Quadratic models are used to model certain real-world situations
such as:
1) Values decrease then
increase
2) Values increase and
then decrease
3) Values depending
on surface area.
4) Objects thrown into
the air.
Application of curve fitting
Suppose a function has the following points f(0) = 5, f(1) = 10
and f(2) = 19. Find an equation of the form f(x) = ax2 + bx + c.
Solution:
plug in each point to get three equations
5 =
0 + 0 + c means c = 5 10 = a + b + 5 means a + b
= 5 19 = 4a + 2b + 5 means 4a +
2b = 14 The last two equations can
be solved simultaneously a + b = 5 2a + b = 7 Subtract the first equation
from the second to get: a = 2 This means that b = 3 Therefore, the equation is
2x2 + 3x + 5 = f(x)
Application
to physics!
An
object thrown into the air is modeled by the equation h(t) = -4.9t2 + vot + ho Where vo is the
initial velocity ho is the initial
height above the ground h(t) is the height after
t seconds. A baseball is thrown with
an upward velocity of 14 m/sec from a building 30m high.
1)
Find its height above the ground t seconds later.
Solution: h(t) = -4.9t2 + 14 t + 30 Simply substitute in the values.
2)
When will the ball reach its peak?
Solution: Find its vertex point. x = -b/2a = -14/-9.8
= 140/98 = 10/7 seconds later.
3)
When will the ball hit the ground?
Solution: This will happen when the height is zero. 0 = -4.9x2 + 14t + 30 49t2 - 140t -300 = 0 (7x + 10)(7x - 30) =
0 x = -10/7 or x = 30/7 Throw out -10/7 Why? Answer is 30/7 seconds.
On to the sample test!! Good luck!
Current quizaroo # 1b
1) Simplify (5 -3i)(2 + 5i)
a) -5 +
19i
b)
-5 + 31i
c)
25+ 19i
d)
25 + 31i
e)
25 - 19i
2) Write in the form a +
bi, the following division problem 1/(3 + 2i)
a) (3/5) - (2/5)i
b) (3/13)
- (2/13)i
c)
(3/5) + (2/5)i
d)
(3/13) + (2/13)i
e)
3 - 2i
3) Solve the quadratic formula by any method: (2x
+ 1)(4x - 3) = (4x - 3)2
a)
4/3, -1
b)
0, -1
c)
-4/3, 1
d)
-3/4, 1
e)
3/4, 2
4) Name in order the
vertex point, axis of symmetry, x-intercepts and y-intercepts for:
y = x2 + 4x + 3
a) (-2,-1), x = -2, (-1, 0) and (-3, 0),
(0, 3)
b) (2, 11), x = 2, (-1, 0) and (-3, 0),
(0, 3)
c) (-2, 1), x = 2, (1, 0) and (3, 0),
(0, 3)
d) (2, 11), x = -2, (1, 0) and (-3, 0),
(0, 3)
e) (-4, 3), x = 2, (-1, 0) and (-3, 0),
(0, 3)
5) If you drive at x miles
per hour and apply your brakes, your stopping distance in feet is approximately
f(x) = x + (x2/25). By how much
does your stopping distance increase if you increase your speed from 30
to 40 mi/h?
Section 1-8: Quadratic
Models 
1) Values decrease then
increase
