Unit 1 Introduction to Precalculus Rectangular Coordinates (Unit 1.1)

Transcription

1 Unit 1 Introduction to Precalculus Rectangular Coordinates (Unit 1.1) William (Bill) Finch Mathematics Department Denton High School

2 Lesson Goals When you have completed this lesson you will: Understand the Cartesian Plane as orthogonal real number lines. Understand coordinate pairs as directed distances from the axes. Derive the distance formula from the Pythagorean Theorem. Apply the midpoint and distance formulas. Understand how to write a complete mathematical solution to a problem using both math symbols and words. Rect Coord 2 / 16

3 Lesson Goals When you have completed this lesson you will: Understand the Cartesian Plane as orthogonal real number lines. Understand coordinate pairs as directed distances from the axes. Derive the distance formula from the Pythagorean Theorem. Apply the midpoint and distance formulas. Understand how to write a complete mathematical solution to a problem using both math symbols and words. Rect Coord 2 / 16

4 Lesson Goals When you have completed this lesson you will: Understand the Cartesian Plane as orthogonal real number lines. Understand coordinate pairs as directed distances from the axes. Derive the distance formula from the Pythagorean Theorem. Apply the midpoint and distance formulas. Understand how to write a complete mathematical solution to a problem using both math symbols and words. Rect Coord 2 / 16

5 Lesson Goals When you have completed this lesson you will: Understand the Cartesian Plane as orthogonal real number lines. Understand coordinate pairs as directed distances from the axes. Derive the distance formula from the Pythagorean Theorem. Apply the midpoint and distance formulas. Understand how to write a complete mathematical solution to a problem using both math symbols and words. Rect Coord 2 / 16

6 Lesson Goals When you have completed this lesson you will: Understand the Cartesian Plane as orthogonal real number lines. Understand coordinate pairs as directed distances from the axes. Derive the distance formula from the Pythagorean Theorem. Apply the midpoint and distance formulas. Understand how to write a complete mathematical solution to a problem using both math symbols and words. Rect Coord 2 / 16

7 Lesson Goals When you have completed this lesson you will: Understand the Cartesian Plane as orthogonal real number lines. Understand coordinate pairs as directed distances from the axes. Derive the distance formula from the Pythagorean Theorem. Apply the midpoint and distance formulas. Understand how to write a complete mathematical solution to a problem using both math symbols and words. Rect Coord 2 / 16

8 Wondering What Precalulus Is? Precalculus: (noun) a varied collection of mathematical fields and topics designed to prepare you for taking Calculus, including: algebra analytic geometry trigonometry miscellaneous other topics Think of Precalculus as a review/extension/connection of past-learned maths with new maths added on. Rect Coord 3 / 16

9 The Cartesian Plane Also called a rectangular coordinate system and named for the French mathematician and philosopher René Descartes ( ). Quad II y-axis Quad III Origin 2 Quad I 3 Quad IV x-axis Directed Distances y-axis x (x, y) y x-axis Rect Coord 4 / 16

10 Scatterplot A scatterplot shows the relationship between two variables plotted on the Cartesian Plane. Manatees are large, slow-moving mammals that live in warm shallow waters. Because they like to float near the surface they are easily injured or killed by motor boats. The next slide has data from Florida from showing the number of powerboat registrations (in thousands) and numbers of manatees killed by boats. Rect Coord 5 / 16

11 Scatterplot Make two scatterplots, the first plotting (year,manatees) and the second plotting (boats,manatees). Year Boats Manatees Year Boats Manatees How would you describe the relationship pictured in each scatterplot? Rect Coord 6 / 16

12 Analytic Geometry Euclidean geometry is based on the works of the Greek mathematician, Euclid (c. 300 B.C.E.). This is mostly what you studied in your high school geometry class. Analytic geometry (also known as coordinate geometry) is the study of geometry using a coordinate system. Modern analytic geometry is generally attributed to Descartes. Rect Coord 7 / 16

13 Pythagorean Theorem For a right triangle with hypotenuse of length c and sides of lengths a and b, then a 2 + b 2 = c 2. b c a Note the converse is also true: if a 2 + b 2 = c 2, then the triangle is a right triangle. Rect Coord 8 / 16

14 Distance Formula Now add a coordinate system to our triangle... y (x 1, y 1 ) y 2 y 1 d (x 1, y 2 ) (x 2, y 2 ) x x 2 x 1 Rect Coord 9 / 16

15 Distance Formula Use the Pythagorean Theorem to find d: y c 2 = a 2 + b 2 d 2 = x 2 x y 2 y 1 2 d = x 2 x y 2 y 1 2 y 2 y 1 (x 1, y 1) (x 1, y 2) d x 2 x 1 (x 2, y 2) x d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 Rect Coord 10 / 16

16 Distance Formula Use the Pythagorean Theorem to find d: y c 2 = a 2 + b 2 d 2 = x 2 x y 2 y 1 2 d = x 2 x y 2 y 1 2 y 2 y 1 (x 1, y 1) (x 1, y 2) d x 2 x 1 (x 2, y 2) x d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 Rect Coord 10 / 16

17 Distance Formula Use the Pythagorean Theorem to find d: y c 2 = a 2 + b 2 d 2 = x 2 x y 2 y 1 2 d = x 2 x y 2 y 1 2 y 2 y 1 (x 1, y 1) (x 1, y 2) d x 2 x 1 (x 2, y 2) x d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 Rect Coord 10 / 16

18 Distance Formula Use the Pythagorean Theorem to find d: y c 2 = a 2 + b 2 d 2 = x 2 x y 2 y 1 2 d = x 2 x y 2 y 1 2 y 2 y 1 (x 1, y 1) (x 1, y 2) d x 2 x 1 (x 2, y 2) x d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 Rect Coord 10 / 16

19 Midpoint Formula The midpoint of the line segment joing the points (x 1, y 1 ) and (x 2, y 2 ) is the average of the x- and y-coordinates. ( x1 + x 2 (x m, y m ) =, y ) 1 + y y (x 1, y 1 ) (x m, y m ) (x 2, y 2 ) x Rect Coord 11 / 16

20 Writing Mathematics Writing the solution to a mathematical problem is a bit like writing an essay: Introduction Initial information such as equation, diagrams, etc. Body Calculations, equations solved, graphs, etc. Conclusion Clear statement of the answer. Note that words are just as important as math symbols. Rect Coord 12 / 16

21 Example 1 Show that the points (4, 0), ( 1, 5), and (2, 1) are vertices of a right triangle. Rect Coord 13 / 16

22 Example 2 In the previous example you showed the given points to be vertices of a right triangle. The hypotenuse had endpoints ( 1, 5) and (4, 0). Find the coordinates of the midpoint of the hypotenuse. Rect Coord 14 / 16

23 Example 3 The volume of a cylinder is given by the formula V = πr 2 h. Solve this formula for h. If a cylindrical can has a volume of 200 cubic centimeters (cm 3 ) and a radius of 4 centimeters, use your previous result to find the height (to the nearest hundredth of a centimeter). r Cat Food h Rect Coord 15 / 16

24 What You Learned You can now: Understand the Cartesian Plane as orthogonal real number lines. Understand coordinate pairs as directed distances from the axes. Derive the distance formula from the Pythagorean Theorem. Apply the midpoint and distance formulas. Understand how to write a complete mathematical solution to a problem using both math symbols and words. Rect Coord 16 / 16

25 What You Learned You can now: Understand the Cartesian Plane as orthogonal real number lines. Understand coordinate pairs as directed distances from the axes. Derive the distance formula from the Pythagorean Theorem. Apply the midpoint and distance formulas. Understand how to write a complete mathematical solution to a problem using both math symbols and words. Rect Coord 16 / 16

26 What You Learned You can now: Understand the Cartesian Plane as orthogonal real number lines. Understand coordinate pairs as directed distances from the axes. Derive the distance formula from the Pythagorean Theorem. Apply the midpoint and distance formulas. Understand how to write a complete mathematical solution to a problem using both math symbols and words. Rect Coord 16 / 16

27 What You Learned You can now: Understand the Cartesian Plane as orthogonal real number lines. Understand coordinate pairs as directed distances from the axes. Derive the distance formula from the Pythagorean Theorem. Apply the midpoint and distance formulas. Understand how to write a complete mathematical solution to a problem using both math symbols and words. Rect Coord 16 / 16

28 What You Learned You can now: Understand the Cartesian Plane as orthogonal real number lines. Understand coordinate pairs as directed distances from the axes. Derive the distance formula from the Pythagorean Theorem. Apply the midpoint and distance formulas. Understand how to write a complete mathematical solution to a problem using both math symbols and words. Rect Coord 16 / 16

29 What You Learned You can now: Understand the Cartesian Plane as orthogonal real number lines. Understand coordinate pairs as directed distances from the axes. Derive the distance formula from the Pythagorean Theorem. Apply the midpoint and distance formulas. Understand how to write a complete mathematical solution to a problem using both math symbols and words. Rect Coord 16 / 16