1.1 The Coordinate Plane
Much should be familiar: read/review definitions
Numbering of quadrants
Ex 1 How/where to plot points -- students review
Ex 2 & for #6: what is a scatter plot?
Ex 3-4: Dist formula: take sqrt of d^2 = …Pythagorean Thm
For applications, identify coordinatess of the relevant points as in Ex 4
Ex 5: Midpoint formula: you average the x-coords and the y-coords
Ex 6: Verifying that ordered pairs are solutions of an equation in x and y
Technique: simple substitution of values for both x and y
Notice that graphically, we are showing that the points are on the curve.
Ex 7,8,9: General
formula of circle: Pythag again, don't take sqrt
(x-c)2 + (y-d)2
= r 2
What does the equation become if the center is at the origin?
FAVORITE QUESTION: Find eqn of circle if endpoints of a diam are (3,-2) and (-9,4)
Ex 10: Unit circle -- radius is 1 ("unit")
1.2 Graphs and Graphing Calculators
Ex 1: Traditional method of graphing by hand -- point-plotting
Time-consuming, and can be misleading if not enough points are graphed (1/x vs x/(x2+1)
Learn cues (e.g.interpret eqn of a circle to get its center and radius)
Ex 2: Big help gr calcr -- plots lots of points (about96 pixels wide for TI-86)
Different ex. from text: graph (x+1)/(12x^2+1).
Important: Try out all the technology tips on your calculator, and go through all the examples.
x-Var key
Selecting/deselecting function to be graphed (shaded = sign)
De-select PLOT 1 etc. on "y=" screen
Find a good viewing window using WINDOW/RANGE function and/or ZOOM
("hidden behavior" is the biggest problem with this function)
Using TRACE -- ugly numbers in the standard window (not convenient decimals)
note y-values -- use this info to determine what to use for y-Min, y-Max
Homework style note:
When giving a calcr graph, show axes and graph window as well as the graph
That is, label the ends of the axes with XMin, XMax, YMin, YMax
When specifying a viewing window:
Give XMin, XMax, YMin, YMax, as inequalities e.g −5≤x≤5...
Also note your XScl and YScl (e.g. Δx=2, Δy=0.5)
How to use max/min finder: Important because zooming and tracing only give you approximations,
which must be refined for any accuracy -- time consuming
Find local max and min values for the example above
Special windows (ZOOM menu): Not just zoom in and out
standard, box, square (built on xMax and xMin), decimal & trig, fit
"fit" can also help you find a complete graph (define -- all crucial points)
You must specify the XMin, XMax, the calculator finds YMin, YMax
Not usable if there are infinite discontinuities (e.g. denominators that approach 0)
1.3 Lines
A line (straight) is completely determined by its slope and y-intercept
Slope (def): m = chg in y / chg in x = y2-y1 / x2-x1 = [not in book] rise/run = [nib] delta y / delta x
Ex: (given two points) by formula, or by counting, (or given graph)
y-int (def): use letter b. Easy if given graph, hard if given 2 points.
Slope-int form y = mx + b
App: graph/analyze eqn
Shape of graph if slope is +, -, 0 [y=b], undef [x=a]
Apps w fixed and variable costs
Do Ex like Ex5 (factory making can openers)
Note slope is "unit cost" i.e. cost for one more item (marginal cost)
Contrast with average cost
Point-slope form
Typical app: build an eqn when you don't already know the y-int.
Either based on point & slope, or 2 points (e.g. from ordered pairs in a problem)
Do Ex 51, including a forecast -- Note you can rescale
Slopes of parallel or [nonvertical] perpendicular lines -- students read and review
App: find eqn of tang line to circle at a given point on the circle, X42
Other form: general (ax+by=c) Crucial thing: no exponents, and no * or / of variables
1.4 Linear Models Hwk: For the data sets in 17-23, find the linear model and write it in the form y = ax+b. Ignore other instructions in your text
Linear model == line of best fit == regression line.
Show calculator steps to enter data and get the regression line coefficients
Note if there are only two points, this calculates the unique line through them (same as section 1.4)
Abs value eqns Ex 8 is like the nasty ex I gave you earlier.
Return to: Merced College; Don Power Updated 08/18/05 by Don Power