Making Graphs from a Table of Values and Understanding the Graphs of Horizontal and Vertical Lines Blue Level Problems

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1 Making Graphs from a Table of Values and Understanding the Graphs of Horizontal and Vertical Lines Blue Level Problems. Coordinate Triangle? We have a triangle ABC, and it has an area of units^. Point A is at (, ) and point B is at (, ). a. Where can point C be? (Read the given parts carefull.) b. What if ABC is isosceles? Now where can C be?. Descartes Triangle Two vertices of a triangle are located at (, ) and (, ). If the area of the triangle is unit², BLUE: What are all possible positions for the third verte? BLACK: What if the triangle is also isosceles?. The point (, ) is reflected about the line =. The image point is then reflected about the line =. The resulting point is (a, b). Compute a + b. INSTRUCTIONS for the REST of this ASSIGNMENT - PLEASE READ CAREFULLY The standard, th grade level epectation for this unit is that ou become comfortable graphing all Straight Line linear functions. In the world of mathematics, there are man other tpes of functions that give us interesting graphs. The rest of this Blue assignment and the Black assignment give ou the chance to learn about some of these other interesting functions. For the rest of these problems, USE a TABLE of VALUES to graph the functions mentioned in the problems. As ou work, challenge our mind to think carefull and strategicall about the input values ou choose for our input/output tables.. Choose input values for which the output value is eas to calculate. Choose input values that ma cause the graph to change its behavior. Choose input values that can help ou determine the graph s general shape of the graph of the tpe of function ou re thinking about. Make our graph, and if ou need more points to determine the shape of the graph, add additional input/output pairs to our table of values. Blue Level Functions Absolute Value Functions: = Black Level Functions/Relations Step Functions: = [] Quadratic (Parabolic) Functions: = ² Cubic Functions: = ³ Inverse (Hperbolic) Functions: = Circle Relations: ² + ² = Eponential Functions: =

2 ABSOLUTE VALUE functions:. The graph of the absolute value is a broken line. Graph the function: =. Graph =. What is the number of square units in the area of the region bounded b the graphs if = and = - +? Quadratic (Parabolic Functions) The graph of a quadratic function is a parabola. As ou complete the following problems, pa attention to how adding a coefficient to ² changes the shape of the graph. Also, notice adding or subtracting a constant changes the shape of the graph. = ² The basic parabola is the one for = ²drawn here. An other parabola will relate to this one in some wa. Use tables of values to draw graphs of the following quadratic functions. 7. Draw graphs of these quadratic functions. a. = ² b. = c. = -² d. =. Draw graphs of these quadratic functions. a. = ²+ b. = ( )² c. = ²- d. = ( + )² 9. The parabolas drawn below are similar to the graph of = ². Write down the equation for each one. a. b. -

3 c. d. - INVERSE RELATIONS The graph of an inverse variation is called a Hperbola. The following problem contains an eample of an inverse variation function.. Check-out Queues: Twelve customers are waiting to have their groceries scanned at a supermarket. The supermarket has several cash registers, but the are not all open. If, for eample, there were si cash registers open, there would be two customers waiting at each one. a. Complete the table. Number of cash registers open () Number of people in the queue () b. Plot the points from the table on the a graph and join them with a smooth curve. Number of people in each queue Number of cash registers open

4 c. What do ou notice about each pair of numbers in the table? Complete each table and graph the equation.. =. =

5 Overview of Simple Hperbolas - Problems An mathematical relation = number has a graph called hperbola. Eamples = = - etc. Eample Answer Draw the graph of = b plotting points and joining them with a smooth curve. First write down all the pairs of numbers that multipl to give. Some will be negative Note: there are other pairs of numbers which are not integers e.g. and., or. and, etc. = Eample Draw the graph of = -. Answer = Cop and complete this table, and then join the points on a graph with two smooth branches of a curve =, i.e. = -

6 . Cop and complete this table, and then join the points on the graph with two smooth branches of a curve = -, i.e. = -. Draw the graphs of these hperbolas. a. = b. = c. = - d. = -9. Draw the graphs of these hperbolas. a. = b. = 7. Match each graph a to d with the correct equation (A) to (D). a. b. c. d. (A) = (B) = (C) = (D) = -

7 Making Graphs from a Table of Values and Understanding the Graphs of Horizontal and Vertical Lines Blue Level Solutions. a. Point C is anwhere along the lines = or = - b. If ABC is isosceles, point C is at (, ) or (, -) Y Point A is at (, ) C C Point B is at (, ) = The area of the triangle is given b: A = ½bh (, ) (, ) A B X Line segment is AB = b = The area, A, is given as units² B substitution the area formula becomes = ½ h; or h = Point C of an triangle of units²whose C C = - base is at AB (as given) can be anwhere along either line of = or = -. An triangle is represented b two green triangles. The two magenta triangles represent the isosceles triangle ABC. In this case point C is at either (, ) or (, -).. BLUE Since the formula for area of a triangle is bh/, h/=, h must be. therefore, on the coordinate plane, since one segment of the triangle is on the ais, the coordinate must merel move spaces to the left or right (making it - or ) therefore, an point on the line = or on the line =- will satisf the equation. BLACK In order to make the triangle isosceles, one of two things must happen: the two sides other than the one with the side length can be equal OR one other side can equal as well. The ke in this is that THE AREA MUST BE. So, the two points that will satisf the first part are (,9), (-,9). This is because the coordinate must be plus or minus, and the coordinate will be the average of those on the endpoints of the given side. In the other scenario, another side must equal. the side could be above the line or below the line, and on either side of the ais. that makes four points. since the side length must be, and the distance from the ais is, a right triangle can be made. using Pthagorean, ^-^=, the root of which is (root). Therefore, the coordinates will be the bottom coordinate MINUS root OR the top coordinate PLUS root. So, the coordinates will be the lower () minus root and the higher () plus root. since the coordinates will be plus or minus, the last coordinates will be: (,-(root)), (,+(root)), (-,+(root)), (-,-(root)). Therefore, the points are: (,9), (-,9), (,-(root)), (,+(root)), (-,+(root)), (-,-(root))

8 . Reflection of (, ) over = results in image point (-, ). Reflection of (-, ) over = results in image point (-, ). The sum of the coordinates of this point is - + =. (-, ) = (, ) = (-, ). - - (-, ) (, ). (, ) (, ) (, ) (, ) - -. The shaded region is a square with diagonal units long. You could use the Pthagorean Theorem to find the length of the side of the square. Or, ou could notice that the diagonals are perpendicular to each other and form four triangles. Each of the triangles is a isosceles right triangle with legs cm. So the area is [½()()] = square units. = = - +

9 7. a. b. c. d.. a. b. c. d a. = ²+ b. = ( + )² c. = ²- d. = ( )². a. Complete the table. Number of cash registers open () Number of people in the queue ()

10 b. Number of people in each queue Number of cash registers open c. Each pair multipl to.. =

11 . = =, i.e. = = -, i.e. =

12 . a. b c. d a. b a. (B) b. (D) c. (C) d. (A)

13 Bibliograph Information Teachers attempted to cite the sources for the problems included in this problem set. In some cases, sources were not known. Problems Bibliograph Information - The Math Dreel ( Math Counts ( Cook, Allen, and Natalia Romalis. Content Area Mathematics for Secondar Teachers The Problem Solver. New York: Christopher-Gordon, Inc.,. 7-, -7 Barton, David. Theta Mathematics: NCEA Level, Pearson,.