Unit 4 Lesson 2 Presentation

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Lesson Quiz

Lesson Presentation

Warm Up

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Warm Up

Find the slope of the line containing each pair of points.

1. (0, 2) and (3, 4) 2. (–2, 8) and (4, 2) 3. (3, 3) and (12, –15)

Solve each equation for y.

–2

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Write a linear equation in slope-intercept and

point-slope form.

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Example 2A: Writing linear Equations in Slope-Intercept Form

Write the equation that describes the line in slope-intercept form.

slope = ; y-intercept = 4

y = mx + b

y = x + 4

Substitute the given values for m and b.

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slope = –9; y-intercept =

y = mx + b Substitute the given values for m and b.

Simply if necessary.

Example 2B: Writing linear Equations in Slope-Intercept Form

Write the equation that describes the line in slope-intercept form.

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slope = 3; y-intercept =

Example 2C: Writing linear Equations in Slope-Intercept Form

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slope = ; y-intercept = –6

Example 2D: Writing linear Equations in Slope-Intercept Form

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slope = 2; (3, 4) is on the line

y = mx + b

Substitute 2 for m, 3 for x, and 4 for y.

4 = 2(3) + b

Step 1 Find the y-intercept.

–2 = b

4 = 6 + b –6 –6

Write the slope-intercept form.

Solve for b. Since 6 is added to b, subtract 6 from both sides to undo the addition.

Example 2E: Writing linear Equations in Slope-Intercept Form

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Example 2E Continued

Step 2 Write the equation.

y = mx + b _{Write the slope-intercept form.} Substitute 2 for m, and –2 for b. y = 2x + (–2)

y = 2x – 2

slope = 2; (3, 4) is on the line

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Check It Out! Example 2

A line has a slope of 8 and (3, –1) is on the

line. Write the equation that describes this line in slope-intercept form.

Step 1 Find the y-intercept. y = mx + b

Substitute 8 for m, 3 for x, and –1 for y.

–1 = 8(3) + b

Write the slope-intercept form.

–25 = b –1 = 24 + b

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Check It Out! Example 2 Continued

Step 2 Write the equation.

y = mx + b _{Write the slope-intercept form.} Substitute 8 for m, and –25 for b. y = 8x + (–25)

y = 8x – 25

A line has a slope of 8 and (3, –1) is on the

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Example 3A: Using Slope-Intercept Form to Graph

Write the equation in slope-intercept form. Then graph the line described by the equation.

y = 3x – 1

y = 3x – 1 is in the form y = mx + b

slope: m = 3 =

y-intercept: b = –1 Step 1 Plot (0, –1).

Step 2 Count 3 units up and 1 unit right and plot another point. Step 3 Draw the line connecting the two points.

•

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Example 3B: Using Slope-Intercept Form to Graph

2y + 3x = 6

Step 1 Write the equation in slope-intercept form by solving for y.

2y + 3x = 6 –3x –3x

2y = –3x + 6 Subtract 3x from both sides. Since y is multiplied by 2,

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Example 3B Continued

Step 2 Graph the line.

slope: m =

y-intercept: b = 3 Plot (0, 3).

• Count 3 units down and 2 units right and plot another point.

• Draw the line connecting the two points. • •

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Check It Out! Example 3a

Write the equation in slope-intercept form.

Then graph the line described by the equation.

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Check It Out! Example 3a Continued

y-intercept: b = 0 Step 1 Plot (0, 0).

Step 2 Count 2 units up and 3 units right and plot another point.

Step 3 Draw the line connecting the two points. •

• y = x + 0 is in the form

y = mx + b. slope:

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Check It Out! Example 3b

6x + 2y = 10

Step 1 Write the equation in slope intercept form by solving for y.

6x + 2y = 10 –6x –6x

2y = –6x + 10 Subtract 6x from both sides. Since y is multiplied by 2,

divide both sides by 2. Write the equation in slope-intercept form.

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Check It Out! Example 3b Continued

y = –3x + 5 is in the form y = mx + b.

y-intercept: b = 0 • Plot (0, 5).

slope: m =

•

• Count 3 units down and 1 unit right and plot another point.

• Draw the line connecting the two points. • Write the equation in slope-intercept form.

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Check It Out! Example 3c

y = –4

Step 1 Plot (0, –4).

y = –4 is in the form y = mx + b.

y-intercept: b = –4 slope: m = 0 = = 0

Since the slope is 0, the line will be a horizontal at y = –4.

•

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Example 4: Application

A closet organizer charges a $100 initial

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Example 4: Application

a. Write an equation that represents the cost as a function of the number of hours.

Cost is $30 for each _hour plus $100 +

A closet organizer charges $100 initial

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Example 4 Continued

A closet organizer charges $100 initial

consultation fee plus $30 per hour. The cost as a function of the number of hours worked is graphed below.

b. Identify the slope and y-intercept and describe their meanings.

The y-intercept is 100. This is the cost for 0 hours, or the initial fee of $100. The slope is 30. This is the rate of change of the cost: $30 per hour.

c. Find the cost if the organizer works 12 hrs. y = 30x + 100

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A caterer charges a $200 fee plus $18 per person served. The cost as a function of the number of guests is shown in the graph.

a. Write an equation that represents the cost as a function of the

number of guests.

Cost is $18 for each _meal plus $200

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A caterer charges a $200 fee plus $18 per person served. The cost as a function of the number of guests is shown in the graph.

b. Identify the slope and y-intercept and describe their meanings.

The y-intercept is 200. This is the cost for 0 people, or the initial fee of $200. The slope is 18. This is the rate of change of the cost: $18 per person.

c. Find the cost of catering an event for 200 guests.

y = 18x + 200

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Example 2: Writing Linear Equations in Point-Slope Form

Write an equation in point-slope form for the line with the given slope that contains the given point.

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Write an equation in point-slope form for the line with the given slope that contains the given point.

a. b. slope = 0; (3, –4)

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Example 3: Writing Linear Equations in Slope-Intercept Form

Write an equation in slope-intercept form for the line with slope 3 that contains (–1, 4).

Step 1 Write the equation in point-slope form:

y – 4 = 3[x – (–1)]

Step 2 Write the equation in slope-intercept form by solving for y.

y – 4 = 3(x + 1) Rewrite subtraction of negative _{numbers as addition.}

Distribute 3 on the right side.

y – 4 = 3x + 3 + 4 + 4

y = 3x + 7

Add 4 to both sides.

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Write an equation in slope-intercept form for the line with slope that contains (–3, 1).

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Rewrite subtraction of negative numbers as addition.

Distribute on the right side.

+1 +1

Step 2 Write the equation in slope-intercept form by solving for y.

Write an equation in slope-intercept form for the line with slope that contains (–3, 1).

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Example 4A: Using Two Points to Write an Equation Write an equation in slope-intercept form for

the line through the two points. (2, –3) and (4, 1)

Step 1 Find the slope.

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Step 3 Write the equation in slope-intercept form.

y = 2x – 7 –3 –3

Example 4A Continued

Write an equation in slope-intercept form for the line through the two points.

(2, –3) and (4, 1)

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Example 4B: Using Two Points to Write an Equation

(0, 1) and (–2, 9)

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Example 4B Continued

(0, 1) and (–2, 9)

y = –4x + 1 + 1 +1

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Check It Out! Example 4a

(1, –2) and (3, 10)

Step 2 Substitute the slope and one of the points into the point-slope form.

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Check It Out! Example 4a Continued

y + 2 = 6x – 6 – 2 – 2 y = 6x – 8 (1, –2) and (3, 10)

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Check It Out! Example 4b

(6, 3) and (0, –1)

Step 2 Substitute the slope and one of the points into the point-slope form.

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Check It Out! Example 4b Continued

+ 3 +3

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Example 5: Problem-Solving Application

The cost to stain a deck is a linear function of the deck’s area. The cost to stain 100, 250, 400 square feet are shown in the

table. Write an equation in slope-intercept form that represents the function. Then

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Understand the Problem 1

1

• The answer will have two parts—an equation in slope-intercept form and the cost to stain an area of 75 square feet.

• The ordered pairs given in the table—(100, 150), (250, 337.50), (400, 525)—satisfy the equation.

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2

2 Make a Plan

You can use two of the ordered pairs to find the slope. Then use point-slope form to write the equation. Finally, write the equation in slope-intercept form.

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Solve 3

3

Step 1 Choose any two ordered pairs from the table to find the slope.

Use (100, 150) and (400, 525).

Step 2 Substitute the slope and any ordered pair from the table into the point-slope form.

y – 150 = 1.25(x – 100) Use (100, 150).

Example 5 Continued

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y – 150 = 1.25(x – 100)

y – 150 = 1.25x – 125 Distribute 1.25.

y = 1.25x + 25 Add 150 to both sides.

Step 4 Find the cost to stain an area of 75 sq. ft. y = 1.25x + 25

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Look Back 4

4

If the equation is correct, the ordered pairs

that you did not use in Step 2 will be solutions. Substitute (400, 525) and (250, 337.50) into the equation.

y = 1.25x + 25

337.50 1.25(250) + 25 337.50 312.50 + 25

337.50 337.50

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Example 5 Continued

y = 1.25x + 25

525 1.25(400) + 25 525 500 + 25

525 525

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What if…? At a newspaper the

costs to place an ad for one week are shown. Write an equation in slope-intercept

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Understand the problem 1

1

• The answer will have two parts—an equation in slope-intercept form and the cost to run an ad that is 21 lines long.

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2

2 Make a Plan

You can use two of the ordered pairs to find the slope. Then use the point-slope form to write the equation. Finally, write the equation in slope-intercept form.

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Solve 3

3

Step 1 Choose any two ordered pairs from the table to find the slope.

Use (3, 12.75) and (5, 17.25).

Step 2 Substitute the slope and any ordered pair from the table into the point-slope form.

Use (5, 17.25).

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y – 17.25 = 2.25(x – 5)

y – 17.25 = 2.25x – 11.25 Distribute 2.25.

y = 2.25x + 6 Add 17.25 to both sides.

Solve 3

3

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Look Back 4

4

If the equation is correct, the ordered pairs

that you did not use in Step 2 will be solutions. Substitute (3, 12.75) and (10, 28.50) into the equation.

y = 2.25x + 6 12.75 2.25(3) + 6 12.75 6.75 + 6

12.75 12.75

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28.50 2.25(10) + 6 28.50 22.50 + 6

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Lesson Quiz: Part I

Write the equation that describes each line in the slope-intercept form.

1. slope = 3, y-intercept = –2 y = 3x – 2

2. slope = 0, y-intercept = y =

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Lesson Quiz: Part II

Write each equation in slope-intercept form. Then graph the line described by the equation.

4. 6x + 2y = 10 y = –3x + 5