1.
(a) Use the quadratic formula to determine, to the nearest tenth, the roots of the equation: x2+ 6x - 2 = 0. (see section 5.2) .
(b) Determine the vertex of the parabola y = x2+ 6x - 2. (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 = x2+ 6x - 2. You may wish to determine a few other "points that the parabola passes through" to obtain a more accurate graph.
2.
Assume you are standing on Venus. 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 8 feet above the ground? (Hint, read section 3.5 carefully. This is another application of the quadratic, recall the x value of the vertex is given by x = -b/2a)
3. Solve the following inequalities. Express your answer using interval notation
(a) x/3 - 2 > x + 2/3
(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 part b.
4.
(a) Perform the indicated operation and express the result in the form a + bi.
(1 + -2 )(1 - -2 )
(b) Simplify. - i 12
(c) Find all real numbers x and y that satisfy the equation: -2x - 1 + 2yi = 4x - 5i. See example 11, section 5.3
5. y = |x| and y = 4
6. The marketing department at Texas Instruments has found that when certain calculators are sold at a price of p dollars, the revenue (in dollars) as a function of the price is: R(p) = -150p2 + 21,000p
(a) What unit price should be established in order to maximize revenue?
(b) If this price is charged what is the maximum revenue.
(c) Graph the function R(p).
7. Solve for x: x = 2
8. 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?
9. 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 bicycle?
Note:
These (and other) sections illustrate the power of the quadratic function in solving a variety of problems. To understand this function you must be able to graph it.
Some key thoughts :
(a) Any function of the form y = ax2 + bx + c where a ≠ 0 is a quadratic.
(b) The formula x = b 2a- 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: 2 -b± b -4ac x = 2a