Rectangular Coordinate Systems and Graphs of Equations

1
Rectangular Coordinate Systems and Graphs of
Equations

• René, René, hes our man,
• If he cant graph it,
• Nobody can. (2.1, 2.2)

2
POD (And who the heck is René?)

• Lets review. Write up here everything you can
  share about the x-y coordinate plane.

3
POD

• Lets review. Write up here everything you can
  share about the x-y coordinate plane.
• Labeling the axes and intervals.
• Quadrants.
• Origin.
• How to plot a point, using ordered pairs.
• What else?

4
Distance formula

• What is it, and how do we use it?

5
Distance formula

• Try it. Find the distance between the points
• A(-3, 6) and B(5,1).

6
Distance formula

• Find the distance between the points
• A(-3, 6) and B(5,1).

7
Midpoint formula

• What is it and how do we use it?

8
Midpoint formula

• Try it. Find the midpoint of the line segment
  connecting the points P1(-2, 3) and P2(4, -2).

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Midpoint formula

• Try it. Find the midpoint of the line segment
  connecting the points P1(-2, 3) and P2(4, -2).

10
Midpoint formula

• Try it again. Verify that the distances from the
  midpoint (1,½) to the endpoints, (-2, 3) and (4,
  -2), are equal.

11
Midpoint formula

• Try it again. Verify that the distances from the
  midpoint (1,½) to the endpoints, (-2, 3) and (4,
  -2), are equal.

12
Another level

• Now, find an equation for the perpendicular
  bisector of the line segment connecting the
  points P1(-2, 3) and P2(4, -2).

13
Another level

• Now, find an equation for the perpendicular
  bisector of the line segment connecting the
  points P1(-2, 3) and P2(4, -2).
• We know a point on that bisector.
• How do we determine the slope?
• What do we plug this information into?

14
Another level

• Now, find an equation for the perpendicular
  bisector of the line segment connecting the
  points P1(-2, 3) and P2(4, -2).
• We know a point on that bisector. (1, ½)
• How do we determine the slope? Its the negative
  reciprocal of the segment. That slope is -5/6.
• The equation (y - ½) 6/5(x 1)

15
Lets graph (put on your red shoes and graph the
blues)

• What do you know about graphs on the coordinate
  plane?

16
Lets graph (put on your red shoes and graph the
blues)

• Graphs as solutions of equations
• Intercepts (how do we find them?)
• x-y charts
• Dependent/independent variables
• Symmetry to either axis, the origin (see p. 108)
• Intersections
• Functions vs. relations (how does this relate to
  symmetry?)
• Domain and range

17
Symmetry

• Each table graph one of these equations on CAS,
  then well look as a class. What are the
  symmetries for each one?

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Symmetry

• At each table, graph each of the equations on
  calculators. What are their symmetries?
• to the origin
• to the y-axis
• to the origin
• to the x-axis not a function!
• substituting y for y leads to the same equation

19
Symmetry

• In general
• Odd functions symmetric to the origin
• f(x) -f(-x)
• Even functions symmetric to the y-axis
• f(x) f(-x)
• Not a function! symmetric to the x-axis
• substituting y for y leads to the
  same equation

20
Intersections

• Estimate the points of intersection for the
  following graphs. How?

21
Intersections

• Estimate the points of intersection for the
  following graphs. How?
• We could use algebra (how?), but lets graph here
  to find out.

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Intersections

• Start by graphing each equation.
• We can do this on the 94 or CAS.
• On the 84, hit 2nd Calc- intersect.

23
Intersections

• Youll be asked to mark the curves involved.
• Hit enter one more time to get the final result.

24
Intersections

• Finding the intersections on CAS isnt a whole
  lot different
• (Youll be glad to know well end here, because
  its a short period. Woo-hoo.)

25
Circles (the wheels on the bus go round and
round)

• Remember what the equation for a circle is?

26
Circles (the wheels on the bus go round and
round)

• Remember what the equation for a circle is?
• What do the variables represent?
• Are we talking functions? Why or why not?

27
Circles (the wheels on the bus go round and
round)

• Remember what the equation for a circle is?
• What do the variables represent?
• (h, k) is the center, r is the radius
• No function. No VLT.

28
Circles

• Use it. Find an equation of the circle with a
  center C(-2, 3) and containing the point D(4, 5).

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Circles

• Use it. Find an equation of the circle with a
  center C(-2, 3) and containing the point D(4, 5).
• Find r with the distance formula.

30
Circles

• Use it. Find an equation of the circle with a
  center C(-2, 3) and containing the point D(4, 5).
• Find r with the distance formula.
• Final equation