Section 1.3 -- The Coordinate Plane
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Transcript of Section 1.3 -- The Coordinate Plane
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MA107 PreCalculusSection 1.3
The Coordinate Plane
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The Coordinate PlaneIf two copies of the number line, one horizontal
and one vertical, are placed so that they intersect at the zero point of each line, a pair of axes is formed.The horizontal number line is called the x-axis and
the vertical number line is called the y-axis.The point where the lines intersect is called the
origin.
We call this a rectangular coordinate plane or a Cartesian coordinate system.
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The Coordinate Plane
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The Coordinate Plane
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Inequalities in Two Dimensions
The graph of an inequality in two variables consists of all ordered pairs that make the inequality a true statement.
Example: Suppose we want to graph the inequality .
Procedure:• Graph the boundary curve
.• Draw a solid curve if
equality is included.• Draw a dashed curve if
equality is not included.• Determine which region(s)
formed by the curve makes the inequality true by testing with one point from inside each region.
• Shade the region(s) that make the inequality true.
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Inequalities in Two Dimensions
Systems of two inequalities:
Idea: Graph both inequalitiesand the region that has been shaded in twice is the region we’re looking for.
Example at left: Graph the solution set of the following system of inequalities:
Click here to see a dynamicexample of linear inequalities.
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DistanceSuppose two points P1 and P2 have coordinates
. What is the distance between P1 and P2?
€
(x1,y1) and (x2,y2)
• The distance from P1 to P2 isthe length of the hypotenuseof a right triangle.• The length of the bottom sideis the same as the distance between x1 and x2 on the x-axis, that is, . The length of the vertical side is the same as the distance between y1 and y2, that is .
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DistanceSo if we let d be the distance between P1 and P2,
by the Pythagorean Theorem ….
Now we take the square rootof both sides. Since distance is positive:
Since we’re squaring in there,we can dispense with theabsolute values and get
Distance Formula
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DistanceExample: Find the distance between the points
(4,-7) and (-1,3).
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DistanceSee Mathematica Player demo on distance.
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MidpointThe midpoint of the line segment connecting the points
P1(x1,y1) and P2(x2,y2) is computed by simply averaging the x- and y-coordinates separately.
Midpoint FormulaThe midpoint between (x1,y1) and (x2,y2) is
Take a moment to find the coordinates of the point half way between and .
Answer:
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Circles in the PlaneA circle is defined as the set of all points that
are the same distance from a given point.The distance is called the radius.The given point is called the center of the circle.
Let (x, y) be any point on a circle with center (h, k) and radius r as shown at left.Since (x, y) must be r units from the center of the circle, the distance formula gives
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Standard Form of the Equation of a Circle
The graph of
is a circle of radius r (r ≥ 0) with center at the point (h, k). If the circle has center at the origin, the equation becomes
The circle is called the unit circle.
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Circles in the PlaneExample:
The center is at the point (2, 3) and the radius is 1 unit.
We can figure out some points on the circle by starting with the center point, (2, 3), and adding or subtracting 1 from each coordinate.
So four examples would be (3, 3), (1, 3), (2, 2), and (2, 4).
Try: Find an equation of a circle with center (-3, 6) and radius 4.
Answer:
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Completing the squareExample: Find the center and radius of the
circle whose equation is .
Since there is an x2-term and an x-term, we have to combine them to form the term by completing the square:We first group the x-terms together:
To turn the thing in parentheses into a perfect square, divide the coefficient of the x-term by 2, square that number and add it to both sides.
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Completing the SquareWe can then factor the x-
terms:
Or to put it in the form of a circle:
We see that h is 1 and k is 0. So the center of the circle is (1,0) and the radius is 2.
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Completing the SquareTo try on your own: Graph the equation
Hint: Divide first by 2.
Circle is centered at (-2,3) and hasradius 1.
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Homeworkpg. 19: 1-19 odd, 25-43 odd, 51, 55, 57