(a) Use the quadratic formula to determine the roots of the equation:
2x2 + x - 1 = 0.
(b) Determine the vertex of the parabola y = 2x2+ x - 1.
(Recall, the x value of the vertex is given by x = -b/2a)
(c) Use parts a and b to sketch the graph of y = 2x2+ x - 1. You may wish to determine a few
other “points that the parabola passes through” to obtain a more accurate graph.
How high will a baseball that is thrown up into the air at 48 feet per second go if it starts its flight at
a height 6 feet above the ground? (Hint: This is another application of the quadratic, recall the x
value of the vertex is given by x = -b/2a)
(b) Solve for x: | x - 1 | = 4
(c) Sketch the graph of the function y = | x - 1 | and use your graph to indicate your solution of
(a) Perform the indicated operation and express the result in the form a + bi.
(b) Simplify. - i
(c) Find all real numbers x and y that satisfy the equation: -3x - 1 + 2yi = 4x – 5i. See example 11,
Find the area bounded by the following (make sure that you compute the area):
y = -x, the x axis and x = 5
A rain gutter is to be made up of rectangular aluminum sheets 12 inches wide by turning up the sides
edges 90 degrees. What depth (of the edges) will provide a maximum cross sectional area and
thereby provide for the greatest flow of water? (Hint, think of quadratic equations and that you want
to maximize the “cross sectional area”.)
The profit P, in dollars, for a company that produces antivirus and systems utilities software is:
P = -0.0002x2 +140x – 250,000 where x is the number of units sold. What sales level will yield a
A worker (on earth) dropped a screwdriver from the top of an elevator shaft. Exactly 5 seconds later
he heard the sound of the screwdriver hitting the bottom of the shaft. How tall is the elevator shaft?
This is easier than you think. Try it. A bicycle manufacturer has determined that when x hundred
bicycles are built, the average cost per bicycle is given by C(x) = 0.1x2
– 0.7 x + 2.425, where C(x) is
in hundreds of dollars. How many bicycles should be built in order to minimize the average cost per
Some key thoughts :
(a) Any function of the form y = ax2 + bx + c where a ≠ 0 is a quadratic.
(b) The formula gives you the x value of the vertex. Substitute this value in the equation (of
part (a)) to find the y value.
(c) To sketch the graph of a quadratic one frequently need only know the vertex (x and y values) and
the roots (found by solving ax2 +bx + c = 0 by factoring or by using the quadratic formula).The
quadratic formula is:
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