Monday, August 13, 2007

Precalculus - Prerequisite 1 - Graphical Representation of Data, P2 - Graphs of Equations, P3 -Lines in a Plane

Prerequisite 1 - Graphical Representation of Data

I) Vocabulary -
A) The Cartesian Plane
- the rectangular coordinate plane that represents ordered pairs fo real numbers by points in a plane.
1) x - axis: the horizontal real number line
2) y - axis; the vertical real number line
3) origin - the point of intersection of the x-axis and the y-axis
4)Quadrants - the two axes divide the plane into four parts - the upper right quadrant is quadrant I, the upper left quadrant is quadrant II, the lower left quadrant is quadrant III, and the lower right quadrant is quadrant IV. They go counterclockwise from the positive side of the x-axis.
5) Ordered pair - each point in the plane corresponds to an ordered pair of real numbers (x, y)
called the coordinates of the point.
a) x-coordinate: represents the directed distance from the y-axis to the point
b) y-coordinate: represents the directed distance from the x-axis to the point.
B) Representing Data Graphically:
1) Scatterplot
2) bar graph
3) line graph
4) histogram
5) Interpreting a model based upon the data:
a) look at the data and know how to find the best-fit line
C) Distance Formula: d = √((x1 - x2)2 + (y1 - y2)2)
this is derived from the pythagorean theorem, where the difference of the x's is represented by "a" and the difference of the y's is represented by "b" and so
AB = a2 + b2 = c2
Example: Given points (-2, -5) and (-3, 4), what is the distance between the two points.
√((-2 - -3)2 + (-5 - 4)2) =
√((1)2 + (-9)2) =
E) Midpoint Formula: the mean, or average, of the x-coordinates and the y-coordinates.therefore the midpoint formula is: ((x1+ x2) /2, (y1+ y2) /2)
example: Given A(-1, 7) and B(3, -3) what is their midpoint?((-1 + 3)/2, (7 + -3)/2) = (2/2, 4/2) = (1, 2)
F) Standard equation of a circle with radius r and center (h, k) is:
r2 = (x - h)2 + (y - k)2
r = √((x - h)2 + (y - k)2)

Prerequisite 2: Graphs of Equations
I) Vocabulary:
A) Solution Point - for an equation in variables x and y, a point (a, b) is a solution point if the substitution of x = a and y = b satisfies the equation.
1) Graph of the equation - the set of all solution points of an equation.
B) How to Sketch the Graph of an Equation by Point Plotting
1) If possible, rewrite the equation so that one of the variables is isolated on one side of the equation.
2) Make a table of several solution points (usually 5 - 7)
3) Plot these points in the coordinate plane.
4) Connect the points with a smooth curve.
C) Using a Graphing Utility to Graph an Equation - to graph an equation involving x and y on a graphing utility, use the following procedure.
1) Rewrite the equation so that "y"is isolated on the left side.
2) Enter the equation into a graphing utility.
3) Determine a viewing window that shows all important features of the graph.
4) Graph the equation.
D) Throughout this course, you will learn that there are many ways to approach a problem.
1) a numerical approach: construct and use a table.
2) a graphical approach: draw and use a graph.
3) an algebraic approach: use the rules of algebra.

Prerequisite 3: Lines in the Plane
I) Vocabulary:
A) the slope of a line
- represents the number of units a line rises or falls vertically for each unit of horizontal change from left to right.
Slope: (nonvertical line): Ratio of the vertical change (the rise, Δ y) to the horizontal change (the run, Δ x)
Formula: slope = m = (Δy/Δx) = (y2 - y1)÷ (x2 - x2)

II) Postulate:
A) In the coordinate plane, 2 nonvertical lines are parallel if and only if they have the same slope. Any two vertical or horizontal lines are parallel.
B) Slope- Intercept Form of the Equation of a Line: y = mx + b where m is the slope of the line and b is the y-intercept (where it crosses the y-axis so point (0,b)
C) Point-slope form of an equation of the line that passes through the point (x1, y1) and has a slope of m is
y - y1 = m(x - x1)
D) Summary of Equations of Lines
1. General form:
Ax + By + C = 0
2. Vertical line: x = a
3. Horizontal line: y = b
4. Slope-intercept form: y = mx + b
5. Point-slope form: y - y1 = m(x - x1)
E) Parallel Lines - two distinct non-vertical lines are parallel if and only if their slopes are equal.
F) Perpendicular Lines - two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals of each other. If m1 = a/b then m2 = -b/a.
The product of the two slopes are -1. (a/b)(-b/a) = -1
m1 - -1/(m2)
Example: if m1 = 2/3 then
m2 = -3/2