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Holt Algebra 2

2-4 Writing Linear Functions 2-4 Writing Linear Functions

Holt Algebra 2

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Holt Algebra 2

2-4 Writing Linear Functions

Warm Up

Write each function in slope-intercept form.

1. 4x + y = 8 2. –y = 3x 3. 2y = 10 – 6x

Determine whether each line is vertical or horizontal.

4. 5. y = 0

y = –4x + 8 y = –3x y = –3x + 5

3 4x =

vertical horizontal

Holt Algebra 2


Use slope-intercept form and point-slope form to write linear functions.Write linear functions to solve problems.

Objectives

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Point-slope form

Vocabulary

Holt Algebra 2


Recall from Lesson 2-3 that the slope-intercept form of a linear equation is y= mx + b, where m is the slope of the line and b is its y-intercept.

In Lesson 2-3, you graphed lines when you were given the slope and y-intercept. In this lesson you will write linear functions when you are given graphs of lines or problems that can be modeled with a linear function.

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Example 1: Writing the Slope-Intercept Form of the Equation of a Line

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

Step 1 Identify the y-intercept.The y-intercept b is 1.

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Example 1 Continued

Step 2 Find the slope.

Choose any two convenient points on the line, such as (0, 1) and (4, –2). Count from (0, 1) to (4, –2) to find the rise and the run. The rise is –3 units and the run is 4 units.

Slope is = = – . riserun

–34

34

3

–44

–3

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Example 1 Continued

Step 3 Write the equation in slope-intercept form.

y = mx + b

34

y = – x + 1 m = – and b = 1. 34

The equation of the line is 34

y = – x + 1.

Holt Algebra 2


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

Step 1 Identify the y-intercept.The y-intercept b is 3.

Check It Out! Example 1

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Step 2 Find the slope.

Choose any two convenient points on the line, such as (–4, 0) and (0, 3). Count from (–4, 0) to (0, 3) to find the rise and the run. The rise is 3 units and the run is 4 units

Check It Out! Example 1 Continued

3

Slope is = . riserun

34

3

4 4

3

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Step 3 Write the equation in slope-intercept form.

Check It Out! Example 1 Continued

y = mx + b

34

y = x + 3 m = and b = 3. 34

The equation of the line is 34

y = x + 3.

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Notice that for two points on a line, the rise is the differences in the y-coordinates, and the run is the differences in the x-coordinates. Using this information, we can define the slope of a line by using a formula.

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If you reverse the order of the points in Example 2B, the slope is still the same.

m = =

Helpful Hint

6 – 165 – 11

– 10– 6

53

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Example 2A: Finding the Slope of a Line Given Two or More Points

Find the slope of the line through (–1, 1) and (2, –5).

Let (x1, y1) be (–1, 1) and (x2, y2) be (2, –5).

Use the slope formula.

The slope of the line is –2.

Holt Algebra 2


Example 2B: Finding the Slope of a Line Given Two or More Points

Find the slope of the line.

x 4 8 12 16

y 2 5 8 11

Let (x1, y1) be (4, 2) and (x2, y2) be (8, 5). Choose any two points.


The slope of the line is .34

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Find the slope of the line shown.

Let (x1, y1) be (0,–2) and

(x2, y2) be (1, –2).

The slope of the line is 0.

Example 2C: Finding the Slope of a Line Given Two or More Points

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Find the slope of the line.x –6 –4 –2

y –3 –1 1

Let (x1, y1) be (–4, –1) and (x2, y2) be (–2, 1).

Choose any two points.



Check It Out! Example 2A

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Let (x1, y1) be (2, –5) and (x2, y2) be (–3, –5).


Check It Out! Example 2B

Find the slope of the line through (2,–5) and (–3, –5).


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Because the slope of line is constant, it is possible to use any point on a line and the slope of the line to write an equation of the line in point-slope form.

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Example 3: Writing Equations of Lines

In slope-intercept form, write the equation of the line that contains the points in the table.

x –8 –4 4 8

y –5 –3.5 –0.5 1

First, find the slope. Let (x1, y1) be (–8, –5) and (x2, y2) be (8, 1).

Next, choose a point, and use either form of the equation of a line.

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Example 3 Continued

Method A Point-Slope Form Rewrite in slope-intercept form.

Substitute.

Simplify. Solve for y.

Distribute.

Using (8, 1):

y – y1 = m(x – x1)

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Method B Slope-intercept Form

Substitute.

Simplify.

Solve for b.

Rewrite the equation using m and b.

Using (8, 1), solve for b.

y = mx + b

b = –2

y = mx + b

1 = 3 + b

The equation of the line is .

Example 3 Continued

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Method A Point-Slope Form

Rewrite in slope-intercept form.

Substitute.

Simplify.

Solve for y.

Distribute.

Write the equation of the line in slope-intercept form with slope –5 through (1, 3).

Check It Out! Example 3a

The equation of the slope is y = –5x + 8.

y – y1 = m(x – x1)

y – (3) = –5(x – 1)

y – 3 = –5(x – 1)

y – 3 = –5(x – 1)

y – 3 = –5x + 5

y = –5x + 8

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Method B Slope-Intercept Form

Check It Out! Example 3bWrite the equation of the line in slope-intercept form through (–2, –3) and (2, 5).

First, find the slope. Let (x1, y1) be (–2,–3) and (x2, y2) be (2, 5).

y = mx + b

5 = (2)2 + b

5 = 4 + b

1 = b

Rewrite the equation using m and b.

y = mx + b y = 2x + 1

The equation of the line is y = 2x + 1.

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Example 4A: Entertainment ApplicationThe table shows the rents and selling prices of properties from a game. Selling Price

($)Rent ($)

75 9

90 12

160 26

250 44

Express the rent as a function of the selling price.

Let x = selling price and y = rent.

Find the slope by choosing two points. Let (x1, y1) be (75, 9) and (x2, y2) be (90, 12).

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To find the equation for the rent function, use point-slope form.

Use the data in the first row of the table.

Simplify.

y – y1 = m(x – x1)

Example 4A Continued

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Example 4B: Entertainment Application

Graph the relationship between the selling price and the rent. How much is the rent for a property with a selling price of $230?

To find the rent for a property, use the graph or substitute its selling price of $230 into the function.

Substitute.

The rent for the property is $40.

y = 46 – 6

y = 40

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Items Cost ($)

4 14.00

7 21.50

18

Express the cost as a linear function of the number of items.

Let x = items and y = cost.

Find the slope by choosing two points. Let (x1, y1) be (4, 14) and (x2, y2) be (7, 21.50).


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To find the equation for the number of items, use point-slope form.

Use the data in the first row of the table.

Simplify.

Check It Out! Example 4a Continued

y – y1 = m(x – x1)

y – 14 = 2.5(x – 4)

y = 2.5x + 4

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Graph the relationship between the number of items and the cost. Find the cost of 18 items.

To find the cost, use the graph or substitute the number of items into the function.

Substitute.

The cost for 18 items is $49.

Check It Out! Example 4b

y = 2.5(18) + 4

y = 45 + 4

y = 49

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By comparing slopes, you can determine if the lines are parallel or perpendicular. You can also write equations of lines that meet certain criteria.

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Holt Algebra 2


Example 5A: Writing Equations of Parallel and Perpendicular Lines

Parallel lines have equal slopes.

Use y – y1 = m(x – x1) with (x1, y1) = (5, 2).

Distributive property.

Simplify.

m = 1.8

y – 2 = 1.8(x – 5)

y – 2 = 1.8x – 9

y = 1.8x – 7

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

parallel to y = 1.8x + 3 and through (5, 2)

Holt Algebra 2


Example 5B: Writing Equations of Parallel and Perpendicular Lines


Simplify.

Use y – y1 = m(x – x1). y + 2 is equivalent to y – (–2).


perpendicular to and through (9, –2)

The slope of the given line is , so the slope of

the perpendicular line is the opposite reciprocal, .

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parallel to y = 5x – 3 and through (1, 4)

Parallel lines have equal slopes.

Use y – y1 = m(x – x1) with (x1, y1) = (5, 2).


Simplify.

m = 5

y – 4 = 5(x – 1)

y – 4 = 5x – 5

y = 5x – 1


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Example 5B: Writing Equations of Parallel and Perpendicular Lines


Simplify.





the perpendicular line is the opposite reciprocal, .

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Simplify.


Check It Out! Example 5b


the perpendicular, line is the opposite reciprocal .



Holt Algebra 2


Lesson Quiz: Part I

y = –2x –1

y = 0.5x – 2

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

1.

2. parallel to y = 0.5x + 2 and through (6, 1)

3. perpendicular to and through (4, 4)

Holt Algebra 2


Lesson Quiz: Part II

Number in Group Cost ($)

4 98

7 134

15 230

4. Express the catering cost as a function of the number of people. Find the cost of catering a meal for 24 people.

f(x) = 12x + 50; $338

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