Cartesian Plane. Analytic Geometry. Unit Name

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1 3.1cartesian Unit Name Analytic Geometry Unit Goals 1. Create table of values in order to graph &/or determine if a relation is linear. Determine slope 3. Calculate missing information for linearelationships. 4. Write and Graph linear functions Goal I can graph using a table of values 3.1 Cartesian Plane Ordered Pairs describe points on the plane uses the form P (x, y) names the point Graph the following: 1) A(7,) ) B( 4, 6) 3) C( 8, 1) 4) D(, 6) 1

2 3.1cartesian Completing T Charts 1) y = 3x + x y Table of Values T Table Gives us points to graph on the Cartesian Plane ) y = x + 5 x y 1 3) y = 3 x + 4 x y 0 3 Seatwork worksheet

3 3.slope Goal I can determine the slope using a graph and points GRADE PITCH Slope INCLINE SLOPE 3. Unit Rate Slope What is unit rate? how much something will change PER unit Change in y Change in x "something" = y value "unit" = x value unit rate = slope = = change in y change in x Δ y Δ x 1

4 3.slope Examples: Determine the slope. Worksheet

5 3.b Slope Goal I can describe "types of slopes" and I can determine slope using the points formula. 3.b Types of Slope y y x x Positive Slope Negative Slope y y x x Zero Slope rising Δy run Δx Undefined Slope rising run Δy Δx Dividing by ZERO is UNDEFINED. 1

6 3.b Slope Slope is written as a "m" Formula for slope y x Find the slope of a line that goes through the points A( 3, ) and B(,4) y x

7 3.b Slope Examples: Find slope 1. B(6,) and C( 5,4). E( 3, 3) and F( 7,9) Worksheet 3

8 3.3linearNonlinear Goal: I will be able to determine if a relationship is linear or nonlinear. (graph, equation, t-table) 3.4 Linear vs. Non-linear Relationships Graphically Equations y = 3x+ Linear y = x + Non-linear y = non-linear An equation is linear when both the x and y have an exponent of 1 when written in the numerator. Examples: Identify if linear (L) or non-linear (NL) 1

9 3.3linearNonlinear T-Charts - from a t-chart, a relationship is linear if the FIRST DIFFERENCES (the pattern in the x & y column) is the exact same. Worksheet

10 3.4 Goal: I will be able to determine the slope from a t-chart. 3.4 Slope from T-Charts If you know that a relation is linear, you can use its table of values to find the slope. Remember that m = Δy/Δx Sometimes the table of values does not go up an equal amount in the x or y columns. This does The following is a table of values for a linear relationship. Find its slope. x y not necessarily mean it is non linear. x y This shows that it doesn't matter which points you use. Just make sure you subtract in the SAME direction.. Determine the slope x y

11 x y x y x y

12 3.5y=mx+b Goal - I will graph lines that are in the form y=mx+b 3.5 Graphing Lines 1. From a t-chart A t-chart is a collection of coordinates for a graph. x y Equations of Lines There are 3-forms of equations for lines 1. y=mx + b (slope-intercept form). y=m(x-x 1 ) + y 1 (point-slope form) 3. Ax + By + C = 0 (general/standard form) y-coordinate -dependent variable y=mx + b x-coordinate -slope -independent -unit rate variable - y-intercept -point where the line crosses the y-axis 1

13 3.5y=mx+b Example: Determine the values of m and b for the following. Equation m b Graphing

14 3.5y=mx+b. y = 3x 6 3. y = x + 4. y = x 6 5. y = 6 6. x = 4 3

15 3.6 Goal I will be able to graph by determining x&y intercepts AND by rearranging equations into the form y=mx+b. 3.6 x & y Intercepts y-intercept - where the graph crosses the y-axis - the x-value is ALWAYS 0. eg (0,-) x-intercept - where the graph crosses the x-axis - the y-value is ALWAYS 0. eg (3,0) y x 1

16 3.6 Working Example: Determine the x and y intercepts for the following equations. a) 3x - 4y = 1 I cannot see the slope or the y intercept not in the form y=mx+b b) 4x + 5y 8 = 0

17 3.6 Rearranging Equations If I want to graph an equation that is not in the form y =mx+b, you can rearrange the equation. example: Rearrange the following equations into the slope-intercept form. y=mx+b P. 43 #1 8 note 3 8 thinking type 3

18 3.7 Equations 1 Goal: I will write the equation of a line given its graph or its slope and y-intercept 3.7 Determining Equations of Lines (Part 1) Section 1: IF you are given the slope and x intercept, you can substitute these values into y=mx + b to get an equation. Examples: Determine the equation of the line. a) m= b=7 b) m= 1 b= 4 Section : If you are given a graph of a line, you can determine valuable information. 1) x-intercept ) y-intercept 3) Slope (draw a triangle) 4) t-chart (collection of coordinates) -using ) and 3), we can substitute these values into y = mx + b slope y y intercept x 4 6 1

19 3.7 Equations 1 Examples: Determine the equation of each line. D A B C Practice

20 Goal -I will be able to find the equation of a line given one point and the slope. 3.8 Point-Slope Form y = m (x x 1 ) + y 1 -slope (x 1, y 1 ) -A point on the line Examples: Determine the equation for the following a) m =, A ( 3, 5) b) m =, B(16, 3) 1 4

21 c) m = 0, C (,6) If the slope is 0, then it is a horizontal line with the equation y = #. Then the point given, you take the y coordinate of the point. d) m =, D (, 7) 1 3

22 Goal: I can determine an equation of a line if given points. 3.9 Equation of a Line Given Two Points Slope A point EXTRA STEP You must find the slope first. m= Examples: Determine the equation for the following. 1) A (, 6) B (8, 1)

23 . C (5, 3) and D (, 4) 3. E ( 7, ) and F ( 7, 10)

Goal -I will write equations that are parallel or perpendicular to given information. 3.10 Parallel and Perpendicular Lines Part One - parallel the lines will never cross. http://www.

24 Goal -I will write equations that are parallel or perpendicular to given information Parallel and Perpendicular Lines Part One - parallel the lines will never cross. Similar - they all have the same slope Different - the y-intercept Conclusion If the slope is the same, then the lines will be PARALLEL eg Find an equation that is parallel to y=-3x + 6

25 Part Two Perpendicular means they will intersect but at a 90 o angle. The slopes are opposite signs and are reciprocals. ***negative reciprocals*** Example: Find an equation of a line that is perpendicular to 1. y = 5x + 3. y = x 9 3 4

26 Examples: 1. Write the equation that is parallel to y = 6x and through the point ( 3, 4). Write the equation that is perpendicular to y 6x 5 = 0 and through the point (, 3)

27 Goal I can integrate properties of linear equations to solve problems Review of What I Should Know 1. Slope: Graph Δy Δx Formula y y 1 x x 1 Equation y = mx + b T table Δy Δx. x & y Intercepts (x, 0) (0, y) 3. Slope & Point y = m( x x 1 ) + y 1 4. Parallel lines are equal and perpendicular lines have slopes that are negative reciprocal Thinking Type Questions 1. The equation of a line is kx + 3y = 0 and passes through the point A(6,-3). Determine the value of k.

28 . A line passes through the points C(10,k) and B(-1, 13). If the line has a slope of -, determine the value of k. 3. The equation of a line is 1x + ay - b = 0. If this same line has a slope of -3 and a y-intercept of 5, determine the values of a and b.

Goal: I will solve systems of equations graphically. 3.1 Systems of Equations A system of equations is where lines intersect. The solution is the POINT OF INTERSECTION.

29 Goal: I will solve systems of equations graphically. 3.1 Systems of Equations A system of equations is where lines intersect. The solution is the POINT OF INTERSECTION. an ordered pair (x,y) revenue profit cost $ loss break even point # of people Examples: Determine the solution to the system of equations. 1) y=x+1 y=-x+4 y x 4

30 ) y=-x+ y y=x-1/ x 4 y 3) x-y=-5 x + y = x 4 4) x+y=3 y 3x+3y= x 4

ANSWER KEY. Checklist Do I understand? Test Date: A-Day 2/7. Algebra 1 Benchmark Review study guide is due on test day!

ANSWER KEY. Checklist Do I understand? Test Date: A-Day 2/7. Algebra 1 Benchmark Review study guide is due on test day! ANSWER KEY Name: Algebra 1 Benchmark Review study guide is due on test day! Test Date: A-Day 2/7 Checklist Do I understand? Unit 1 Solving Equations and Inequalities Two Step Equations Properties of Real