formula, we first write equation (i) in the form
- 07 t^2 4 t 70 0 (ii ) Here, a = 0.07, b = 4, c = −70, giving
- 046954 71. 189811
- 14 4 35. 6 2 ( 0. 07 ) 4 42 4 ( 0. 07 )( 70 ) or t 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 y^ 0 feet above ground, with an initial velocity v^ 0 ft/sec, then its height above ground after t seconds is given by 0 0 y ( t ) 16 t^2 v t y . 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?
Quadratic Functions and Modeling
- The population (in thousands) for Alpha City, t years after January 1, 2004 is modeled by the quadratic function P^ (^ t ) ^0.^3 t^2 ^6 t ^80 .In what month of what year does Alpha City’s population reach twice its initial (1/1/2004) population?
- The population (in thousands) for Beta City, t years after January 1, 2005 is modeled by the quadratic function P ( t ) 0. 7 t^2 12 t 200 .How long will it take Beta City’s population to reach 350 thousand?
- The population (in thousands) for Gamma City, t years after January 1, 2002 is modeled by the quadratic function (^ )^1.^521300. P t t^2 t How long will it take Gamma City’s population to reach 500 thousand?
- The population (in thousands) for Delta City, t years after January 1, 2003 is modeled by the quadratic function P (^^ t ) ^0.^5 t^2 ^7 t ^90 .In what month of what year does Alpha City’s population reach twice its initial (1/1/2003) population?
- The population (in thousands) for Omega City, t years after January 1, 2002 is modeled by the quadratic function (^ )^0.^255100. P t t^2 t In what month of what year does Alpha City’s population reach 200 thousand?
- 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.
- 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.
- 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?
- A ball is dropped from the top of a 960 feet tall building. How long does it take the ball to hit the ground?
- 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?
Quadratic Functions and Modeling Fitting Quadratic Models to Data: Find the quadratic model 2 P ( t ) P 0 bt at (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.
- Iowa City, IA t (years) 1970 1980 1990 2000 P (people) 46,850 50,508 59,735 62,
- Arizona t (years) 2000 2001 2002 2003 2004 P (thous) 5,131 5,320 5,473 5,581 5,
- Florida t (years) 1970 1980 1990 2000 2004 P (thous) 6,789 9,746 12,938 15,982 17,
- Georgia t (years) 1970 1980 1990 2000 2004 P (thous) 4,590 5,463 6,478 8,186 8,
- Nevada t (years) 1970 1980 1990 2000 2004 P (thous) 489 801 1,202 1,998 2,
- U.S. t (years) 1970 1980 1990 2000 2004 P (millions) 203 227 249 281 294 Answers to Exercises
