Graphing in the Coordinate Plane Chapter 10 Graphing in the Coordinate Plane

What You’ll learn To graph points on a plane 10-1 Graphing Points What You’ll learn To graph points on a plane

Rene Descartes develops Coordinate System Coordinate Plane is a grid formed by a horizontal number line called the X axis and a vertical number line called the Y axis Ordered Pair (x,y) gives the coordinates of the location of a point X-coordinate is the first number of horizontal units from the origin Y-coordinate is the second number of vertical units for the origin -1,1 X axis Y axis origin Ordered pair

The X and Y axes divide the coordinate plane into 4 Quadrants II I III IV

Writing Coordinates The smiley face is 2 units to the right of the y axis, so the x coordinate is 2 The smiley face is 2 units above the x axis, so the y coordinate is 2 The ordered pair for the location of the fly is (2,2)

Graph point A (3,-5) Text page 524 1-12

Horizontal and Vertical Lines In a coordinate plane: lines that are parallel to the x-axis are horizontal Lines that are parallel to the y-axis are vertical Assign Pr 10-1

10-2 Graphing Linear Equations What You’ll Learn: To find ordered pairs that are solutions of linear equations To graph linear equations

Determine whether each ordered pair is a solution of y = x + 5 (40,45) y = x + 5 45 = 40 + 5 45 = 45 (21,27) y = x + 5 27 = 21 + 5 27 = 26 Substitute for x and y in the equation

Determine whether each ordered pair is a solution of y = 3x - 1 (4, 11) (7, 12) (17, 23) Text page 529 # 1-8; 32-37

Making a Table of Values y = x + 1 to graph a line (x,y) -4 -4 + 1 -3 -4,-3 -2 -2 + 1 -1 -2,-1 0 + 1 1 0,1 1 + 1 2 1,2 3 3 + 1 4 3,4 Solutions of the equation Graph points, draw line Choose values for x

Using the graph find 2 more solutions of y = ½ x - 2 Graph the linear equation y = ½ x-2 How many solutions does a linear equation have?? Text page 529 # 9-28; pr 10-2

Graph each linear equation 2y + 10 = 2x Does anyone remember how? Slope Intercept Form? y = mx + b

Practice Graph y – 2 = ¼ x Graph y = 5/6x + 3 Graph 2x + 3y = -6

10-3 Finding the Slope of a line What You’ll Learn: To find and use the slope of a line Investigation text page 533

Slope of a line Slope is a ration that describes the steepness of a line Slope = rise run Rise – compares the vertical change a line Run – the horizontal change of a line

Finding Slope Negative Slope Positive slope

Finding Slope Slope = rise / run -6 6/-6 1/-1 or -1 +6 Pr 10-3; page 538 practice quiz, quiz 10-1 to 10-3

10-4 Exploring Nonlinear Relationships What You’ll Learn: To graph nonlinear equations

Equations Linear Equations Nonlinear Equations Represented by a straight line Nonlinear Equations Represented by a nonstraight line y = x2 is an example of a curve called a parabola

Graphing a Parabola: Graph y = -x2 using integer values of x from –3 to 3 Make values of table x -x2 y (x,y) -3 -(-3)2 -9 -3,-9 -2 -(-2)2 -4 -2,-4 -1 -(-1)2 -1,-1 -(0)2 0,0 1 -(1)2 1,-1 2 -(2)2 2,-4 3 -(3)2 3,-9 Graph ordered pairs, then connect

Graph a Parabola Graph y = 2x2 using integer values of x from –3 to 3 Why is the graph of y = 2x2 only in Quadrants I and II? Is (-1,-2) a solution?? 10-4 a, b

Graph absolute value equations (V shape) : y = /x/ (x,y) -2 /-2/ 2 -2,2 -1 /-1/ 1 -1,-1 /0/ 0,0 /1/ 1,1 /2/ 2,2

V Shape Equations Graph y = 2 /x/ using integers –2 to 2 Determine whether (4, 8) is a solution Text page 542 # 1-24;10-4 c,d; pr 10-4

What You’ll learn: To graph translations

Graphing Translations A transformation is a change of the position, shape, or size of a figure. There are 3 types of transformations: Translation: slide, flips, and turns Move every point of a figure in the same direction and distance Image The end result after a transformation Prime rotation Used to identify and image point A` as prime A

Translating a Point Translate F(4,1) up 2 units and 3 units to the left’ what are the coordinates of the image F’?? To translate use arrow notation F(4,1) F’(1,3)

Practice

Translating a Figures To translate a geometric figure Translate each vertex (point) of the figure Connect the image points to finish The translated image should be the same dimensions as the original image

Translate ABC: right 2 units, down 2 A( )>>A’( ) B( )>>B’( ) C( )>>C’ ( ) Text page 552 1-13; Pr 10-6 Write a rule for the translation using arrow notation

Write a Rule of Translation for MNP

10-7 Symmetry and Reflections What You’ll Learn: To identify lines of symmetry To graph reflections

Identify Symmetry A line of symmetry can e drawn through the figure so that one side is a mirror image of the other 2 lines of symmetry 1 line of symmetry

Reflections A reflection is a transformation that flips a figure over a line called line of reflection When a figure is reflected the image is congruent to the original image

Graphing Reflections 3,1 Y reflection Original -1,3 1,3 -3,1 1,1 -1,1 Text page 556 #1-18; 27-30; pr 10-7 -1,-1 1,-1 3,-1 -3,-1 X reflection X – Y reflection 1,-3 -1,-3

Lets Write the Rules for Reflections Over the Y Axis (X,Y) ( X , Y) Over the X Axis Over the Y and X Axis

10-8 Rotations What you’ll Learn: To identify Rotational Symmetry To rotate a figure about a point

Rotation Rotation is a transformation that turns a figure about a fixed point called the center of rotation Rotation of symmetry is if a figure can be rotated 180 degrees or less and match the original image Rotations are made counter clockwise A` for 90 degrees A`` for 180 degrees A``` for 270 degrees

Do the following figures have rotational symmetry ?? Text page 561 # 1-6 Angle of Rotation: 360 / 5 = 72

Angle of Rotation

“Image rotation” 360 90 90 90 90 270 180

“Image rotation” 360 90 90 90 90 270 180 original Pr 10-8; re 10-1 to 10-4; 10-6 to 10-8, chapter test 270 180 original