ObjectiveThe student will be able to:

find the slope of a line given 2 points and a graph.

MFCR- Lesson 2-2

10-13-14

10-13-14 Bellwork – formative #14

1. Find the missing coordinate so the ordered pairs are solutions to: ½ x + y = 4

• a. (0, ?)

• b. (?, 0)

• c. (-4, ?)

2. Find the x and y intercepts of 2x+8=0, then graph.

3. Find the x and y intercepts of 2x-2y-6=0, then graph.

What is the meaning of this sign?

1. Icy Road Ahead

2. Steep Road Ahead

3. Curvy Road Ahead

4. Trucks Entering Highway Ahead

What does the 7% mean?

7% is the slope of the road. It means the road drops 7 feet vertically for every 100 feet

horizontally.

7%

So, what is slope???Slope is the steepness of a line.

7 feet

100 feet

Slope can be expressed different ways:

A line has a positive slope if it is going uphill from left to right.

A line has a negative slope if it isgoing downhill from left to right.

2 1

2 1

( ) vertical change

( ) horizontal change

y y risem

x x run

When given the graph, it is easier to apply “rise over run”.

1) Determine the slope of the line.

Start with the lower point and count how much you rise and run to get to the other

point!

Determine the slope of the line.

6

3

run

3

6= =

1

2

rise

Notice the slope is positive AND the line increases!

2) Find the slope of the line that passes through the points (-2, -2) and (4, 1).

(1 ( 2))

(4 ( 2))m

2 1

2 1

( )

( )

y ym

x x

(1 2)

(4 2)

When given points, it is easier to use the formula!

y2 is the y coordinate of the 2nd ordered pair (y2 = 1)

y1 is the y coordinate of the 1st ordered pair (y1 = -2)

13

6 2

Did you notice that Example #1 and Example #2 were the same problem

written differently?

(-2, -2) and (4, 1)6

31

2slope

You can do the problems either way!Which one do you think is easiest?

Find the slope of the line that passes through (3, 5) and (-1, 4).

1. 4

2. -4

3. ¼

4. - ¼

3) Find the slope of the line that goes through the points (-5, 3) and (2, 1).

m y2 y1

x2 x1

1 3

2 ( 5)m

1 3

2 5m

2

7m

Determine the slope of the line shown.

1. -2

2. -½

3. ½

4. 2

Determine the slope of the line.

The line is decreasing (slope is negative).

2

-1

rise

run

2

1 2

Find points on the graph. Use two of them and apply rise over run.

What is the slope of a horizontal line?

The line doesn’t rise!

m 0

number0

All horizontal lines have a slope of 0.

What is the slope of a vertical line?

The line doesn’t run!

All vertical lines have an undefined slope.

m number

0undefined

Remember the word “VUXHOY”

Vertical lines

Undefined slope

X = number; This is the equation of the line.

Horizontal lines

O - zero is the slope

Y = number; This is the equation of the line.

Find the Slope of the Line

• Given the points (5, 1) & (6, 4)

m = = = 3

• Given the points (-6, -2) & (-1, 0)

• m = 4 1

6 5

2 0

6 1

3

1

2 2 2

6 1 5 5

Exit Ticket – Find the slope of each line. #1 and #2

10-14-14 Bellwork

• Find the slope of the line that passes through the given points.

1.(1,-8) and (-5, -4)

2.(5, -2) and (5, 5)

3.(1,6) and (-7,6)

10-14-14 Bellwork solutions

1. -2/3

2. Undefined

3. 0

Lesson 2-2 continued Parallel Slopes

Perpendicular Slopes

Parallel Lines• What do you know about the slope of

parallel lines?

• Parallel lines have the SAME SLOPE!

• What is the slope of the line parallel to y = -2x + 4?

• m = -2

Perpendicular Lines• What do you know about the slope of

perpendicular lines?

• Perpendicular lines have slopes that are opposite reciprocals!

• What is the slope of the line perpendicular to y = -2x + 4?

• m = 1/2

Summary

• Parallel slopes are the same.

• Perpendicular slopes are opposite reciprocals.

Classify as parallel or perpendicular

1. Line 1: (4, -1), (-3, 6)

Line 2: (-1,3), (2,0)

2. Line 1: (0,0), (2,3)

Line 2: (-2, 5), (0,-2)

3. Line 1: (2, 5), (4, 9)

Line 2: (-1, 4), (3, 2)

CW: Int. Algebra Book

• Pages 125-127# 8-54 evens

Bellwork 10-15-14

1. What is the slope of the line perpendicular to y = -2x + 4?

2. What is the slope of the line parallel to

y = -2x + 4?

Friday’s CW solutions 2.2 slope worksheets

Lesson 2-3 ObjectiveThe student will be able to:

1) write equations using slope-intercept form.

2) identify slope and y-intercept from an equation

Important!!!

This is one of the big concepts in Algebra 1. You need to thoroughly

understand this!

Slope – Intercept Form

y = mx + b

m represents the slope

b represents the y-intercept

Writing Equations

When asked to write an equation, you need to know two things – slope (m) and y-intercept (b).

There are three types of problems you will face.

Writing Equations – Type #1Write an equation in slope-intercept form of the

line that has a slope of 2 and a y-intercept of 6.

To write an equation, you need two things:

slope (m) =

y – intercept (b) =

We have both!! Plug them into slope-intercept form

y = mx + b

y = 2x + 6

2

6

Write the equation of a line that has a y-intercept of -3 and a slope of -4.

1. y = -3x – 4

2. y = -4x – 3

3. y = -3x + 4

4. y = -4x + 3

Writing Equations – Type #2Write an equation of the line that has a slope of 3

and goes through the point (2,1).

To write an equation, you need two things:

slope (m) =

y – intercept (b) =

We have to find the y-intercept!! Plug in the slope and ordered pair into

y = mx + b

1 = 3(2) + b

3

???

Writing Equations – Type #21 = 3(2) + b

Solve the equation for b

1 = 6 + b

-6 -6

-5 = b

To write an equation, you need two things:

slope (m) =

y – intercept (b) =

y = 3x - 5

3

-5

Writing Equations – Type #3Write an equation of the line that goes through the points (-2,

1) and (4, 2). To write an equation, you need two things:

slope (m) =y – intercept (b) =

We need both!! First, we have to find the slope. Plug the points into the slope formula.

Simplify2 1

4 ( 2)m

??????

1

6m

Writing Equations – Type #3Write an equation of the line that goes through the

points (-2, 1) and (4, 2). To write an equation, you need two things:

slope (m) =y – intercept (b) =

It’s now a Type #2 problem. Pick one of the ordered pairs to plug into the equation. Which one looks easiest to use?

I’m using (4, 2) because both numbers are positive.2 = (4) + b

1

6???

1

6

Writing Equations – Type #3

2 = (4) + bSolve the equation for b

2 = + b

To write an equation, you need two things:

slope (m) =

y – intercept (b) =

1

62

32

3

11

3b

2

3

1

6 11

3

1 11

6 3y x

Write an equation of the line that goes through the points (0, 1) and (1, 4).

1. y = 3x + 4

2. y = 3x + 1

3. y = -3x + 4

4. y = -3x + 1

To find the slope and y-intercept of an equation, write the equation in slope-intercept form: y = mx + b.

Find the slope and y-intercept.

1) y = 3x – 7

y = mx + b

m = 3, b = -7

Find the slope and y-intercept.

2) y = x

y = mx + b

y = x + 0

3) y = 5

y = mx + b

y = 0x + 5

2

3m =

b = 0

2

3

2

3

m = 0b = 5

-3 -3 -3

Find the slope and y-intercept.4) 5x - 3y = 6

Write it in slope-intercept form. (y = mx + b)

5x – 3y = 6

-3y = -5x + 6

y = x - 25

3m =

b = -2

5

3

Write it in slope-intercept form. (y = mx + b)

2y + 2 = 4x

2y = 4x - 2

y = 2x - 1

Find the slope and y-intercept. 5) 2y + 2 = 4x

2 2 2

m = 2b = -1

Find the slope and y-intercept of y = -2x + 4

1. m = 2; b = 4

2. m = 4; b = 2

3. m = -2; b = 4

4. m = 4; b = -2

Graphing a Line Given a Point & Slope

• Graph a line though the point

(2, -6) with m=2/3

• Graph (2, -6)

• Count up 2 for the

rise, and to the right

3 for the run

• Plot the point, repeat, then connect

x

y

Graphing Lines• m = - ½ b = 3

• Plot y-intercept (b)

• Use the slope to

find two more

points

• Connect

x

y

Graph the Line

• (3, -3) m = undefined

x

y

POINT-SLOPE FORM

• Chapter 2-3

WarPPPPPpm-UpBELLWORK – Thurs. 10-24-13

Write an equation of the line in slope-intercept form.

2. passes through (–2, 2) and (1, 8)

ANSWER

ANSWER

1. passes through (3, 4), m = 3

y = 2x + 6

y = 3x – 5

Warm-Up

3. A carnival charges an entrance fee and a ticket fee. One person paid $27.50 and brought 5 tickets.

Another paid $45.00 and brought 12 tickets. How much will 22 tickets cost?

ANSWER $70

Example 1

Write an equation in point-slope form of the line that passes through the point (4, –3) and has a slope of 2.

y – y1 = m(x – x1) Write point-slope form.

y + 3 = 2(x – 4) Substitute 2 for m, 4 for x1, and –3 for y1.

Guided Practice

Write an equation in point-slope form of the line that passes through the point (–1, 4) and has a slope of –2.

1.

y – 4 = –2(x + 1)ANSWER

Example 2

y + 2 = (x – 3).2 3

Graph the equation

SOLUTION

Because the equation is in point-slope form, you know that the line has a slope of and passes through the point (3, –2).

2 3

Plot the point (3, –2). Find a secondpoint on the line using the slope.Draw a line through both points.

Guided Practice

–Graph the equation2.

ANSWER

y – 1 = (x – 2).

Example 3

Write an equation in point-slope form of the line shown.

SOLUTION

STEP 1

=y2 – y1

x2 – x1

m = 3 – 1 –1 – 1

=2

–2= –1

Find the slope of the line.

Example 3

Method 1 Method 2

Use (–1, 3). Use (1, 1).y – y1 = m(x – x1) y – y1 = m(x – x1)

y – 3 = –(x +1) y – 1 = –(x – 1)

STEP 2Write the equation in point-slope form. You can use either given point.

CHECK

Check that the equations are equivalent by writing them in slope-intercept form.

y – 3 = –x – 1y = –x + 2

y – 1 = –x + 1y = –x + 2

Guided Practice

Write an equation in point-slope form of the line that passes through the points (2, 3) and (4, 4).

3.

y – 3 = (x – 2) or1

2y – 4 = (x – 4)

1 2

ANSWER

Example 5

WORKING RANCH

The table shows the cost of visiting a working ranch for one day and night for different numbers of people. Can the situation be modeled by a linear equation? Explain. If possible, write an equation that gives the cost as a function of the number of people in the group.

Lesson Quiz

ANSWER y + 4 = –2(x – 6)

Write an equation in point-slope form of the line that passes through (6, –4) and has slope 2.

1.

Write an equation in point-slope form of the line that passes through (–1, –6) and (3, 10).

2.

ANSWER y + 6 = 4(x + 1) or y –10 = 4(x–3)

Lesson Quiz

A travel company offers guided rafting trips for $875 for the first three days and $235 for each additional day. Write an equation that gives the total cost (in dollars) of a rafting trip as a function of the length of the trip. Find the cost for a 7-day trip.

3.

ANSWER

C = 235t + 170, where C is total cost and t is time (in days); $1815

Write Equations and Parallel and Perpendicular Lines

Warm-Up

Are the lines parallel? Explain.

2. –x = y + 4, 3x + 3y = 5

ANSWER

ANSWER

1. y – 2 = 2x, 2x + y = 7

Yes; both slopes are –1.

No; one slope is 2 and the other is –2.

Example 1

SOLUTION

Write an equation of the line that passes through (–3, –5) and is parallel to the line y = 3x – 1.

STEP 1

Identify the slope. The graph of the given equation has a slope of 3. So, the parallel line through (–3, –5) has a slope of 3.

Example 1

STEP 2Find the y-intercept. Use the slope and the given point.

y = mx + b

–5 = 3(–3) + b

4 = b

Write slope-intercept form.

Substitute 3 for m, 3 for x, and 5 for y.

Solve for b.

STEP 3

Write an equation. Use y = mx + b.

y = 3x + 4 Substitute 3 for m and 4 for b.

Guided Practice

1. Write an equation of the line that passes through

(–2, 11) and is parallel to the line y = –x + 5.

y = –x + 9ANSWER

Example 2

Determine which lines, if any, are parallel or perpendicular.Line a: y = 5x – 3

Line b: x + 5y = 2

Line c: –10y – 2x = 0

SOLUTION

Find the slopes of the lines.

Line a: The equation is in slope-intercept form. The slope is 5.

Write the equations for lines b and c in slope-intercept form.

Example 2

Line b: x + 5y = 2

5y = – x + 2

Line c: –10y – 2x = 0

–10y = 2x

y = – x15xy = 2

515 +–

ANSWER

Lines b and c have slopes of – , so they are

parallel. Line a has a slope of 5, the negative reciprocal

of – , so it is perpendicular to lines b and c.

15

15

Guided Practice

Determine which lines, if any, are parallel or perpendicular.Line a: 2x + 6y = –3

Line b: y = 3x – 8

Line c: –1.5y + 4.5x = 6

ANSWER

parallel: b and c; perpendicular: a and b, a and c

Example 3

SOLUTION

Line a: 12y = –7x + 42

Line b: 11y = 16x – 52

Find the slopes of the lines. Write the equations in slope-intercept form.

The Arizona state flag is shown in a coordinate plane. Lines a and b appear to be perpendicular. Are they?

STATE FLAG

Example 3

Line a: 12y = –7x + 42

Line b: 11y = 16x – 52

y = – x + 1242 7

12

1152

y = x –1611

ANSWER

The slope of line a is – . The slope of line b is .

The two slopes are not negative reciprocals, so lines a and b are not perpendicular.

712

1611

Guided Practice

3. Is line a perpendicular to line b? Justify your answer using slopes.

Line a: 2y + x = –12

Line b: 2y = 3x – 8

ANSWER

No; the slope of line a is – , the slope of line b is . The slopes are not negative reciprocals so the lines are not perpendicular.

12

32

Example 4

SOLUTION

Write an equation of the line that passes through (4, –5) and is perpendicular to the line y = 2x + 3.

STEP 1

Identify the slope. The graph of the given equation has a slope of 2. Because the slopes of perpendicular lines are negative reciprocals, the slope of the perpendicular line through (4, –5) is .1

2–

Example 4

STEP 2 Find the y-intercept. Use the slope and thegiven point.

Write slope-intercept form.

–5 = – (4) + b12

Substitute – for m, 4 for x, and

–5 for y.

12

y = mx + b

–3 = b Solve for b.

STEP 3 Write an equation.

y = mx + b Write slope-intercept form.

y = – x – 312 Substitute – for m and –3 for b.1

2

Guided Practice

4. Write an equation of the line that passes through (4, 3) and is perpendicular to the line y = 4x – 7.

y = – x + 414ANSWER

Lesson Quiz

1. Write an equation of the line that passes through the point (–1, 4) and is parallel to the line y = 5x – 2.

y = 5x + 9

ANSWER

Write an equation of the line that passes through the point (–1, –1) and is perpendicular to the line y = x + 2.1

4–

2.

y = 4x + 3

ANSWER

Lesson Quiz

3. Path a, b and c are shown in the coordinate grid. Determine which paths, if any, are parallel or perpendicular. Justify your answer using slopes.

ANSWER

Paths a and b are perpendicular because their slopes, 2 and are negative reciprocals. No paths are parallel.1

2–

Graphing Lines

• Graph the line perpendicular to y = 2x + 3 that goes through the

point (-2, 3).• The slope of the line is 2 so the slope of the perpendicular line is -1/2.• m = -1/2 b = 3

Graphing Lines

• Graph the line parallel to x = -1 that goes through the point

(3, -3).

• The slope of the line is undefined (VUX) so the slope of the parallel line is undefined.

Writing Equations in Standard Form

• MFCR Lesson 2-3

Warm-Up

ANSWER

ANSWER

1. (1, 4), (6, –1)

y + 2 = 3(x + 1) or y – 7 = 3(x – 2)

y – 4 = –(x – 1) or y + 1 = –(x – 6)

2. (–1, –2), (2, 7)

Write an equation in point-slope form of the line that passes through the given points.

Example 1

To write another equivalent equation, multiply each side by 0.5.

4x – 12y = 8

To write one equivalent equation, multiply each side by 2.

SOLUTION

Write two equations in standard form that are equivalent to 2x – 6y = 4.

x – 3y = 2

Example 2

SOLUTION

y – y1 = m(x – x1)

Calculate the slope.STEP 1

–3m =1 – (–2)

1 – 2=

3–1 =

Write an equation in point-slope form. Use (1, 1).

Write point-slope form.

y – 1 = –3(x – 1) Substitute 1 for y1, 3 for m

and 1 for x1.

Write an equation in standard form of the line shown.

STEP 2

Example 2

Rewrite the equation in standard form.

3x + y = 4 Simplify. Collect variable terms on one side, constants on the other.

STEP 3

Guided Practice

Write an equation in standard form of the line through (3, –1) and (2, –3).

2.

–2x + y = –7ANSWER

Example 3

SOLUTION

Write an equation of the specified line.

The y-coordinate of the given point on the blue line is –4. This means that all points on the line have a y-coordinate of –4. An equation of the line is y = –4.

a.

The x-coordinate of the given point on the red line is 4. This means that all points on the line have an x-coordinate of 4. An equation of the line is x = 4.

b.

Blue linea. Red lineb.

Guided Practice

Write equations of the horizontal and vertical lines that pass through the given point.

3. (–8, –9)

y = –9, x = –8ANSWER

4. (13, –5)

y = –5, x = 13ANSWER

Lesson Quiz

Write an equation in standard form of the line that passes through the given point and has the given slope m or that passes through the two given points.

ANSWER 2x + y = –4

1.

(1, –6), m = –2

2. (–4, –3), (2, 9)

ANSWER –2x + y = 5