DRILL Given each table write an equation to find “y” in terms of x. 2) Find the value of x: X 1 2 3 4 Y 7 11 15 19

Chapter 5 Analyzing Linear Equations 5-1 Patterns and Slope Connection

Linear Patterns In order for a pattern to be linear the common difference in “y” divided by the common difference in “x” must be the same for all given values.

DRILL X 1 3 6 10 Y 5 13 25 41 Is this pattern linear? Why/Why Not Solve for x: X 1 3 6 10 Y 5 13 25 41

Slope Slope is defined in numerous ways some of which are: 1) 2) Change in “y” Change in “x” 3)

Slope Formula The formula for finding slope is: Where the coordinates of two points are (x1, y1) and (x2, y2)

Types of Slopes Undefined _ +

DRILL (2, 5) and (4, 13) (-3, 3) and (7, -2) (-4, 0) and (4, 24) Find the Slope Given the following points: (2, 5) and (4, 13) (-3, 3) and (7, -2) (-4, 0) and (4, 24) (5, 7) and (5, 13)

7.4 Slope Objectives: To find the slope of a line given two points on the line To describe slope for horizontal and vertical lines

Slope -4 -2 2 4 6 8 10 -6 -8 -10 Slope = run rise

Example 1 Graph the line containing points (2,1) and (7,6) and find the slope. -4 -2 2 4 6 8 10 -6 -8 -10 Slope = run rise

Practice Graph the line containing these points and find their slopes. 1) (-2,3) (3,5) 2) (0,-3) (-3,2)

Slope Slope = or

Example 2 Find the slope of the line containing points (1,6) and (5,4).

Practice Find the slope of the lines containing these points. 1) (2,2) (8,9) 2) (-2,3) (2,1) 3) (5,-11) (-9,4)

Slope of a Horizontal Line What about the slope of a horizontal line? -4 -2 2 4 6 8 10 -6 -8 -10 What is the rise? What is the run? 6 Slope = The slope of ANY horizontal line is 0.

Slope of a Vertical Line What about the slope of a vertical line? -4 -2 2 4 6 8 10 -6 -8 -10 What is the rise? 5 What is the run? Slope = The slope of ANY vertical line is undefined.

Example 1 Find the slope of the line y = -4. Slope = -4 -2 2 4 6 8 10 -6 -8 -10 Slope =

Example 2 Find the slope of the line x = 7. Slope = -4 -2 2 4 6 8 10 -6 -8 -10 Slope = The line has no slope.

Practice Find the slopes, if they exist, of the lines containing these points. 1) (9,7) (3,7) 2) (4,-6) (4,0) 3) (2,4) (-1,5)

5 minutes Warm-Up Graph. Then find the slope. y = 3x + 2 2) y = -2x +5

7.5.1 Equations and Slope Objectives: To find the slope and y-intercept of a line from an equation

Slope-Intercept Equation Find the slope of the line y = 2x - 4. 8 y = 2x - 4 6 4 x y 2 -4 -8 -6 -4 -2 2 4 6 8 1 -2 -2 rise: -4 2 -4 run: -2 -6 -8 The y-intercept is -4.

Slope-Intercept Equation y = mx + b y = 4x + 8 slope y = 2x - 3 y-intercept slope = 4 y-intercept = 8 slope = 2 y-intercept = -3

Example 1 Find the slope and y-intercept of the line y = -4x + 4. The slope is -4. The y-intercept is 4.

Example 2 Find the slope and y-intercept of the line y = 5x - 7. The slope is 5. The y-intercept is -7.

Practice Find the slope and y-intercept of each line. y = x + 3

Example 3 Find the slope and y-intercept of the line 3x + 4y = 12. solve for y -3x -3x 4y = 12 – 3x 4 4 slope y-intercept = 3

Practice Find the slope and y-intercept of each line. y = -x - 3 2) 8x + 2y = 10 3) 3y – 6x = 12

Homework p.326 #1-10,19-27 odds

Warm-Up 4 minutes Find the slope and y-intercept. y = 3x + 4 3) 2x + 7y = 9 4) 4x = 9y + 7

Objectives: To graph lines using the slope-intercept equation 7.5.2 Equations and Slope Objectives: To graph lines using the slope-intercept equation

Example 1 Graph y = 3x + 2. What is the slope of this line? 8 What is the slope of this line? 6 4 2 -8 -6 -4 -2 2 4 6 8 -2 What is the y-intercept? -4 b = 2 -6 -8

Example 2 Graph What is the slope of this line? 8 What is the slope of this line? 6 4 2 -8 -6 -4 -2 2 4 6 8 -2 What is the y-intercept? -4 b = -3 -6 -8

Practice Graph each line. y = 3x - 5 2) y = -2x + 4 3)

Example 3 Graph 3x + 4y = 12. 3x + 4y = 12 solve for y -3x -3x

Example 3 Graph 3x + 4y = 12. slope y-intercept 8 6 4 2 -8 -6 -4 -2 2

Practice Graph each line. 7x + 2y = 4 2) 5y – 10 = 4x

Homework p.326 #29,31,37,39 *Use graph paper for the graphs

6 minutes Warm-Up 1. Find the slope of the line containing the points (-2,5) and (4,6). 2. Find the slope of the line y = x – 9. 3. Find the slope of the line 3y – 4x = 9.

7.6.1 Finding an Equation of a Line Objectives: To write an equation of a line using the slope-intercept equation

The Slope-Intercept Equation y = mx + b slope y-intercept Create an equation of a line with a slope of -3 and a y-intercept of 4. y = -3x + 4 y = 4 – 3x 3x = 4 - y -4 = -y – 3x

Example 1 y = mx + b 4 = 3 (-2) + b 4 = -6 + b +6 +6 10 = b Write an equation for the line with slope 3 that contains the point (-2,4) y = mx + b solve for b 4 = 3 (-2) + b substitute simplify 4 = -6 + b +6 +6 10 = b y = 3x + 10

Practice Write an equation for the given line that contains the given point and has the given slope. 1) (5,10); m = 4 2) (-3,8); m = 2

Example 2 Write an equation for the line containing the points (1,5) and (2,8).

Example 2 y = mx + b 5 = 3 (1) + b 5 = 3 + b -3 -3 2 = b y = 3x + 2 Write an equation for the line containing the points (1,5) and (2,8). y = mx + b substitute 5 = 3 (1) + b simplify 5 = 3 + b -3 -3 2 = b y = 3x + 2

Practice Write an equation for the line that contains the given points. 1) (-4,1) (-1,4) 2) (-3,5) (-1,-3)

Homework p.331 #3,5,7,15,19

6 minutes Warm-Up 1. Write an equation for the line with slope -2 and containing the point (-3,0). 2. Write an equation for the line containing the points (0,0) and (4,2).

7.6.2 Finding an Equation of a Line Objectives: To write an equation of a line using the point-slope equation

The Point-Slope Equation y – y1 = m(x – x1) Create an equation of a line with a slope of -3 that contains the point (7,2). y – 2 = -3(x – 7) y – 2 = -3x + 21 +2 +2 y = -3x + 23

Example 1 Write an equation for the line with slope 7 that contains the point (3,4). y – y1 = m(x – x1) y – 4 = 7(x – 3) y – 4 = 7x - 21 +4 +4 y = 7x - 17

Practice Write an equation for the line with the given point and slope. 1) (-3,0), m = -3 2) (4,3), m = ¾

Example 2 Write an equation for the line containing (5,7) and (2,1). First, find the slope: y – y1 = m(x – x1) y – 7 = 2(x – 5) y – 7 = 2x - 10 +7 +7 y = 2x - 3

Practice Write an equation for a line containing the following points. 1) (12,16) (1,5) 2) (-3,5) (-1,-3)

Example 3 Write an equation for the line shown below. First, find any two points on the line. (-3,-3) and (1,-1) 4 2 -4 -2 2 4 y – y1 = m(x – x1) -2 -4 y + 3 = ½(x + 3) y + 3 = ½x + 1½ -3 -3 y = ½x – 1½

Homework p.331 #9,13,21,23,25 Quiz Tomorrow

Warm-Up 5 minutes 1. Graph the line y = 3x + 4. 3. What is the slope of the lines in the equations above?

7.8.1 Parallel and Perpendicular Lines Objectives: To determine whether the graphs of two equations are parallel

Parallel Lines Parallel lines are lines in the same plane that never intersect. Parallel lines have the same slope. -8 -6 -4 -2 2 4 6 8

Example 1 Determine whether these lines are parallel. y = 4x -6 and The slope of both lines is 4. So, the lines are parallel.

Example 2 Determine whether these lines are parallel. y – 2 = 5x + 4 and -15x + 3y = 9 +2 +2 +15x +15x y = 5x + 6 3y = 9 + 15x 3 3 y = 3 + 5x y = 5x + 3 The lines have the same slope. So they are parallel.

Example 3 Determine whether these lines are parallel. y = -4x + 2 and +2y + 2y 2y - 5 = 8x +5 +5 2y = 8x + 5 2 2 Since these lines have different slopes, they are not parallel.

Practice Determine whether the graphs are parallel lines. 1) y = -5x – 8 and y = 5x + 2 2) 3x – y = -5 and 5y – 15x = 10 3) 4y = -12x + 16 and y = 3x + 4

Example 4 Write the slope-intercept form of the equation of the line passing through the point (1, –6) and parallel to the line y = -5x + 3. slope of new line = -5 y – y1 = m(x – x1) y – (-6) = -5(x – 1) y + 6 = -5x + 5 y = -5x - 1

Practice Write the slope-intercept form of the equation of the line passing through the point (0,2) and parallel to the line 3y – x = 0.

Homework p.340 #3-11 odds

4 minutes Warm-Up Determine whether the graphs of the equations are parallel lines. 3x – 4 = y and y – 3x = 8 2) y = -4x + 2 and -5 = -2y + 8x

7.8.2 Parallel and Perpendicular Lines Objectives: To determine whether the graphs of two equations are perpendicular

Perpendicular Lines Perpendicular lines are lines that intersect to form a 900 angle. -8 -6 -4 -2 2 4 6 8 The product of the slopes of perpendicular lines is -1.

Example 1 Determine whether these lines are perpendicular. and y = -3x - 2 m = -3 Since the product of the slopes is -1, the lines are perpendicular.

Example 2 Determine whether these lines are perpendicular. y = 5x + 7 and y = -5x - 2 m = -5 Since the product of the slopes is not -1, the lines are not perpendicular.

Practice Determine whether these lines are perpendicular. 1) 2y – x = 2 and y = -2x + 4 2) 4y = 3x + 12 and -3x + 4y – 2 = 0

Example 3 Write an equation for the line containing (-3,-5) and perpendicular to the line y = 2x + 1. First, we need the slope of the line y = 2x + 1. m = 2 Second, we need to find out the slope of the line that is perpendicular to y = 2x + 1. Lastly, we use the point-slope formula to find our equation.

Practice Write an equation for the line containing the given point and perpendicular to the given line. 1) (0,0); y = 2x + 4 2) (-1,-3); x + 2y = 8

Homework p.340 #13,15,21,27,29,31,33

DRILL What is the equation of a line containing the two points (2, 4) and (5, 16)? What is the slope and y-intercept of the line y = -3x – 8 ? 3) Write the equation of a line given the slope is ½ and the y-intercept is -5.

The Best-Fit Line Linear Regression PGCC CHM 103 Sinex

Work with your group to make a prediction for the height at: Age (months) Height (inches) 18 76.1 19 77 20 78.1 21 22 78.8 23 79.7 24 79.9 25 81.1 26 81.2 27 82.8 28 29 83.5 Work with your group to make a prediction for the height at: • 21 months • 28 months • 20 years

Line of Best Fit Definition - A Line of Best is a straight line on a Scatterplot that comes closest to all of the dots on the graph. A Line of Best Fit does not touch all of the dots. A Line of Best Fit is useful because it allows us to: Understand the type and strength of the relationship between two sets of data Predict missing Y values for given X values, or missing X values for given Y values

Equation For Line of Best Fit y = 0.6618x + 64.399 X (months) Formula Y (inches) 21 0.6618(21) + 64.399 28 0.6618(28) + 64.399 240 0.6618(240) + 64.399 78.3 82.9 223.3

Predicting Data with Scatterplots Interpretation - Making a prediction for an unknown Y value based on a given X value within a range of known data Extrapolation - Making a prediction for an unknown Y value based on a given X value outside of a range of known data More accurate: Interpretation Less accurate: Extrapolation

How do you determine the best-fit line through data points? Fortunately technology, such as the graphing calculator and Excel, can do a better job than your eye and a ruler! y-variable x-variable

y = mx + b y = mx The Equation of a Straight Line where m is the slope or Dy/Dx and b is the y-intercept In some physical settings, b = 0 so the equation simplifies to: y = mx

Linear regression minimizes the sum of the squared deviations y = mx + b y-variable deviation = residual = ydata point – yequation x-variable

Linear Regression Minimizes the sum of the square of the deviations for all the points and the best-fit line Judge the goodness of fit with r2 r2 x100 tells you the percent of the variation of the y-variable that is explained by the variation of the x-variable (a perfect fit has r2 = 1)

Goodness of Fit: Using r2 r2 is low r2 is high y-variable How about the value of r2? x-variable

Strong direct relationship 99.1% of the y-variation is due to the variation of the x-variable

Noisy indirect relationship Only 82% of the y-variation is due to the variation of the x-variable - what is the other 18% caused by?

When there is no trend! No relationship!

In Excel When the chart is active, go to chart, and select Add Trendline, choose the type and on option select display equation and display r2 For calibration curves, select the set intercept = 0 option Does this make physical sense?

Does the set intercept = 0 option make a difference? Using the set intercept = 0 option lowers the r2 value by a small amount and changes the slope slightly

A = mc or A = 0.89c The equation becomes 99.1% of the variation of the absorbance is due to the variation of the concentration.

Drill Write the equation of a line that passes through the two points (-2, 4) and (4, 7). Find the value of “y” when “x” is 12, if y = 14 when x = 2. What is the equation of a line that passes through the point (-9, 4) and has a slope of 1/3?

ECR Find the equation for the line of best fit. What is the slope of the line and what does it mean? Predict the distance from home after 8 hours. Hours Driven Distance From House 1 42 2 58 3 78 4 90 5 109