"The essence of all things is numbers." -- (Pythagoras, ca 550 B.C.)
The origin of some mathematical words.
Calculus: From the Latin word calculus, meaning "small stone." Stones were used in many ancient counting processes.
Circumference: From the Latin word circumferre, meaning "to carry around."
Hypotenuse: From the Greek words hypo and teinein, meaning "to stretch under."
Theorem: From the Greek word theoros, meaning "spectacle, to look at." To the ancient Greeks, theorems were statements to be observed as laws.
What did Herkimer call the mafia leader who retired and went into farming?
Answer: The Sodfather.
Things Herky would like to know:
Is it true that people who have the last laugh are also the slowest thinkers?
Do atheists have financial problems because they are a non-prophet organization.
Reading: Section 5.1, pages 249-252.
Written: Pages 253-254/17-19, and 21-31 (odds).
In problems 21,23, and 25, graph the parabola on your calculator and estimate the vertex and axis of symmetry. In problems 27, 29, and 31, establish the vertex and axis of symmetry directly from the written form of the parabola.
Mathematical word analysis:
QUADRATIC: This is the Latin root for "to make square." A quadratic expression in x, for instance, has 2 as the highest power of x.
Quadratic functions are very important in mathematics. This quadratic function is constantly involved in our everyday lives:
s = (-1/2)gt2 + vot + so
In this equation, the variables are s (distance) and t (time), g is a constant (gravity), and v0 and s0 are constants relating to velocity and distance.
A quadratic function can be presented in different forms. For instance...
y = (x+6)(x-4) is a quadratic function written in factored form . This allows us to immediately identify the x-intercepts at -6 and 4. That is, the points (-6,0) and (4,0) are on the function. (You do relate to the word function , don't you?) The function could, of course, also be written y = x2 +2x - 24.
A very useful form for a quadratic function is illustrated by y = (x-3)2 + 5. This form allows you to determine maximum or minimum values for y very easily. For instance, one can observe that the smallest possible value y can assume if 5, and that occurs when x = 3. The point (3,5) is the vertex of the parabola represented by the function, and the parabola opens upward. In this case, the domain of the function is all real numbers, and the range is all values 5 or greater.
If the function is y = -2(x-7)2 + 88, then the maximum value for y is 88. The point (7,88) is the vertex of the parabola. The domain of the function is all real numbers, and the range is all numbers that do not exceed 88.
Problem: Identify the vertex and axis of symmetry for the quadratic function
y = 7(x - 12)2 + 3
Solution (with communication):
The axis of symmetry is the vertical line x = 12, and the vertex is at the point (12,3).
Note: Since 7(x - 12)2 cannot be negative, the smallest value of y will occur with this expression is zero, and that occurs when x = 12.
Problem: Find values for c and d such that the quadratic function y = (x - c)2 + d has a minimum value y = -34 when x = 78.
Solution (with communication):
We will get a minimum value for y when (x - c)2 = 0. Since this must happen at x = 78, this necessitates c = 78. So the function must have the form
y = (x - 78)2 + d
For y to have a minimum value of -34 when x = 78, we must have
-34 = (78 - 78)2 + d ==> d = -34.
Hence, the requested quadratic function is
y = (x - 78)2 - 34.