Representing Equations Graphing in the Coordinate Plane

What You’ll learn To graph points on a plane 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); B (-3,-2); C (-2, 4) Text page 524 1-12; Pr 10-1

Ordered Pair (3, 5) Domain of a relation is the set of first coordinates (or x values) Range is the set of second coordinates (y values)

Relations and Functions Relation: a set of ordered pairs Function: a special type of relation that pairs each domain with exactly on range

Representations of Relations Table Graph Mapping Diagram

Identifying Functions State the domain and range, then whether the relation is a function Alg. book p. 173 # 1-14

Linear Equations A function is linear if it can be written in standard form Ax + By = C When an equation is written in standard form: x and y both have an exponent of 1 x and yare not multiplied together and y do not appear in denominators, exponents, or radicals

Linear or Nonlinear? y = x + 3 3xy + x = 1 y – 2 = 3x x² + y = - 1

Graphing Linear Equations using a table 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)

Graphing Linear Equations Step 1 Make table Step 2 Insert x values Step 3 Find y values Step 4 Graph the ordered pairs and draw a line through the points X Equation = y y (X, Y)

Graph: y = x + 1 x X + 1 Y (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

Graph points from table

Practice Graph y – 2 = ¼ x Graph y = 5/6x + 3 Graph 2x + 3y = -6 Alg I bk p. 235 #’s 30-49

Graphing Linear Equations using Intercepts What You’ll Learn: To find X and Y Intercept To graph linear equations

EX. 1 Graphing using the Intercepts Graph 2x + 3y = 6 using the x and y intercepts. Step 1: Find the intercepts Find the y intercept letting x = 0 2x + 3y = 6 2(0) + 3y = 6 3y = 6 y = 2

Graphing using the Intercepts Graph 2x + 3y = 6 using the x and y intercepts. Step 1: Find the intercepts Find the x intercept letting y = 0 2x + 3y = 6 2x+ 3(0) = 6 2x + 0 = 6 x = 3

Step 2 Graph the equation Plot points (0,2) & (3,0)

Practice On graph paper, graph the following using x and y intercepts: 4x + 2y = 8 3y + x = 9 3x – 4y = 24 Alg I bk p. 240 #’s 1-21

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

Finding Slope Negative Slope Positive slope

Slopes of Horizontal and Vertical Lines m = 0 m = undefined

Which Slope is Steeper?

Which Slope is Steeper?

Find The Slopes Slope Formula (m) m = y2 – y1 m = (-3) – 3 1 – (-2) x2 - x1 m = (-3) – 3 1 – (-2) m = - 6 = -2 or -2 3 1 -2,3 1,-3

Find the Slope of the following Lines that contain the following Points (-3, 7) ; (3, 3) (0, -2) ; (4,0) (-5,-3) ; (-2,-3) Verify you answers by graphing each pair of coordinates Alg 1 bk Text. P. 248 4-11; p257 #1-10

Review: For y = -1/3x + 3 ; graph and find the slope of the line Find 3 solutions to graph Use X/Y chart Graph order pairs and connect points Find slope m = y2 – y1 x2 – x1 Verify by counting on graph

Using the y - Intercept Remember Slope is the ratio of vertical change and horizontal change (Rise/Run) The y - intercept is the y-coordinate of the point where a line crosses the y-axis

Below is the graph of y = -2/3 x +6 Find the y intercept of the line Find the Slope of the line What do you notice about the Equation?

Slope Intercept Form An equation written in the form y = mx + b is in slope intercept form. The graph is a line with slope “m” and y- intercept “b”

Identify Slope (m) and y-intercept (b) Find the slope of each equation: Find the y-intercept of each equation: y = 6x y = -x – 4 y = 2x – 1 y = - x y = - 4x + 2 5 y = 6x y = -x – 4 y = 2x – 1 y = - x y = - 4x + 2 5

Identify Slope (m) and y-intercept (b) Find the slope of each equation Find the y intercept of each equations y + ½ x = -6 -1/4x = y + 3 y – x = -1 y + ½ x = -6 -1/4x = y + 3 y – x = -1 Text p. 139 # 1-6; 25-27

Using slope intercept to graph a line Step 1: Find the slope and y-intercept Step 2: Graph the y-intercept Step 3: use slope to graph next 2 points Step 4: Draw line through the points

Graph y =1/3x + 4

Graph the following Equations y = ½ x – 3 y + 2 = 3/4x Text p. 140 #7-12 odd; 21-23

Writing an Equation for a Line Write and equation for a line with a slope of ½ and y - intercept of -6 Remember you formula y = mx + b Insert your given/found values y = ½ x - 6

Writing Equations for a Line a. m = 4, b = -2 b. m = - 3/2 , b = 5 c. m = ¼ , b = 0

Write equation from graph Steps: y = mx + b Step 1: Find the y - intercept Step 2: Find the slope Count or use formula Step 3: Write equation in slope intercept form m = 3/1 b = -1 Alg. I bk p. 272 #’s 1-12;14-22

3-4 Key Concepts Write an equation Graph y = -2x + 4

Practice

Point Slope Form Objective: To graph a line and write a linear equation using point slop form

What you should know Find the slope of a line containing points (0,2) and (3,4) Write the following in slope-intercept form, find m and b y – 5 = 3(x +2)

Point -Slope Form of a Linear Equations y – y1 = m ( x – x1) Point ( x, y) Slope: Rise/Run

Example 1: Writing an Equation in Point - Slope Form Remember: y – y1 = m ( x – x1) m = 5/2 ; (-3,0) m = - 7 ; (4,2) m = 0 ; (-2,-3)

Example 2a: Using Point-Slope Form to Graph Graph: y – 1 = 3 (x – 1)

Example 2b: Using Point-Slope Form to Graph Graph: y + 2 = - 1/2(x – 3) Rewrite to match the formula

Homework Text page 279 #’s 1-6

Example 3a: Writing an Equation in Slope Intercept Form Remember: y – y1 = m ( x – x1 ) y = m ( x – b) Containing: m = -4 ; (-1, - 2) Step 1: Write equation in point slope form Step 2: Transform equation in slope intercept form (get y by itself)

Example 3b: Writing an Equation in Slope Intercept Form Containing: point (1, -4) and (3, 2) Step 1: Find the slope Step 2: Write in point slope Step 3: Transform into slope intercept Do with both points

Example 3c: Writing an Equation in Slope Intercept Form Remember: y – y1 = m ( x – x1 ) y = m ( x – b) Containing: X intercept = -2 ; Y Intercept = 4 Step 1: Write ordered pairs of intercepts Step 2: Find Slope Step 3: Write in point slope form Step 4: Transform equation in slope intercept form

Example 4: Using Two Points to find Intercepts Containing: (4, 8) ; (- 1 , - 12) Step 1: Find Slope Step 2: Write in point slope form Step 3: Transform into slope intercept form Find X & Y Intercepts

Homework Text page 279 #’s 7-15

Parallel / Perpendicular Parallel Lines – Two lines that do not intersect Perpendicular – Two lines the do intersect to form right angles

Theorem Two lines are parallel if their slopes are equal Two lines are perpendicular if the product of their slopes are equal -1

3x – y = 4 y = 3x + 1 What are these two lines? Parallel/Perpendicular Put each in y int. form 3x – y = 4 >> y = 3x – 4 y = 3x +1 Identify slopes 3/1 & 3/1 Identify relationship Lines are Parallel because the slopes are the same

Identify which lines are Parallel and Perpendicular Ln 1: y = 2x – 6 Ln 2: 4x – 2y = 1 Ln 3: x – y + 2 = 0 Ln 4: 2y = 10 – x Page 370 1-15

Writing Equations to Parallel Lines Write and equation in slope intercept form for the line that passes through (4,5) and is parallel to y = 5x + 10. Step 1: Find the slope of the line Step 2: Write the equation in point slope form Step 3: Transform the equation to slope intercept form

Write and equation in slope intercept form for the line Passes through point (4, 10) and is parallel y = 3x + 8 Passes through point (2, -1) and is parallel y = ½ x

Writing Equations to Perpendicular Lines Write and equation in slope intercept form for the line that passes through (4,5) and is Perpendicular to y = 5x + 10. Step 1: Find the slope of the line (opposite reciprocal) Step 2: Write the equation in point slope form Step 3: Transform the equation to slope intercept form

Write and equation in slope intercept form for the line Passes through point (4, 10) and is perpendicular to y = 3x + 8 Passes through point (2, -1) and is perpendicular to y = ½ x

Homework Text p. 297-298 #’s 22-26; 34,35,37,38

Solving Linear Systems by Graphing Two or more linear equations form a system of linear equations. A solution of the system is any ordered pair that is a solutions of each equation Review problems from p. 325 on board as warm up activity

Solving Systems by Graphing Ex. 1) Tell whether the ordered pair is a solution of a given system (4,1) ; x + 2y = 6 & x – y = 3 substitute x & y values in each equation (-1,2) ; 2x + 5y = 8 & 3x – 2y = 5

Finding a Solution to Systems of Linear Equations (4, 2) y = ½ x y = 2x - 6

Ex.1)Solve the system of equations y= x - 3 and y = - x – 1

Ex 2).Solve the system of equations x + y= 0 and y = ½ x + 1

Homework Page 332 #’s 1-7 Review #’s 9-15

Solving Systems by Substitution What You will Learn: To solve systems on equations with 2 variables by substitution

Solving Systems by Substitution Ex.1a solve for each variable by using substitution y = 2x y = x + 5 Step 1: pick a variable to isolate (get by itself) Step 2: Substitute the equation of isolated variable in 2nd equation Step 3: Solve for the variable Step 4: Substitute the value of the variable to solve for the other variable in the original equation Step 5: Write each variable as ordered pair (x, y)

Solving Systems by Substitution Ex.1b solve for each variable by using substitution 2x + y = 5 y = x - 4 Step 1: pick a variable to isolate (get by itself) Step 2: Substitute the equation of isolated variable in 2nd equation Step 3: Solve for the variable Step 4: Substitute the value of the variable to solve for the other variable in the original equation Step 5: Write each variable as ordered pair (x, y)

Solving Systems by Substitution Ex.1c solve for each variable by using substitution x + 4y = 6 x + y = 3 Step 1: pick a variable to isolate (get by itself) Step 2: Substitute the equation of isolated variable in 2nd equation Step 3: Solve for the variable Step 4: Substitute the value of the variable to solve for the other variable in the original equation Step 5: Write each variable as ordered pair (x, y)

Homework Text page 340 #’s 1-3 Review

Solving Systems by Substitution Ex.2 solve for each variable by using substitution and distributive property 4y – 5x = 9 x - 4y = 11 Step 1: pick a variable to isolate (get by itself) Step 2: Substitute the equation of isolated variable in 2nd equation Step 3: Solve for the variable using distributive property Step 4: Substitute the value of the variable to solve for the other variable in the original equation Step 5: Write each variable as ordered pair (x, y)

Homework Text page 340 #’s 4-6 Review #’s 9, 10 , 13, 15, 19, 22

Solving Systems by Elimination Ex.1 Elimination using addition x – 2y = - 19 5x + 2y = 1 Step 1: Align like term, pick a variable to eliminate Step 2: add equations to eliminate variable Step 3: Solve for the variable Step 4: Substitute the value of the variable to solve for the other variable in the original equation Step 5: Write each variable as ordered pair (x, y)

Solving Systems by Elimination Ex.2 Elimination using subtraction 3x + 4y = 18 -2x + 4y = 8 Step 1: Align like term, pick a variable to eliminate Step 2: subtract equations to eliminate variable(add opposites of entire equation) Step 3: Solve for the variable Step 4: Substitute the value of the variable to solve for the other variable in the original equation Step 5: Write each variable as ordered pair (x, y)

Solving Systems by Elimination Ex.3 Elimination using multiplication (common multiple) 2x + y = 3 -x + 3y = -12 Step 1: Align like term, pick a variable to eliminate Step 2: Multiply equation to make common multiple to eliminate Step 3: Solve for the variable Step 4: Substitute the value of the variable to solve for the other variable in the original equation Step 5: Write each variable as ordered pair (x, y)

Homework Text page 347 #’s 1-9 Review #’s 11-19

Solving Special Systems Consistent System A system with at least one solution or when two lines intersect at least one point Independent system Has exactly 1 solution Dependent system Has infinite solutions Inconsistent System When 2 lines do not intersect, therefore there is no ordered pair to satisfy both equations (parallel lines)

Ex.1 Systems with no solutions Show that the systems have no solution y = x – 1 -x + y = 2 Step 1: Write the equations to line up the like terms Step 2: Add terms Step 3: identify relationship

Ex.2 Systems with infinite solutions Show that the systems have no solution y = 2x +1 2x – y = -1 Step 1: Write the equations to line up the like terms Step 2: Add terms Step 3: identify relationship

Homework Text page 335 #’s 8-10 Review Text page 336 #’s 20-22

Practice