CRASH COURSE IN PRECALCULUS

Transcription

1 CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai

2 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

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.

4 Coordinate (Cartesian) Plane

5 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.

6

7 The Distance Formula Definition By using the Pythagorean theorem, define distance between the points A(x 1, y 1 ) and B(x 2, y 2 ) in the plane is d(a, B) = (x 2 x 1 ) 2 + (y 2 y 1 ) 2

8 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!

9 Line Equations Definition The slope m of a nonvertical line that passes through the points A(x 1, y 1 ) and B(x 2, y 2 ) is m = y 2 y 1 x 2 x 1 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 (x 1, y 1 ) and has slope m is y y 1 = m(x x 1 ). Horizonal Line Equation: y = c, where c R is fixed.

10 Line Equations The line equation that through the points A(x 1, y 1 ) and B(x 2, y 2 ) in the plane is y y 1 = ( y 2 y 1 x 2 x 1 ) (x x 1 ) which is called Two-Points Form of a line equation. y y 1 = m(x x 1 ) y = mx + b, b = y 1 mx 1. 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 a 2 + b 2 0) is a line. Conversely, Every line is the graph of a linear equation.

11 Facts about The Slopes of Two Lines Theorem 1. Two nonvertical lines are parallel if and only if they have the same slope.

12 Facts about The Slopes of Two Lines 2. Two lines have slopes m 1 and m 2 are perpendicular if and only if m 1 m 2 = 1. The horizonal lines y = c 1 are perpendicular the the vertical lines x = c 2 for any fixed c 1, c 2 R.

13 Graphing Regions in Coordinate Plane Describe and sketch regions given by the following each set: 1. {(x, y) R 2 2x y 3}

14 Graphing Regions in Coordinate Plane 2. {(x, y) R 2 2x + y < 3}

15 Graphing Regions in Coordinate Plane 3. {(x, y) R 2 2x y 3 and 2x + y < 3}

16 Digression The coordinate of intersecting point of two lines are found by solving the system of two linear equations 2x 3y = 3 (1) 2x + 3y = 3 (2) (1) + (2) 4x = 6 x = 3 2. With x = 3 2 plugging into (1) 3 3y = 3 y = 0. Hence ( 3, 0) is the coordinate of the intersecting point. 2

17 Writing the set of region given by the graph The line on the left L 1 passes through ( 3 2, 0) and (0, 3) and the line on the right L 2 passes through ( 5 2, 0) and (0, 5), using the two-points form of line a equation:

18 Writing the set of region given by the graph L 1 y 3 = (x 0) x 2y = 6 L 2 y ( 5) = 0 ( 5) (x 0) x 2y = 10 Since the region lies between them and the lines are solid, the set is {(x, y) R 2 6 x 2y 10} = {(x, y) R 2 x 2y 2 8} because 6 x 2y 10 8 (x 2y) 2 8 x 2y 2 8. See

19 Writing the set of region given by the graph

20 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.

21 Examples 1. Graph x 2 + y 2 = 25

22 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) = (x + 1) 2 + (y 3) 2 3 [x ( 1)] 2 + (y 3) 2 = ( 3) 2. Hence the equation represents a circle of radius 3 and centered at ( 1, 3).

23 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 l (called directrix).

24 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 l 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 l 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 + p 2 = y 2 + 2py + p 2 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 l 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.

25 Examples of Parabolas

26 Examples of Parabolas

27 Examples of Parabolas

28 Examples of Parabolas

29 Examples of Parabolas

30 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 F 1 and F 2 (called foci) is a positive constant 2a > d(f 1, F2) = 2c > 0.

31 Equations of Ellipses Let the 2a > 0 be the sum of distance the sum of whose distances from two fixed point F 1 and F 2. Then if P(x, y) is any point on the ellipse, then we have Consider the following cases: d(p, F 1 ) + d(p, F 2 ) = 2a 1. If the foci F 1 ( c, 0), F 2 (c, 0) (c > 0) are placed on the x-axis, using the Distance Formula, we see that the equation is x 2 a 2 + y 2 b 2 = 1 where b 2 = a 2 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.

32 Equations of Ellipses 2. If the foci F 1 (0, c), F 2 (0, c) (c > 0) are placed on the x-axis, the equation is x 2 b 2 + y 2 a 2 = 1 The major axis is vertical of length 2a and the minor axis is horizontal of length 2b.

33 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 F 1 and F 2 (called foci) is a positive constant 2a > 0.

34 Equations of Hyperbolas Let the 2a > 0 be the sum of distance the sum of whose distances from two fixed point F 1 and F 2. Then if P(x, y) is any point on the hyperbola, then we have Consider the following cases: d(p, F 1 ) d(p, F 2 ) = ±2a 1. If the foci F 1 ( c, 0), F 2 (c, 0) (c > a) are placed on the x-axis, using the Distance Formula, we see that the equation is x 2 a 2 y 2 b 2 = 1 where b 2 = c 2 a 2 (with b > 0 and so a > b). The transverse axis is horizontal of length 2a. See

35

36 Equations of Hyperbolas 2. If the foci F 1 (0, c), F 2 (0, c) (c > 0) are placed on the x-axis, the equation is x 2 b 2 + y 2 a 2 = 1 The transverse axis is vertical of length 2a. See

37