Graph the following ordered pairs on the same coordinate plane: Warm-up 2/25/08 Graph the following ordered pairs on the same coordinate plane: A (0,1) B (2,3) C (-1,2) D (-1,-1)

Warm-up 2/9/07 7c + 2 = -37 -5.5714 -2c -8 = 12 -10 2/3c + 11 = 5 -9 Solve each equation for c. 7c + 2 = -37 -2c -8 = 12 2/3c + 11 = 5 c/-4 = 9/3 30% of c = 120 c% of 90 = 150 -5.5714 -10 -9 -12 400 166.7

30 MINUTES TO FINISH TESTS REMINDERS I WILL BE OUT TOMORROW, YOU WILL HAVE SOME REVIEW WORKSHEETS THAT WILL BE FOR A GRADE 30 MINUTES TO FINISH TESTS IF YOU DON’T FINISH TEST, YOU WILL NEED TO COME BACK DURING LUNCH OR AFTER SCHOOL WEDNESDAY

GRAPHING AND WRITING LINEAR EQUATIONS Unit 5 GRAPHING AND WRITING LINEAR EQUATIONS UNIT 5 PROJECT

Topic: Graphing and Writing Linear Equations Key Learning(s): Find slope, graph lines using a variety of methods, write equations of lines Unit Essential Question (UEQ): How do you find the slope of a line? How do you write an equation of any given line?

Concept I: Slope and Rate of Change (5.1, 5.2) Lesson Essential Question: How do you calculate the slope of a line? How do you draw a line through a point with a given slope? How do you find rates of change from tables and graphs? Vocabulary: Slope, rate of change, linear function

Slope-Intercept form (5.4) Concept II: Slope-Intercept form (5.4) Lesson Essential Question (LEQ): How do you use slope and the y-intercept to draw graphs and write equations? Vocabulary: Y-intercept Slope-intercept form

Concept III: Writing the equation of a line Lesson Essential Question (LEQ): How do you write an equation given the slope and a point? How do you write an equation given two points from a graph or a table? Vocabulary:

Concept IV: Using the x-intercept Lesson Essential Question (LEQ): How do you graph equations using the x-and y-intercepts? How do you write equations in standard form? How do you use the x-intercept of a linear equation to solve a related one-variable equation? Vocabulary: X-intercept; point-slope form; break-even point

Concept V: Parallel and Perpendicular Lines Lesson Essential Question (LEQ): How do you write equations for parallel and perpendicular lines? How do you use slope to determine if two lines are parallel, perpendicular, or neither? Vocabulary: Parallel lines, perpendicular lines

Concept VI: Direct & Inverse Variation (5.3) Lesson Essential Question (LEQ): How do you relate slope to constant of variation? How do you solve inverse variations? How does direct variation compare to inverse variation? Vocabulary: Linear equation; constant of variation; direct variation; inverse variation

Concept VII: Scatter Plots and Equations of Lines Lesson Essential Question (LEQ): How do you find the equation of a trend line? How can you use a calculator to find the line of best fit? Vocabulary: Line of best fit, correlation coefficient

§5.1: Slope EQ: How do you calculate the slope of a given line? What are some things that have slope? The slope of a line is the measure of its steepness or “tilt”

6 line2 5 line4 4 line3 3 2 1 line1 -6 -5 -4 -3 -2 -1 Imagine each of these lines is a hill. Which one has the least slope? (line 1) This line has a slope of zero. (it’s flat) Line 2 is like the edge of a cliff. It has an undefined slope. X = 0 is undefined. Line 3 has a steep slope, but not too bad. This line is y = x. All the y and x values are the same. Which lines do you think have positive slopes? Negative? Think about a car, driving left to right on the page. If it goes up hill, the line has a positive slope. If it goes down hill, the line has a negative slope.

The slope of a line is given as a ratio of the rise to the run. To find the slope of a line, you need to find two points on the line. Imagine yourself walking from one point to the other. (left to right). You cannot count diagonally, you must go up or down, then across. Then, write the slope as a ratio (fraction).

6 5 4 3 A 2 1 -6 -5 -4 -3 -2 -1 D Slope of line AD is 6/5. It is positive.

6 5 4 A 3 2 B 1 -6 -5 -4 -3 -2 -1 When finding the slope of a line, look for two points where the line crosses two lines (to make it easy). Draw two points, then find the slope between those two points. THE SLOPE OF LINE AB IS -2/2 = -1. THE SLOPE OF THIS LINE IS NEGATIVE (-1). THIS MAKES SENSE BECAUSE IT GOES DOWNHILL.

A surveyor places two stakes, A and B, on the side of a hill A surveyor places two stakes, A and B, on the side of a hill. Stake A is 20 feet lower than stake B. If the horizontal distance between the two stakes is 200 feet, what is the slope of the hill? Draw a diagram. Slope = 20/200 = 1/10 or 0.1

Slope Formula There is a formula to find the slope of two points without having to graph a line. If you have two points, A and B, we will name the coordinates for A(x1,y1) and the coordinates for B(x2,y2) to distinguish between the two points. The formula: slope of AB = y2 – y1 x2 – x1

This slope formula is just finding the difference in the y-coordinates rise the difference in the x-coordinates run Ex. Find the slope between point A(-2,-3) and B(1,4). *Make sure you subtract them in the same order! Formula: -3 – 4 = -7 = 7 -2 – 1 -3 3

Given a point and slope Graph the point Count the slope from that point Find your 2nd point Draw the line Ex) Draw a line through the point (1,2) with slope -1/2.

Find slope and equation To find the equation of a line given two points Find the slope Determine what the line is Y = vs. x = Ex) Find the slope & equation between (1,2) and (4,2). Ex) Find the slope & equation between (4,-1) and (4,2)

Section 5.1 p.217 – 218 #1 – 15 odd 18 – 21 all 25 – 33 odd Assignment Section 5.1 p.217 – 218 #1 – 15 odd 18 – 21 all 25 – 33 odd

Warm-up 2/27/08 In your textbooks, read over p. 219 on changing between units (dimensional analysis) and do # 1 – 12.

5.2: Rates of change How do you find rates of change from tables and graphs? Rate of change (slope) How do you find slope? How do you find the rate of change? Linear function

5.4: Slope-Intercept Form LEQ: How do you use slope and the y-intercept to draw graphs and write equations? Slope intercept form Y = mx + b “m” is the slope “b” is the y-intercept (where it crosses the “y-axis”)

How to graph them Put the “y-intercept” point on the y-axis From that point… Count the slope to find another point (or two) Draw the line connecting the points.

Changing an equation to slope intercept form: In order to use an equation in slope-intercept form, you must ensure that it is first IN slope intercept form. If it is not: Solve the equation so that “y” is by itself on one side Everything else on the other side can be combined/rearranged to mx + b form.

Steps: Re-write the equation in “function” form Identify the y-intercept Graph the y-intercept Identify the slope Use slope to plot more points Draw the line through the points

Section 5.2 p. 222 – 223 # 1 – 21 odd Section 5.4 p. 233 #1 – 23 odd Assignment Section 5.2 p. 222 – 223 # 1 – 21 odd Section 5.4 p. 233 #1 – 23 odd

Warm-up 2/28/08 Find the slope of the line passing through (-2,-3) and (3, -1/2). Graph a line with a slope of -1/2 which passes through the point (2,1).

Return Papers Go over 5.1, 5.2, 5.4 Questions?

5.1 Quiz

§5.5: Writing Equations of Lines LEQ: Given various points, how can you find the equation of a line? Your friend called you one night with a question about her algebra homework. You decide that the best way to help her is to have her draw the line of an equation on a graph. How would you describe the line without actually showing her?

Graphic Organizer Worksheet 5.5 Practice! Graphic Organizer Worksheet 5.5

Summary On a piece of scratch paper: Write an equation of a line through point (-3, -1) with a slope of -2. Write an equation for the line that passes through point (-3,2) and (3,1) Decide if the following relationship between the x- and y-values is linear. If it is, write an equation for the relationship.

Homework Section 5.5 p. 239 # 1 – 39 odd

Warm-up 2/29/08 Write the equation of the line.

Write the equation of the line:

Write the equation of the line:

Check HW (answers) Finish 5.5 WS 5.5 Summary

Checkpoint 1 Unit 5 project Due Date: Tuesday, March 11

How do you put a table into a calculator and graph it? p. 240 in book

5.6: Scatter plots and equations of lines LEQ: Given various points, how do you find the equation of a line? Scatter plots Correlation Positive Negative None Correlation Coefficient

p. 243 – 244 #1 – 6, 8 – 10, 12 Do #12 by hand Assignment p. 244 #11 Practice p. 243 – 244 #1 – 6, 8 – 10, 12 Do #12 by hand Assignment p. 244 #11

Warm-up 3/3/08 Write the equation of the line with given slope and y-intercept: M = ¾ ; b = 8 M = -7; b = ½ Write the equation of the line through the given points (4,3) (-2,1) (5,-4) (0,2)

Quiz?

§5-7: Standard Form 5.9: Using x-intercepts LEQ: What requirements does the standard form of an equation have? In the slope intercept of an equation: Y is the M is the B is the

Standard form Standard form of an equation is in the form: Ax + By = C Basically the x and y-terms are on one side of the equation and the number term is on the other sides of the equation. It is called standard form because it is most often used.

Using standard form to graph equations Standard form can be used to easily graph equations using the x- and y-intercepts Do not try to use the method of x- and y-intercepts unless the equation is in standard form! “cover up” method reasoning

Find the x- and y-intercepts then graph each equation 5x + 2y = 10 6.5x – 4y = 52 -x + 3y = -15 8x – y = 104 2; 5 8; -13 15; -5 13; -104

Break-even point When starting a business, people want to know their break even point The break even point is the point at which their income equals their expenses Suppose you invested $140 to start a business selling T-shirts. You sell each shirt for $7.50. Find the break even point. IE) When does 7.50t = 140? How many t-shirts must you sell?

Guided Practice In Textbooks p. 248 – 249 #12, 14, 24 – 30 Even, 32

Assignment Section 5.7 p. 248 – 249 # 1 – 11 odd #15 – 31 odd On 15 – 21, only find the intercepts

Warm-up 3/4/08 Change each equation to Ax + By = C form. Y = 5x – 1 Write the equation of the line thru the points. (1, -4),(7,0) (1/2, -4),(-11, -4)

Heads up Throughout the next month, there will be various visitors in the room Behavior Questions 3rd block mid-terms will be next Tuesday, so START STUDYING NOW! No late work will be accepted more than 2 weeks after the original due date unless there are special circumstances.

Any 5.7 Homework Questions?

How do you know if a given point is on a line? Ex. Does the point (5,2) lie on the line given by the equation x + y = 7? To determine if the point is on the line, put the x-value from the ordered pair in for x and the y-value into for y and solve. If the equation is true, the point is on the line. x + y = 7 (5) + (2) = 7? Yes. So, this point does lie on the line given.

Assignment Section 5.9 p. 258 – 259 #4 – 6, 11 – 21

§5.8: Parallel & Perpendicular Lines LEQ: What is true about all parallel and perpendicular lines? What does it mean to be parallel? What does it mean to be perpendicular?

Parallel or Perpendicular? 2x + y = 15 3y = -6x + 30 How do you know?

Parallel or Perpendicular? Y = 4x + 3/4 Y = -1/4x + 4 How do you know?

Generalization Parallel lines Perpendicular lines Have the exact same slope, but a different y-intercept Perpendicular lines Have opposite, reciprocal slopes Y-intercept can be same or different

Find the slope of a line parallel to the equation Y = -2/3x – 1 2x – 3y = -2 X = 5 X = 0 7x + 2y = 12 -2/3 2/3 Undef. -7/2

Y = -3x Y = x Y = -x/5 – 7 X = 3 3x + 5y = 7 Y = 0.25x 1/3 -1 5 5/3 -4 Find the slope of the line perpendicular to the graph of each equation. Y = -3x Y = x Y = -x/5 – 7 X = 3 3x + 5y = 7 Y = 0.25x 1/3 -1 5 5/3 -4

Practice Guided Practice p. 254 # 42 – 46 even Assignment p. 253 – 254 #1 – 30 (mult of 3), 37 41 – 49 odd

Warm-up 3/5/08 For each equation, write the equation of a line that is perpendicular and one that is parallel to the given line. Y = 3x – 2 4x + 5y = 10 Y = 2/3x + 1/6 Y = 1.6x - 0.4

Last quick section:

§5.3: Direct Variation EQ: What is consistent in direct variation equations? How are they related to linear functions? If a relationship can be expressed in the form y = kx, the two variables are said to vary directly, or be proportional. (They increase or decrease together)

k is the constant of variation -rate of change (y’s divided by x’s) y = kx k is the constant of variation -rate of change (y’s divided by x’s) -slope of the graphed line *NOTE: the y-intercept is 0. Direct variation – A linear function that can be written in the form y = kx where k≠0. Complete p.64 #5a, b, c Complete #65 #6

Relationships among linear equations 2y = x – 1 x + y = 7 3y = 5 Linear Equations 3x + 2y = 6 X = 3 Linear Equations in Function form Y = -5x - 2 Is a direct variation always a linear function? Under what circumstances, if any, would a linear function be a direct variation? Direct Variation f(x) = -3x

Identify the Slope

Identify the Slope

Identify the Slope

Identify the Slope

Identify the Slope

Write the equation of a direct variation that passes through the point

Write the equation of a direct variation that passes through the point

Write the equation of a direct variation that passes through the point

Write the equation of a direct variation that passes through the point

Write the equation of a direct variation that passes through the point

Multiply to find the missing value Include units in the answer

Multiply to find the missing value Include units in the answer

Multiply to find the missing value Include units in the answer

IN CLASS Test Review

Homework Wrap up p. 261 – 263 # 1 - 34

WARM-UP 3/6/08 Provide a step-by step procedure of how each method works. Slope and y-intercept Slope and a point Two points Using x and y-intercept Ax + By = C form What is a real world situation that can be modeled with a linear equation.

Reminders Test Review Warm-ups I’ll be out tomorrow You will have practice mid-term to work on Take it seriously I will grade them over the weekend and try to give you feedback on Monday.

Warm-up 3/10/08 A line goes through (1,3) and has slope 3/2. Graph the line. What is true about a line that rises as it goes from left to right? What would a direct variation with a constant of variation of 2/3 look like? What are the x- and y-intercepts of the line 8x + 4y = 4.

Reminder: Mid-term TOMORROW!!! Agenda Reminder: Mid-term TOMORROW!!! Pass back unit 5 tests Go over unit 5 tests Re-takes Study for mid-term!