Graphing Lines with a Table

• Select (or use pre-selected) values for x

• Substitute those x values in the equation and solve for y

• Graph the x and y values as ordered pairs

• Connect points with a line

Example

• Graph y = 2x - 1

Example

• Graph y = 2x

Example

• Graph 2x + 3y = 4

Time to work

• Worksheet!

Ch 7

Linear Equations

7.1

Slope

Slope

• Slope – The ratio of the rise, or vertical change, to the run,

or horizontal change

riseslope = m = run

Example

• Determine the slope of each line.

Example

• Determine the slope of each line.

Rate of Change

• In real-life applications, slope is the rate of change (how much a value is changing)

Example

• The graph below shows the distance traveled by Rebecca and Ian during a day-long bicycle ride. Find the slope of each line. To what does the slope refer?

Example

• A line contains the points whose coordinates are listed in the table. Determine the slope of the line.

Slope Formula

Example

• Determine the slope of each line. The line through the points at (3, 8) and (3, 4)

Example

• Determine the slope of each line. The line through the points at (-4, 1) and

(-3, -2)

Example

• Determine the slope of each line. The line through the points at (2, 5) and (3, 9)

Example

• Determine the slope of each line. The line through the points at (-8, 1) and (4, 1)

Types of Slope

Assignments

• #1 – due today– P287: 1, 2, 4 – 12 even

• #2 – due next time– P288: 13 – 27, 34, 35, 36

7.2

Writing Equations in Point-Slope Form

Point-Slope Form

• Replace the m, x1, and y1 with the values given

Example

• Write the point-slope form of the equation of the line passing through the given point and having the given slope.

(-2, 7), m = -1/3

Example

• Write the point-slope form of the equation of the line passing through the given point and having the given slope.

(4, 0), m = 4

Example

• Write the point-slope form of the equation of the line passing through the given point and having the given slope.

(-3, 2), m = 2

Example

• Write the point-slope form of the equation of the line passing through the given point and having the given slope.

(5, 4), m = -2/3

Writing from a graph

• You can also write an equation in point-slope form from a graph

• First find the slope of the line by counting

• Then pick a point (any point on the line will work)

• Plug those values into the formula

Example

• Write the point-slope form of an equation of the line below.

Example

• Write the point-slope form of an equation of the line below.

Example

• Write the point-slope form of an equation for the line passing through (1, 4) and (3, -5) Hints: find the slope first / it doesn’t matter which

point you use.

Assignments

• #1 – due today P293: 3 – 13

• #2 – due next time P293: 15 – 37

7.3

Writing Equations in Slope-Intercept Form

Intercepts

• y-intercept – The point on the y-axis where the line crosses that

axis

• x-intercept – The point on the x-axis where the line crosses that

axis

Slope-Intercept Form

• Another form, besides point-slope

• This form helps you graph!

y = mx + b

• m – slope

• b – y-intercept (point where it crosses y-axis)

Example

• Write an equation in slope-int form of each line with the given slope and y-int.

m = 3, b = -1

Example

• Write an equation in slope-int form of each line with the given slope and y-int.

m = -2/3, b = 0

Example

• Write an equation in slope-int form of each line with the given slope and y-int.

m = 0, b = -4

Example

• Write an equation in slope-int form of each line with the given slope and y-int.

m = 2, b = 1

Example

• Write an equation in slope-int form of each line with the given slope and y-int.

m = -5/3, b = 0

Example

• Write an equation in slope-int form of each line with the given slope and y-int.

m = 0, b = -8

Example

• Write an equation of the line in slope-intercept form for the situation:

Slope 1 and passes through (2, 5)

Example

• Write an equation of the line in slope-intercept form for the situation:

Slope -3 and passes through (1, -4)

Example

• Write an equation of the line in slope-intercept form for the situation:

Passing through (-4, 4) and (2, 1)

Example

• Write an equation of the line in slope-intercept form for the situation:

Passing through (6, 2) and (3, -2)

Example

• Write an equation of the line in slope-intercept form for the situation:

Slope is ¾ and passes through (8, -2)

Example

• Write an equation of the line in slope-intercept form for the situation:

Passes through (2, 4) and (0, 5)

Assignments

• #1 – due today P299: 4 – 12

• #2 – due next time P299: 14 – 40 even, 41 – 45, 49 – 50, 53 – 57

7.4

Scatter Plots

Scatter Plots

• Scatter Plot – Graph where two sets of data are plotted as

ordered pairs on the same coordinate plane

Used to see if there is a trend, pattern, or relationship among the variables

Scatter Plots

Example

• Determine whether the scatter plot shows a positive relationship, negative relationship, or no relationship. If there is a relationship, describe it.

• The scatter plot shows the number of years of experience and the salary for each employee in a small company.

Example• Determine whether the

scatter plot shows a positive relationship, negative relationship, or no relationship. If there is a relationship, describe it.

• The scatter plot shows the word processing speeds of 12 students and the number of weeks they have studied word processing.

Example

• Determine whether the scatter plot shows a positive relationship, negative relationship, or no relationship. If there is a relationship, describe it.

Example

• The table shows the average number of minutes a pediatric dentist spends during each appointment instructing the patient in proper dental care, and the number of cavities for each patient.

Example

• Make a scatter plot of the data. Let the horizontal axis represent instruction time and let the vertical axis represent the number of cavities.

• Does the scatter plot show a relationship between instruction time and cavities? Explain.

• Describe the independent and dependent variables. Then state the domain and the range.

Assignments

• P305: 4 – 8, 10 – 17, 19 – 23

7.5

Graphing Linear Equations

Graphing with Intercepts

• What are intercepts?– Point where the line crosses the x- and y-axes

•Find the intercepts and plot them, draw a line between•Point of y-intercept is always (o, y)•Point of x-intercept is always (x, 0)

Example

• Determine the x-intercept and y-intercept of the graph of the line 2y – x = 8. Then graph.

Example

• Determine the x-intercept and y-intercept of the graph of the line 3x – 2y = 12. Then graph.

Example

• Determine the x-intercept and y-intercept of the graph of the line x + y = 2. Then graph.

Example

• Determine the x-intercept and y-intercept of the graph of the line 3x + y = 3. Then graph.

Example

• Determine the x-intercept and y-intercept of the graph of the line 4x – 5y = 20. Then graph.

Example

• Suppose to ship a package it costs $2.05 for the first pound and $1.55 for each additional pound. This can be represented by y = 2.05 + 1.55x. Determine the slope and y-intercept of the graph of the equation.

Example

• Determine the slope and y-intercept of the graph 6x – 9y = 18.

Example

• Determine the slope and y-intercept of the graph of 4x + 3y = 6.

Example

• Graph the equation using slope intercept form.

2 53

y x

Example

• Graph the equation using slope intercept form.

1 25

y x

Example

• Graph the equation using slope intercept form.

1 32

y x

Example

• Graph the equation using slope intercept form.

3 4x y

Example

• Graph the equation using slope intercept form.

3y

Example

• Graph the equation using slope intercept form.

4x

Example

• Graph the equation using slope intercept form.

1y

Example

• Graph the equation using slope intercept form.

3x

Assignments

• P314: 7 – 11, 24 – 34 even, 36 – 38, 43 – 49

7.6

Families of Linear Graphs

Review

• Slope formula:

• Point-Slope Form:

• Slope-Intercept Form:

Linear Graphs

Example

• Graph the pair of equations. Describe any similarities or differences. Explain why they are a family of graphs.

1 221 12

y x

y x

Example

• Graph the pair of equations. Describe any similarities or differences. Explain why they are a family of graphs.

5 11

y xy x

Example

• Graph the pair of equations. Describe any similarities or differences. Explain why they are a family of graphs.

2 12 5

y xy x

Example

• Graph the pair of equations. Describe any similarities or differences. Explain why they are a family of graphs.

13 1

y xy x

Example

• Gretchen and Max each have a savings account and plan to save $20 per month. The current balance in Gretchen’s account is $150 and the balance in Max’s account is $100. Then y = 20x + 150 and y = 20x + 100 represent how much money each has in their account, respectively, after x months. Compare and contrast the graphs of the equations.

Parent Graphs

• The simplest of graphs in a family

• Questions: How does changing the slope affect the line?

How does changing the y-int affect the line?

Example

• Change y = -3x – 1 so that the graph of the new equation fits each description. Same y-intercept, less steep positive slope.

Same slope, y-intercept is shifted down 2 units.

Example

• Change y = 2x + 1 so that the graph of the new equation fits each description.– Same slope, shifted down 1 unit

– Same y-intercept, less steep positive slope

Assignments

• #1 – due today P319: 1, 4 – 10 even

• #2 – due next time P319: 12 – 30 even, 31, 34 – 39

7.7

Parallel and Perpendicular Lines

Parallel

• Two lines are parallel if they never intersect

• What would have to be true about the lines so that they would never intersect?

• They have the same slope!!

Parallel Lines

Example

• Determine whether the graphs of the equations are parallel.

3 49 3 12y xx y

Example

• Determine whether the graphs of the equations are parallel.

27 2y x

x y

Example

• Determine whether the graphs of the equations are parallel.

3 32 6 5y x

y x

Parallelogram

• A four-sided figure with two sets of parallel sides

Example

• Determine whether figure EFGH is a parallelogram.

Example

• Determine whether figure ABCD is a parallelogram.

Example

• Write an equation in slope-intercept form of the line that is parallel to the graph of and passes through the point at (-3, 1).

2 33

y x

Example

• Write an equation in slope-intercept form of the line that is parallel to the graph of and passes through the point at (2, 3).

6 4y x

Example

• Write an equation in slope-intercept form of the line that is parallel to the graph of and passes through the point at (2, 0).

3 2 9x y

Perpendicular Lines

Example

• Determine whether the graphs of the equations are perpendicular.

2 41 32

y x

y x

Example

• Determine whether the graphs of the equations are perpendicular.

1 255 1

y x

y x

Example

• Determine whether the graphs of the equations are perpendicular.

4 34 5y x

y x

Example

• Write an equation in slope-intercept form of the line that is perpendicular to the graph of . and passes through the point at (2, -3).2 5y x

Example

• Write an equation in slope-intercept form of the line that is perpendicular to the graph of and passes through the point at (0, 0).

Example

• Write an equation in slope-intercept form of the line that is perpendicular to the graph of . and passes through the point at (3, 0).2 3 2x y

Assignments

• #1 – due today P325: 2 – 14

• #2 – due today P326: 16 – 38 even, 42 – 50