Rectangular Coordinate Systems and Graphs of Equations

René, René, he’s our man,If he can’t graph it,Nobody can. (2.1, 2.2)

POD (And who the heck is René?)

Let’s review. Write up here everything you can share about the x-y coordinate plane.

POD

Let’s 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?

Distance formula

What is it, and how do we use it?

Distance formula

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

2212

21 yyxx

Distance formula

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

43.9

89

2564

1653 22

d

Midpoint formula

What is it and how do we use it?

Midpoint formula

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

2,

22121 yyxx

Midpoint formula

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

2

1,1

2

1,2

2

2

)2(3,

2

42

Midpoint formula

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

Midpoint formula

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

4

2594259

212)14(2

13122

222

Another level

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

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?

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? It’s the negative reciprocal of the segment. That slope is -5/6.

The equation: (y - ½) = 6/5(x – 1)

Let’s graph (put on your red shoes and graph the blues)

What do you know about graphs on the coordinate plane?

Let’s graph (put on your red shoes and graph the blues)

Graphs as solutions of equationsIntercepts (how do we find them?)x-y chartsDependent/independent variablesSymmetry– to either axis, the origin (see p. 108)IntersectionsFunctions vs. relations (how does this relate to

symmetry?)Domain and range

Symmetry

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

2

4 3

2

xy

xy

xy

xy

SymmetryAt 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

xy

xy

xy

xy

3

2

4

SymmetryIn 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

Intersections

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

12

32

xy

xy

Intersections

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

We could use algebra (how?), but let’s graph here to find out.

12

32

xy

xy

Intersections

Start by graphing each equation.We can do this on the 94 or CAS.

On the 84, hit 2nd Calc- intersect.

12

32

xy

xy

Intersections

You’ll be asked to mark the curves involved.

Hit enter one more time to get the final result.

12

32

xy

xy

Intersections

Finding the intersections on CAS isn’t a whole lot different

(You’ll be glad to know we’ll end here, because it’s a short period. Woo-hoo.)

12

32

xy

xy

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

Remember what the equation for a circle is?

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?

222 )( rkyhx

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.

222 )( rkyhx

Circles

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

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.

222 )3()2( ryx

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:

222 )3()2( ryx

40436

)53()42( 22

40)3()2( 22 yx