Download CRASH COURSE IN PRECALCULUS

CRASH COURSE IN PRECALCULUS
Shiah-Sen Wang
The graphs are prepared by Chien-Lun Lai
Based on : Precalculus: Mathematics for Calculus
by J. Stuwart, L. Redin & S. Watson,
6th edition, 2012, Brooks/Cole
Chapter 1. Section 1.10 and Chapter 11 Section 11.1-11.3.
LECTURE 3. PLANE GEOMETRY
The purpose of this lecture is to review some high school plane
geometry:
Algebra ⇐⇒ Geometry.
We will discuss lines and quadratic curves (conic sections) in
the plane.
Coordinate (Cartesian) Plane
Construction of Coordinates
As we identify points on a line with the real numbers, we can
identify points in a (Euclidean) plane with ordered pairs of
numbers to form the coordinate plane or Cartesian plane,
which is done by drawing two perpendicular real lines that
intersect at 0 on each line. The horizontal line with positive
direction to the right is called the x-axis and the other line with
positive direction pointing upward is called the y -axis. The
point of intersection of the two axes is called the origin O.
These two axes divide the plane into four quadrants, labeled I,
II, III and IV. Any point P on the plan can be labeled by a
unique ordered pairs of numbers (a, b). a is called the
x-coordinate of P and b is called the y-coordinate of P.
The Distance Formula
Definition
By using the Pythagorean theorem, define distance between
the points A(x1 , y1 ) and B(x2 , y2 ) in the plane is
√
d(A, B) = (x2 − x1 )2 + (y2 − y1 )2
Graphs Equations in Two Variables
In this section, we only treat lines and quadratic curves of
standard forms.
The more general cases will will be discussed in Calculus!
Line Equations
Definition
The slope m of a nonvertical line that passes through the
points A(x1 , y1 ) and B(x2 , y2 ) is
m=
y2 − y1
x2 − x1
Consequently, the slope of a horizontal line is 0.
The slope of a vertical line is not defined.
Vertical Line Equation: x = c, where c is a fixed real number.
Point-Slope Equation: The line equation that through the point
(x1 , y1 ) and has slope m is
y − y1 = m(x − x1 ).
Horizonal Line Equation: y = c, where c ∈ R is fixed.
Line Equations
The line equation that through the points A(x1 , y1 ) and
B(x2 , y2 ) in the plane is
y − y1 = (
y2 − y1
) (x − x1 )
x2 − x1
which is called Two-Points Form of a line equation.
y − y1 = m(x − x1 ) ⇐⇒ y = mx + b, b = y1 − mx1 .
The (red) equation is called Slope-Intercept Form of a line
equation with slope m and y-intercept b. Together
Theorem
The graph of a linear equation
ax + by + c = 0 (a and b are not both 0 ⇔ a2 + b2 ≠ 0)
is a line. Conversely, Every line is the graph of a linear
equation.
Facts about The Slopes of Two Lines
Theorem
1. Two nonvertical lines are parallel if and only if they have
the same slope.
Facts about The Slopes of Two Lines
2. Two lines have slopes m1 and m2 are perpendicular if and
only if m1 m2 = −1.
The horizonal lines y = c1 are perpendicular the the
vertical lines x = c2 for any fixed c1 , c2 ∈ R.
Graphing Regions in Coordinate Plane
Describe and sketch regions given by the following each set:
1. {(x, y ) ∈ R2 ∣ 2x − y ≤ 3}
Graphing Regions in Coordinate Plane
2. {(x, y ) ∈ R2 ∣ 2x + y < 3}
Graphing Regions in Coordinate Plane
3. {(x, y ) ∈ R2 ∣ 2x − y ≤ 3 and 2x + y < 3}
Digression
The coordinate of intersecting point of two lines are found by
solving the system of two linear equations
⎧
⎪
⎪2x − 3y = 3
⎨
⎪
⎪
⎩2x + 3y = 3
(1)
(2)
3
.
2
3
With x = plugging into (1) Ô⇒ 3 − 3y = 3 Ô⇒ y = 0. Hence
2
3
( , 0) is the coordinate of the intersecting point.
2
(1) + (2) Ô⇒ 4x = 6 Ô⇒ x =
Writing the set of region given by the graph
The line on the left L1 passes through ( −3
2 , 0) and (0, 3) and the
5
line on the right L2 passes through ( 2 , 0) and (0, −5), using the
two-points form of line a equation:
Writing the set of region given by the graph
L1 ∶ y − 3 =
⎛ 0−3 ⎞
(x − 0) ⇔ x − 2y = −6
⎠
⎝ −3
−
0
2
L2 ∶ y − (−5) =
⎛ 0 − (−5) ⎞
(x − 0) ⇔ x − 2y = 10
⎝ 52 − 0 ⎠
Since the region lies between them and the lines are solid, the
set is
{(x, y ) ∈ R2 ∣ −6 ≤ x − 2y ≤ 10} = {(x, y) ∈ R2 ∣ ∣x − 2y − 2∣ ≤ 8}
because
−6 ≤ x − 2y ≤ 10 ⇔ −8 ≤ (x − 2y) − 2 ≤ 8 ⇔ ∣x − 2y − 2∣ ≤ 8. See
Writing the set of region given by the graph
Circles
The equation of a circle with center at (a, b) and radius r is
(x − a)2 + (y − b)2 = r 2
This is called the standard form for equation of the circle. In
particular, if the center is O(0, 0), then the equation is
x 2 + y 2 = r 2.
Examples
1. Graph x 2 + y 2 = 25
Examples
2. Show that the equation x 2 + y 2 + 2x − 6y + 7 = 0 represent a
circle, and find the center and radius of the circle.
0 = x 2 + y 2 + 2x − 6y + 7
= (x 2 + 2x + 1) + (y 2 − 6y + 9) + (7 − 1 − 9)
√ 2
= (x + 1)2 + (y − 3)2 − 3 ⇔ [x − (−1)]2 + (y − 3)2 = ( 3) .
√
Hence the equation represents a circle of radius 3 and
centered at (−1, 3) .
Parabolas
Geometric Definition of a Parabola
A parabola is the set of points in the plane that are
equidistance from a fixed point F (called focus) and a fixed line
` (called directrix).
Equations of Parabolas
With vertex V at the origin O(0, 0), two cases to consider here:
1. The focus F (0, p) and the directrix ` equation is given by
y = −p. If P(x,
√ y) is a point on the parabola, then
d(F , P) = x 2 + (y − p)2 and
√ the distance from P to ` is
∣y − (−p)∣ = ∣y + p∣. Hence x 2 + (y − p)2 = ∣y + p∣ ⇔
x 2 + (y − p)2 = ∣y + p∣2 = (y + p)2 ⇔
x 2 + y 2 − 2py + p2 = y 2 + 2py + p2 ⇔ x 2 = 4py. Two
possibilities: (a) p > 0, the parabola opens upward; (b)
p < 0, the parabola opens downward.
2. The focus F (p, 0) and the directrix ` equation is given by
x = −p. If P(x, y ) is a point on the parabola, as before, we
can have y 2 = 4px. Two possibilities: (a) p > 0, the parabola
opens rightward; (b)p < 0, the parabola opens leftward.
Examples of Parabolas
Examples of Parabolas
Examples of Parabolas
Examples of Parabolas
Examples of Parabolas
Ellipse
Geometric Definition of a Ellipse
A ellipse is the set of points in the plane that the sum of whose
distances from two fixed point F1 and F2 (called foci) is a
positive constant 2a > d(F1 , F 2) = 2c > 0.
Equations of Ellipses
Let the 2a > 0 be the sum of distance the sum of whose
distances from two fixed point F1 and F2 . Then if P(x, y) is any
point on the ellipse, then we have
d(P, F1 ) + d(P, F2 ) = 2a
Consider the following cases:
1. If the foci F1 (−c, 0), F2 (c, 0) (c > 0) are placed on the
x-axis, using the Distance Formula, we see that the
equation is
x2 y2
+
=1
a2 b 2
where b2 = a2 − c 2 (with b > 0 and so a > b). The major axis
is horizontal of length 2a and the minor axis is vertical of
length 2b.
Equations of Ellipses
2. If the foci F1 (0, −c), F2 (0, c) (c > 0) are placed on the
x-axis, the equation is
x2 y2
+
=1
b 2 a2
The major axis is vertical of length 2a and the minor axis is
horizontal of length 2b.
Hyperbolas
Geometric Definition of a Hyperbola
A hyperbola is the set of points in the plane that the difference
of whose distances from two fixed point F1 and F2 (called foci)
is a positive constant 2a > 0.
Equations of Hyperbolas
Let the 2a > 0 be the sum of distance the sum of whose
distances from two fixed point F1 and F2 . Then if P(x, y) is any
point on the hyperbola, then we have
d(P, F1 ) − d(P, F2 ) = ±2a
Consider the following cases:
1. If the foci F1 (−c, 0), F2 (c, 0) (c > a) are placed on the
x-axis, using the Distance Formula, we see that the
equation is
x2 y2
−
=1
a2 b 2
where b2 = c 2 − a2 (with b > 0 and so a > b). The
transverse axis is horizontal of length 2a. See
Equations of Hyperbolas
2. If the foci F1 (0, −c), F2 (0, c) (c > 0) are placed on the
x-axis, the equation is
−
x2 y2
+
=1
b 2 a2
The transverse axis is vertical of length 2a. See