5.1 write linear equations in slope-intercept form · write linear equations in slope-intercept...

Copyright © Holt McDougal. All rights reserved. Lesson 5.1 • Algebra 1 Notetaking Guide 119

Write Linear Equations inSlope-Intercept FormGoal p Write equations of lines.

5.1

Your Notes

Write an equation of the line with a slope of 24 and a y-intercept of 6.

Solutiony 5 mx 1 b Write slope-intercept form.

y 5 x 1 Substitute for m and for b.

Example 1 Use slope and y-intercept to write an equation

Checkpoint Write an equation of the line with the given slope and y-intercept.

1. Slope is 8; 2. Slope is 2 } 3 ;

y-intercept is 25. y-intercept is 22.

3. Slope is 23; 4. Slope is 2 5 } 2 ;


Use the slope-intercept form (y 5 mx 1 b) to write an equation of a line if slope and y-intercept are given.


Write Linear Equations inSlope-Intercept FormGoal p Write equations of lines.

5.1

Your Notes

Write an equation of the line with a slope of 24 and a y-intercept of 6.

Solutiony 5 mx 1 b Write slope-intercept form.

y 5 24 x 1 6 Substitute 24 for m and 6 for b.

Example 1 Use slope and y-intercept to write an equation

Checkpoint Write an equation of the line with the given slope and y-intercept.

1. Slope is 8; 2. Slope is 2 } 3 ;


y 5 8x 1 (25) y 5 2 } 3 x 1 (22)

3. Slope is 23; 4. Slope is 2 5 } 2 ;


y 5 23x 1 7 y 5 2 5 } 2 x 1 9

Use the slope-intercept form (y 5 mx 1 b) to write an equation of a line if slope and y-intercept are given.

Your Notes

120 Lesson 5.1 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Write an equation of the line shown.

SolutionStep 1 Calculate the slope.

x

y

1

3

12121

23

23 3

(3, 22)

(0, 4)

m 5 y2 2 y1 } x2 2 x1

5 2

2

5 5

Step 2 Write an equation of the line. The line crosses the y-axis at . So, the y-intercept is .

y 5 mx 1 b Write slope-intercept form.

y 5 x 1 Substitute for m and for b.

Example 2 Write an equation of a line given two points

You can write an equation of a line if you know the y-intercept and any other point on the line.

5. Write an equation of the line shown.

x

y1

12121

23

25

3 5

(5, 23)

(0, 26)

Checkpoint Complete the following exercise.

Your Notes


Write an equation of the line shown.


x

y

1

3

12121

23

23 3

(3, 22)

(0, 4)

m 5 y2 2 y1 } x2 2 x1

5 3

22

0

4

2

2

5 3

26 5 22

Step 2 Write an equation of the line. The line crosses the y-axis at (0, 4) . So, the y-intercept is 4 .


y 5 22 x 1 4 Substitute 22 for m and 4 for b.

Example 2 Write an equation of a line given two points

You can write an equation of a line if you know the y-intercept and any other point on the line.

5. Write an equation of the line shown.

y 5 3 } 5 x 2 6

x

y1

12121

23

25

3 5

(5, 23)

(0, 26)


Your Notes

Write an equation for the linear function f with the values f(0) 5 4 and f(2) 5 12.

Solution

Step 1 Write f(0) 5 4 as and f(2) 5 12 as .

Step 2 Calculate the slope of the line that passes through and .

m 5 y2 2 y1 } x2 2 x1

5 2

2

5

5

Step 3 Write an equation of the line. The line crosses the y-axis at (0, ). So, the y-intercept is .


y 5 Substitute for m and for b.

The function is .

Example 3 Write a linear function

Homework

6. Write an equation for the linear function with the values f(0) 5 3 and f(3) 5 15.



Your Notes

Write an equation for the linear function f with the values f(0) 5 4 and f(2) 5 12.

Solution

Step 1 Write f(0) 5 4 as (0, 4) and f(2) 5 12 as (2, 12).

Step 2 Calculate the slope of the line that passes through (0, 4) and (2, 12).

m 5 y2 2 y1 } x2 2 x1

5 2

12

0

4

2

2

5 2

8

5 4

Step 3 Write an equation of the line. The line crosses the y-axis at (0, 4 ). So, the y-intercept is 4 .


y 5 4x 1 4 Substitute 4 for m and 4 for b.

The function is f (x) 5 4x 1 4.

Example 3 Write a linear function

Homework

6. Write an equation for the linear function with the values f(0) 5 3 and f(3) 5 15.

y 5 4x 1 3



5.2 Use Linear Equations inSlope-Intercept FormGoal p Write an equation of a line using points

on the line.

WRITING AN EQUATION OF A LINE IN SLOPE-INTERCEPT FORM

Step 1 Identify the slope . You can use the to calculate the slope if you know

two points on the line.

Step 2 Find the . You can substitute the and the of a point (x, y) on the line into y 5 mx 1 b. Then solve for .

Step 3 Write an equation using .

Your Notes

Write an equation of the line that passes through the point (1, 2) and has a slope of 3.

Solution

Step 1 Identify the slope. The slope is .

Step 2 Find the y-intercept. Substitute the slope and the coordinates of the given point into y 5 mx 1 b. Solve for b.


5 ( ) 1 b Substitute for m, for x, and for y.

5 b Solve for .

Step 3 Write an equation of the line.



Example 1 Write an equation given the slope and a point

Be careful not to mix up the x- and y-values when you substitute.

Your NotesYour Notes


5.2 Use Linear Equations inSlope-Intercept FormGoal p Write an equation of a line using points

on the line.

WRITING AN EQUATION OF A LINE IN SLOPE-INTERCEPT FORM

Step 1 Identify the slope m . You can use the slope formula to calculate the slope if you know two points on the line.

Step 2 Find the y-intercept . You can substitute the slope and the coordinates of a point (x, y) on the line into y 5 mx 1 b. Then solve for b .

Step 3 Write an equation using y 5 mx 1 b .

Your Notes

Write an equation of the line that passes through the point (1, 2) and has a slope of 3.

Solution

Step 1 Identify the slope. The slope is 3 .

Step 2 Find the y-intercept. Substitute the slope and the coordinates of the given point into y 5 mx 1 b. Solve for b.


2 5 3 ( 1 ) 1 b Substitute 3 for m, 1 for x, and 2 for y.

21 5 b Solve for b .




Example 1 Write an equation given the slope and a point

Be careful not to mix up the x- and y-values when you substitute.

Your NotesYour Notes


Your Notes

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


Write an equation of the line that passes through (2, 23) and (22, 1).


m 5 y2 2 y1 } x2 2 x1

5 2

2

5 5

Step 2 Find the y-intercept. Use the slope and the point (2, 23).


23 5 ( ) 1 b Substitute for m, for x, and

for y.

5 b Solve for b.




Example 2 Write an equation given two points

You can also find the y-intercept using the coordinates of the other given point.


Your Notes

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

y 5 4x 2 6


Write an equation of the line that passes through (2, 23) and (22, 1).


m 5 y2 2 y1 } x2 2 x1

5 22

1

2

(23)

2

2

5 24

4 5 21

Step 2 Find the y-intercept. Use the slope and the point (2, 23).



21 5 b Solve for b.




Example 2 Write an equation given two points

You can also find the y-intercept using the coordinates of the other given point.


Your Notes

2. Write an equation for the line that passes through (28, 213) and (4, 2).



HOW TO WRITE EQUATIONS IN SLOPE-INTERCEPT FORM

1. Given slope m and y-intercept b. Substitute and in the equation

.

2. Given slope m and one point. Substitute and the of the

point in . Solve for . Write the .

3. Given two points. Use the points to find the slope . Then

substitute and the of in . Solve for . Write

the .

Homework


Your Notes


y 5 5 } 4 x 2 3


y 5 2 3 } 2 x 2 1 } 2


HOW TO WRITE EQUATIONS IN SLOPE-INTERCEPT FORM

1. Given slope m and y-intercept b. Substitute m and b in the equation

y 5 mx 1 b .

2. Given slope m and one point. Substitute m and the coordinates of the

point in y 5 mx 1 b . Solve for b . Write the equation .

3. Given two points. Use the points to find the slope m . Then

substitute m and the coordinates of one point in y 5 mx 1 b . Solve for b . Write the equation .

Homework


5.3 Write Linear Equations inPoint-Slope FormGoal pWrite linear equations in point-slope form.

VOCABULARY

Point-slope form

POINT-SLOPE FORM

The point-slope form of the equation of the nonvertical line through a given point (x1, y1) with a slope of m is

.

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

Solution y 2 y1 5 m(x 2 x1) Write point-slope form.

y 2 5 (x 2 ) Substitute for m, for x1, and for y1.

Example 1 Write an equation in point-slope form


Your Notes

5.3 Write Linear Equations inPoint-Slope FormGoal pWrite linear equations in point-slope form.

VOCABULARY

Point-slope form The equation of a nonvertical line y 2 y1 5 m(x 2 x1) that passes through a given point (x1, y1) with a slope m

POINT-SLOPE FORM

The point-slope form of the equation of the nonvertical line through a given point (x1, y1) with a slope of m is y 2 y1 5 m(x 2 x1) .

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

Solution y 2 y1 5 m(x 2 x1) Write point-slope form.

y 2 2 5 2 (x 2 3 ) Substitute 2 for m, 3 for x1, and 2 for y1.

Example 1 Write an equation in point-slope form


Your Notes

Your Notes

Graph the equation y 2 2 5 1 } 2 (x 2 2).

SolutionBecause the equation is in point-slope form, you know

that the line has a slope of and passes through the point .

Plot the point Find a second point on the line using the . Draw a line through the points.

x

y

1

3

5

12121

23 3 5

Example 2 Graph an equation in point-slope form

1. Write an equation in point-slope form of the line that passes through the point (23, 5) and has a slope of 4.

2. Graph the equation y 1 1 5 2(x 2 1).

x

y

1

3

12121

23

23 3

Checkpoint Complete the following exercises.


Your Notes

Graph the equation y 2 2 5 1 } 2 (x 2 2).

SolutionBecause the equation is in point-slope form, you know

that the line has a slope of 1 } 2 and passes through the point (2, 2) .

Plot the point (2, 2) Find a second point on the line using the slope . Draw a line through the points.

x

y

1

3

5

12121

23 3 5

12

(2, 2)(4, 3)

Example 2 Graph an equation in point-slope form

1. Write an equation in point-slope form of the line that passes through the point (23, 5) and has a slope of 4.

y 2 5 5 4(x 1 3)

2. Graph the equation y 1 1 5 2(x 2 1).

x

y

1

3

12121

23

23 3

12 (2, 1)

(1, 21)



Your Notes

Homework


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

x

y

1

3

2121

23

23 3

(4, 2)

(22, 23)

SolutionStep 1 Find the slope of the line.

m 5 y2 2 y1 } x2 2 x1

5 2

2

5 5

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

Method 1 Use (22, 23). Method 2 Use (4, 2).

y 2 y1 5 m(x 2 x1) y 2 y1 5 m(x 2 x1)

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

y 5 x y 5 x

y 5 y 5

Example 3 Use point-slope form to write an equation

Your Notes

Homework


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

x

y

1

3

2121

23

23 3

(4, 2)

(22, 23)

SolutionStep 1 Find the slope of the line.

m 5 y2 2 y1 } x2 2 x1

5 22

23

4

2

2

2

5 26

25 5

5 } 6

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

Method 1 Use (22, 23). Method 2 Use (4, 2).

y 2 y1 5 m(x 2 x1) y 2 y1 5 m(x 2 x1)

y 1 3 5 5 } 6 (x 1 2) y 2 2 5 5 }

6 (x 2 4)

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

y 1 3 5 5 } 6 x 1 5 } 3 y 2 2 5 5 } 6 x 2 10 } 3

y 5 5 } 6 x 2 4 } 3 y 5 5 } 6 x 2 4 } 3

Example 3 Use point-slope form to write an equation

Your Notes

128 5.3 Focus On Functions • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Relate Arithmetic Sequences to Linear Functions Goal p Identify, graph, and write the general form

of arithmetic sequences.

VOCABULARY

Sequence

Arithmetic sequence

Common difference

Tell whether the sequence is arithmetic. If it is, find the next two terms.

26, 1, 8, 15, 22, …Solution

The first term is a1 5 . Find the of terms.

a2 2 a1 5 1 2 (26) 5 a32 a2 5 2 5

a4 2 a 5 2 5 a 2 a 5 2 5

The terms have a , d 5 . The sequence arithmetic. The next two terms are a6 5 and a7 5 .

Example 1 Identify an arithmetic sequence

Graph the sequence 26, 1, 8, 15, 22, …

Make a table pairing each term with its position number.

Solution

Position, x 1 2 3Term, y −6

Plot the pairs of numbers as points.

Example 2 Graph a sequence

The sequence in this example is a function. The domain is the set of positive numbers. The range is the set of terms.

Focus On FunctionsUse after Lesson 5.3

x

y

1

5

Your Notes

128 5.3 Focus On Functions • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Relate Arithmetic Sequences to Linear Functions Goal p Identify, graph, and write the general form

of arithmetic sequences.

VOCABULARY

Sequence an ordered list of numbers

Arithmetic sequence sequence with a constant difference between consecutive terms

Common difference the constant difference in an arithmetic sequence

Tell whether the sequence is arithmetic. If it is, find the next two terms.

26, 1, 8, 15, 22, …Solution

The first term is a1 5 26. Find the difference of consecutive terms.

a2 2 a1 5 1 2 (26) 5 7 a32 a2 5 8 2 1 5 7

a4 2 a 3 5 15 2 8 5 7 a 5 2 a4 5 22 2 15 5 7

The terms have a common difference , d 5 7 . The sequence is arithmetic. The next two terms are a6 5 29 and a7 5 36 .

Example 1 Identify an arithmetic sequence

Graph the sequence 26, 1, 8, 15, 22, …

Make a table pairing each term with its position number.

Solution

Position, x 1 2 3 4 5 Term, y −6 1 8 15 22

Plot the pairs of numbers as points.

Example 2 Graph a sequence

The sequence in this example is a function. The domain is the set of positive numbers. The range is the set of terms.

Focus On FunctionsUse after Lesson 5.3

x

y

1

5

Homework

Your Notes

Copyright © Holt McDougal. All rights reserved. 5.3 Focus On Functions • Algebra 1 Notetaking Guide 129

1. Tell whether the sequence is arithmetic. If it is, find the next two terms. 3, 22, 27, 212, 217, …

2. Graph the sequence 24, 21, 2, 5, 8,…. Identify the domain and range.

Domain:

Range:


KEY CONCEPT

Rule for an Arithmetic Sequence

The nth of an arithmetic with first term a1 and d is given by an 5 1 ( 2 ) .

Write a rule for the nth term of the sequence 25, 0, 5, 10, 15, …. Find a50.

Solution

The first term a1 5 . The common difference is d 5 .Substituting into the general rule an 5 1 ( 2 )

gives the rule an 5 1 ( 2 ) .

a50 5 1 ( 2 ) a50 5

Example 3 Write a rule for the nth term of a sequence

x

y

1

1

3. Write a rule for the nth term of the sequence and find a30 for: 18, 7, 24, 215, −26, …

a30 5


Homework

Your Notes

Copyright © Holt McDougal. All rights reserved. 5.3 Focus On Functions • Algebra 1 Notetaking Guide 129

1. Tell whether the sequence is arithmetic. If it is, find the next two terms. 3, 22, 27, 212, 217, …

Yes; next terms: 222, 227

2. Graph the sequence 24, 21, 2, 5, 8,…. Identify the domain and range.

Domain: 1, 2, 3, 4, 5,…

Range: 24, 21, 2, 5, 8,…


KEY CONCEPT

Rule for an Arithmetic Sequence

The nth term of an arithmetic sequence with first term a1 and common difference d is given by an 5 a1 1 ( n 2 1 ) d .

Write a rule for the nth term of the sequence 25, 0, 5, 10, 15, …. Find a50.

Solution

The first term a1 5 25 . The common difference is d 5 5 .Substituting into the general rule an 5 a1 1 ( n 2 1 ) d

gives the rule an 5 25 1 ( n 2 1)5.

a50 5 25 1 ( 50 2 1 )5 a50 5 240

Example 3 Write a rule for the nth term of a sequence

x

y

1

1

3. Write a rule for the nth term of the sequence and find a30 for: 18, 7, 24, 215, −26, …

an 5 18 1 (n 2 1)(211) a30 5 2301


5.4 Write Linear Equations inStandard FormGoal p Write equations in standard form.

Write two equations in standard form that are equivalent to 4x 1 2y 5 12.

SolutionTo write one equivalent equation, multiply each side by .

To write one equivalent equation, multiply each side by .

Example 1 Write equivalent equations in standard form

1. Write two equations in standard form that are equivalent to 6x 2 4y 5 6.



Your Notes


5.4 Write Linear Equations inStandard FormGoal p Write equations in standard form.

Write two equations in standard form that are equivalent to 4x 1 2y 5 12.

SolutionTo write one equivalent equation, multiply each side by 0.5 .

2x 1 y 5 6

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

8x 1 4y 5 24

Example 1 Write equivalent equations in standard form


3x 2 2y 5 3; 12x 2 8y 5 12


24x 1 2y 5 23; 224x 1 12y 5 218


Your Notes


Your Notes


Write an equation in standard form of the line shown.

x

y

1

3

2121

23

3 5 71

(2, 4)

(6, 24)


m 5 y2 2 y1 } x2 2 x1

5 2

2

5

5

Step 2 Write an equation in point-slope form. Use (2, 4).

y 2 y1 5 m(x 2 x1) Write point-slope form.

y 2 5 (x 2 ) Substitute for y1, for m, and

for x1.

Step 3 Rewrite the equation in standard form.

y 2 5 x 1 Distributive property

y 1 x 5 Collect variable terms on one side, constants on the other.

Example 2 Write an equation from a graph

All linear equations can be written in standard form, Ax 1 By 5 C.

Your Notes


Write an equation in standard form of the line shown.

x

y

1

3

2121

23

3 5 71

(2, 4)

(6, 24)


m 5 y2 2 y1 } x2 2 x1

5 6

24

2

4

2

2

5 4

28

5 22

Step 2 Write an equation in point-slope form. Use (2, 4).

y 2 y1 5 m(x 2 x1) Write point-slope form.

y 2 4 5 22 (x 2 2 ) Substitute 4 for y1, 22 for m, and 2 for x1.

Step 3 Rewrite the equation in standard form.

y 2 4 5 22 x 1 4 Distributive property

y 1 2 x 5 8 Collect variable terms on one side, constants on the other.

Example 2 Write an equation from a graph

All linear equations can be written in standard form, Ax 1 By 5 C.

Your Notes

3. Write an equation in standard form of the line through (3, 21) and (2, 24).


Write an equation of the specified line.

a. Line A

x

y

1

212321

23

25

1

(3, 2)

(24, 26)Line B

Line Ab. Line B

Solutiona. The x-coordinate of the

given point on Line A is. This means that all

points on the line have an x-coordinate of . An equation of the line is .

b. The y-coordinate of the given point on Line B is . This means that all points on the line have a y-coordinate of . An equation of the line is

.

Example 3 Write an equation of a line


Your Notes

3. Write an equation in standard form of the line through (3, 21) and (2, 24).

y 2 3x 5 210


Write an equation of the specified line.

a. Line A

x

y

1

212321

23

25

1

(3, 2)

(24, 26)Line B

Line Ab. Line B

Solutiona. The x-coordinate of the

given point on Line A is 3 . This means that all points on the line have an x-coordinate of 3 . An equation of the line is x 5 3 .

b. The y-coordinate of the given point on Line B is 26 . This means that all points on the line have a y-coordinate of 26 . An equation of the line is y 5 26 .

Example 3 Write an equation of a line


Your Notes

Find the missing coefficient in the equation of the line shown. Write the completed equation.

x

y

1

3

5

21

23

23 3 51

(2, 1)

Ax 1 5y 5 23

SolutionStep 1 Find the value of A. Substitute the coordinates of

the given point for x and y in the equation.

Ax 1 5y 5 23 Write equation.

A( ) 1 5( ) 5 23 Substitute for x and for y.

A 1 5 23 Simplify.

A 5 Subtract from each side.

A 5 Divide by .

Step 2 Complete the equation.

x 1 5y 5 23 Substitute for A.

Example 4 Complete an equation in standard form

4. Write equations of the horizontal and vertical lines that pass through (210, 5).

5. Find the missing coefficient in the equation of the line that passes through (22, 2). Write the completed equation.

6x 1 By 5 4


Homework


Your Notes

Find the missing coefficient in the equation of the line shown. Write the completed equation.

x

y

1

3

5

21

23

23 3 51

(2, 1)

Ax 1 5y 5 23

SolutionStep 1 Find the value of A. Substitute the coordinates of

the given point for x and y in the equation.

Ax 1 5y 5 23 Write equation.

A( 2 ) 1 5( 1 ) 5 23 Substitute 2 for x and 1 for y.

2 A 1 5 5 23 Simplify.

2 A 5 28 Subtract 5 from each side.

A 5 24 Divide by 2 .

Step 2 Complete the equation.

24 x 1 5y 5 23 Substitute 24 for A.

Example 4 Complete an equation in standard form

4. Write equations of the horizontal and vertical lines that pass through (210, 5).

Horizontal: y 5 5; Vertical: x 5 210

5. Find the missing coefficient in the equation of the line that passes through (22, 2). Write the completed equation.

6x 1 By 5 4

B 5 8; 6x 1 8y 5 4


Homework


Your Notes

5.5 Write Equations of Parallel and Perpendicular LinesGoal p Write equations of parallel and perpendicular lines.

VOCABULARY

Converse

Perpendicular lines

PARALLEL LINES

If two nonvertical lines have the same , then they are .

If two nonvertical lines are , then they have the same .

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

SolutionStep 1 Identify the slope. The graph of the given equation

has a slope of . So, the parallel line through (2, 4) has a slope of .

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



5 b Solve for b.

Step 3 Write an equation. Use y 5 mx 1 b.


Example 1 Write an equation of a parallel line


Your Notes

5.5 Write Equations of Parallel and Perpendicular LinesGoal p Write equations of parallel and perpendicular lines.

VOCABULARY

Converse A statement in which the hypothesis and conclusion of a conditional statement are interchanged

Perpendicular lines Two lines in a plane that intersect each other and form a right angle

PARALLEL LINES

If two nonvertical lines have the same slope , then they are parallel .

If two nonvertical lines are parallel , then they have the same slope .

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


has a slope of 4 . So, the parallel line through (2, 4) has a slope of 4 .




24 5 b Solve for b.

Step 3 Write an equation. Use y 5 mx 1 b.


Example 1 Write an equation of a parallel line


Your NotesPERPENDICULAR LINES

If two nonvertical lines have the slopes that are , then the lines are .

If two nonvertical lines are , then their slopes are .


Determine which of the following lines, if any, are parallel or perpendicular:

Line a: 12x 2 3y 5 3

Line b: y 5 4x 1 2

Line c: 4y 1 x 5 8

Solution

Find the slopes of the lines.

Line b: The equation is in slope-intercept form. The slope is .

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

Line a: 12x 2 3y 5 3

23y 5 1 3

y 5

Line c: 4y 1 x 5 8

4y 5 1 8

y 5

Lines a and b have a slope of , so they are .

Line c has a slope of , the negative reciprocal of , so it is to lines a and b.

Example 2 Determine parallel or perpendicular lines

Your NotesPERPENDICULAR LINES

If two nonvertical lines have the slopes that are negative reciprocals , then the lines are perpendicular .

If two nonvertical lines are perpendicular , then their slopes are negative reciprocals .


Determine which of the following lines, if any, are parallel or perpendicular:

Line a: 12x 2 3y 5 3

Line b: y 5 4x 1 2

Line c: 4y 1 x 5 8

Solution

Find the slopes of the lines.

Line b: The equation is in slope-intercept form. The slope is 4 .

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

Line a: 12x 2 3y 5 3

23y 5 212x 1 3

y 5 4x 2 1

Line c: 4y 1 x 5 8

4y 5 2x 1 8

y 5 2 1 } 4 x 1 2

Lines a and b have a slope of 4 , so they are parallel .

Line c has a slope of 2 1 } 4 , the negative reciprocal of 4 , so it is perpendicular to lines a and b.

Example 2 Determine parallel or perpendicular lines

Your Notes

1. Write an equation of the line that passes through (24, 6) and is parallel to the line y 5 23x 1 2.

2. Determine which of the following lines, if any, are parallel or perpendicular.

Line a: 4x 1 y 5 2

Line b: 5y 1 20x 5 10

Line c: 8y 5 2x 1 8


Determine if the following lines are perpendicular.

Line a: 6y 5 5x 1 8

Line b: 210y 5 12x 1 10

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

Line a: 6y 5 5x 1 8

y 5

Line b: 210y 5 12x 1 10

y 5

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

The two slopes negative reciprocals, so lines a and b perpendicular.

Example 3 Determine whether lines are perpendicular


Your Notes

1. Write an equation of the line that passes through (24, 6) and is parallel to the line y 5 23x 1 2.

y 5 23x 2 6

2. Determine which of the following lines, if any, are parallel or perpendicular.

Line a: 4x 1 y 5 2

Line b: 5y 1 20x 5 10

Line c: 8y 5 2x 1 8

Lines a and b are parallel with a slope of 24. Line c is perpendicular to lines a and b with

a slope of 1 } 4 .


Determine if the following lines are perpendicular.

Line a: 6y 5 5x 1 8

Line b: 210y 5 12x 1 10

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

Line a: 6y 5 5x 1 8

y 5 5 } 6 x 1 4 }

3

Line b: 210y 5 12x 1 10

y 5 2 6 } 5 x 2 1

The slope of line a is 5 } 6 . The slope of line b is 2 6 }

5 .

The two slopes are negative reciprocals, so lines a and b are perpendicular.

Example 3 Determine whether lines are perpendicular


Your Notes

Write an equation of the line that passes through (23, 4)

and is perpendicular to the line y 5 1 } 3 x 1 2.


has a slope of . Because the slopes of

perpendicular lines are negative reciprocals, the slope of the perpendicular line through (23, 4) is .




5 b Solve for b.

Step 3 Write an equation.



Example 4 Write an equation of a perpendicular line

3. Determine whether line a through (1, 3) and (3, 4) is perpendicular to line b through (1, 23) and (2, 25). Justify your answer using slopes.

4. Write an equation of the line that passes through (4, 22) and is perpendicular to the line y 5 5x 1 2.



Homework

Your Notes

Write an equation of the line that passes through (23, 4)

and is perpendicular to the line y 5 1 } 3 x 1 2.


has a slope of 1 } 3 . Because the slopes of

perpendicular lines are negative reciprocals, the

slope of the perpendicular line through (23, 4) is 23 .




25 5 b Solve for b.

Step 3 Write an equation.



Example 4 Write an equation of a perpendicular line

3. Determine whether line a through (1, 3) and (3, 4) is perpendicular to line b through (1, 23) and (2, 25). Justify your answer using slopes.

Line a: m 5 1 } 2 ; Line b: m 5 22; perpendicular

4. Write an equation of the line that passes through (4, 22) and is perpendicular to the line y 5 5x 1 2.

y 5 2 1 } 5 x 2 6 }

5



Homework

5.6 Fit a Line to DataGoal p Make scatter plots and write equations to

model data.

VOCABULARY

Scatter plot

Correlation

Line of fit

CORRELATION

• If y tends to increase as x increases, the paired data are said to have a correlation.

• If y tends to decrease as x increases, the paired data are said to have a correlation.

• If x and y have no apparent relationship, the paired data are said to have correlation.

Describe the correlation of data graphed in the scatter plot.

a.

x

y

2

6

2 6

b.

x

y

2

6

2 6

Solution

a. b. correlation correlation

Example 1 Describe the correlation of data

Your Notes


5.6 Fit a Line to DataGoal p Make scatter plots and write equations to

model data.

VOCABULARY

Scatter plot A graph used to determine whether there is a relationship between paired data

Correlation The relationship between two data sets

Line of fit A model used to represent the trend in data showing a positive or negative correlation

CORRELATION

• If y tends to increase as x increases, the paired data are said to have a positive correlation.

• If y tends to decrease as x increases, the paired data are said to have a negative correlation.

• If x and y have no apparent relationship, the paired data are said to have relatively no correlation.

Describe the correlation of data graphed in the scatter plot.

a.

x

y

2

6

2 6

b.

x

y

2

6

2 6

Solution

a. negative b. relatively no correlation correlation

Example 1 Describe the correlation of data

Your Notes


Your Notes

a. Make a scatter plot of the data in the table.

x 1 1.5 2 2 3 3.5 4

y 3 1 1 20.5 21 20.5 22

b. Describe the correlation of the data.

Solutiona. Treat the data as ordered

x

y

1

21

3

121 3

pairs. Plot the ordered pairs as in a coordinate plane.

b. The scatter plot shows a correlation.

Example 2 Make a scatter plot

USING A LINE OF FIT TO MODEL DATA

Step 1 Make a of the data.

Step 2 Decide whether the data can be modeled by a .

Step 3 Draw a line that appears to the data closely. There should be approximately as many points the line as it.

Step 4 Write an equation using points on the line. The points do not have to represent actual data pairs, but they must lie on the line of fit.


Your Notes

a. Make a scatter plot of the data in the table.

x 1 1.5 2 2 3 3.5 4

y 3 1 1 20.5 21 20.5 22

b. Describe the correlation of the data.

Solutiona. Treat the data as ordered

x

y

1

21

3

121 3

pairs. Plot the ordered pairs as points in a coordinate plane.

b. The scatter plot shows a negative correlation.

Example 2 Make a scatter plot

USING A LINE OF FIT TO MODEL DATA

Step 1 Make a scatter plot of the data.

Step 2 Decide whether the data can be modeled by a line .

Step 3 Draw a line that appears to fit the data closely. There should be approximately as many points above the line as below it.

Step 4 Write an equation using two points on the line. The points do not have to represent actual data pairs, but they must lie on the line of fit.


Your Notes

Game Attendance The table shows the average attendance at a school's varsity basketball games for various years. Write an equation that models the average attendance at varsity basketball games as a function of the number of years since 2000.

Year 2000 2001 2002 2003 2004 2005 2006

Avg. Game Attendance 488 497 525 567 583 621 688

Solution

Step 1 Make a

x

y

0 1 2 3 4 5 6

4500

500

550

600

650

700

Avera

ge a

tten

dan

ce

Years since 2000

Game Attendance

of the data. Let x represent the number of years since 2000. Let y represent average game attendance.

Step 2 Decide whether the data can be modeled by a line. Because the scatter plot shows a correlation, you can fit a line to the data.

Step 3 Draw a line that appears to fit the points in the scatter plot .

Step 4 Write an equation using two points on the line. Use (1, 493) and (5, 621).

Find the of the line.

m 5 y2 2 y1 } x2 2 x1

5 2

2

5

5

Example 3 Write an equation to model data


Your Notes

Game Attendance The table shows the average attendance at a school's varsity basketball games for various years. Write an equation that models the average attendance at varsity basketball games as a function of the number of years since 2000.

Year 2000 2001 2002 2003 2004 2005 2006

Avg. Game Attendance 488 497 525 567 583 621 688

Solution

Step 1 Make a scatter plot

x

y

0 1 2 3 4 5 6

4500

500

550

600

650

700

Avera

ge a

tten

dan

ce

Years since 2000

Game Attendance

of the data. Let x represent the number of years since 2000. Let y represent average game attendance.

Step 2 Decide whether the data can be modeled by a line. Because the scatter plot shows a positive correlation, you can fit a line to the data.

Step 3 Draw a line that appears to fit the points in the scatter plot closely .

Step 4 Write an equation using two points on the line. Use (1, 493) and (5, 621).

Find the slope of the line.

m 5 y2 2 y1 } x2 2 x1

5 5

621

1

493

2

2

5 4

128

5 32

Example 3 Write an equation to model data


Your Notes

Homework

1. Make a scatter plot of the data in the table. Describe the correlation of the data.

x 1 2 2 3 4 5

y 5 5 6 7 8 8

x

y

1

3

5

7

121 3 5 7

2. Use the data in the table to write an equation that models y as a function of x.

x 1 2 3 4 5 6

y 65 76 82 86 92 97


Find the y-intercept of the line. Use the point (5, 621).



5 b Solve for b.

An equation of the line of fit is .

The average attendance y of varsity basketball games can be modeled by the function where x is the number of years since 2000.


Your Notes

Homework

1. Make a scatter plot of the data in the table. Describe the correlation of the data.

x 1 2 2 3 4 5

y 5 5 6 7 8 8

x

y

1

3

5

7

121 3 5 7

positive correlation

2. Use the data in the table to write an equation that models y as a function of x.

x 1 2 3 4 5 6

y 65 76 82 86 92 97

y 5 6x 1 62


Find the y-intercept of the line. Use the point (5, 621).



461 5 b Solve for b.

An equation of the line of fit is y 5 32x 1 461.

The average attendance y of varsity basketball games can be modeled by the function y 5 32x 1 461 where x is the number of years since 2000.


5.7 Predict with Linear ModelsGoal p Make predictions using best-fitting lines.

VOCABULARY

Best-fitting line

Interpolation

Extrapolation

Zero of a function

Your Notes


5.7 Predict with Linear ModelsGoal p Make predictions using best-fitting lines.

VOCABULARY

Best-fitting line The line that most closely follows a trend in data

Interpolation Use of a line or its equation to approximate a value between two known values

Extrapolation Use of a line or its equation to approximate a value outside the range of known values

Zero of a function A zero of a function y 5 f (x) is an x-value for which f (x) 5 0 (or y 5 0).

Your Notes


Your Notes


NFL Salaries The table shows the average National Football League (NFL) player's salary (in thousands of dollars) from 1997 to 2001.

Year 1997 1999 2000 2001

Average Player's Salary (in thousands of dollars) 585 708 787 986

a. Make a scatter plot of the data.

b. Find an equation that models the average NFL player's salary (in thousands of dollars) as a function of the number of years since 1997.

c. Approximate the average NFL player's salary in 1998.

Solutiona. Enter the data into lists on

a graphing calculator. Make a scatter plot, letting the number of years since 1997 be the (0, 2, 3, 4) and the average player's salary be the .

b. Perform

X=1 Y=649

Y1=94X+555

using the paired data. The equation of the best-fitting line is y 5 .

c. Graph the best-fitting line. Use the trace feature and the arrow keys to find the value of the equation when x 5 .

The average NFL player's salary in 1998 was thousand dollars.

Example 1 Interpolate using an equation

The slope, of the best-fitting line indicates that an NFL player’s salary increased by about thousand dollars per year.

Your Notes


NFL Salaries The table shows the average National Football League (NFL) player's salary (in thousands of dollars) from 1997 to 2001.

Year 1997 1999 2000 2001


a. Make a scatter plot of the data.

b. Find an equation that models the average NFL player's salary (in thousands of dollars) as a function of the number of years since 1997.

c. Approximate the average NFL player's salary in 1998.

Solutiona. Enter the data into lists on

a graphing calculator. Make a scatter plot, letting the number of years since 1997 be the x-values (0, 2, 3, 4) and the average player's salary be the y-values .

b. Perform linear regression

X=1 Y=649

Y1=94X+555

using the paired data. The equation of the best-fitting line is y 5 94x 1 555 .

c. Graph the best-fitting line. Use the trace feature and the arrow keys to find the value of the equation when x 5 1 .

The average NFL player's salary in 1998 was 649 thousand dollars.

Example 1 Interpolate using an equation

The slope, 94 of the best-fitting line indicates that an NFL player’s salary increased by about 94 thousand dollars per year.

Your Notes

NFL Salaries Look back at Example 1.

a. Use the equation from Example 1 to approximate the average NFL player's salary in 2002 and 2003.

b. In 2002, the average NFL player's salary was actually 1180 thousand dollars. In 2003, the average NFL player's salary was actually 1259 thousand dollars. Describe the accuracy of the extrapolations made in part (a).

Solution Y1(5) 1025Y1(6) 1119 a. Evaluate the equation of the

best-fitting line from Example 1 for x 5 and x 5 .

The model predicts the average NFL player's salary as thousand dollars in 2002 and thousand dollars in 2003.

b. The differences between the predicted average NFL player's salary and the actual average NFL player's salary in 2002 and 2003 are thousand dollars and thousand dollars, respectively. The equation of the best-fitting line gives a less accurate prediction for the years outside of the given years.

Example 2 Extrapolate using an equation

RELATING SOLUTIONS OF EQUATIONS, ZEROS OF FUNCTIONS, AND x-INTERCEPTS OF GRAPHS

In Chapter 3,you learnedto solve an equation like

4x 2 4 5 0:

4x 2 4 5 0

4x 5

x 5

The solution of 4x 2 4 5 0 is .

In Chapter 4,you found the

of the graph ofa function like y 5 4x 2 4:

x

y1

2121

23

31

Now you arefinding the zero of a function like

f(x) 5 4x 2 4:

f(x) 5 0

5 0

x 5

The zero of f(x) 5 4x 2 4 is .


Your Notes

NFL Salaries Look back at Example 1.

a. Use the equation from Example 1 to approximate the average NFL player's salary in 2002 and 2003.

b. In 2002, the average NFL player's salary was actually 1180 thousand dollars. In 2003, the average NFL player's salary was actually 1259 thousand dollars. Describe the accuracy of the extrapolations made in part (a).

Solution Y1(5) 1025Y1(6) 1119 a. Evaluate the equation of the

best-fitting line from Example 1 for x 5 5 and x 5 6 .

The model predicts the average NFL player's salary as 1025 thousand dollars in 2002 and 1119 thousand dollars in 2003.

b. The differences between the predicted average NFL player's salary and the actual average NFL player's salary in 2002 and 2003 are 155 thousand dollars and 140 thousand dollars, respectively. The equation of the best-fitting line gives a less accurate prediction for the years outside of the given years.

Example 2 Extrapolate using an equation

RELATING SOLUTIONS OF EQUATIONS, ZEROS OF FUNCTIONS, AND x-INTERCEPTS OF GRAPHS

In Chapter 3,you learnedto solve an equation like

4x 2 4 5 0:

4x 2 4 5 0

4x 5 4

x 5 1

The solution of 4x 2 4 5 0 is 1 .

In Chapter 4,you found the x-intercept of the graph ofa function like y 5 4x 2 4:

x

y1

2121

23

31

Now you arefinding the zero of a function like

f(x) 5 4x 2 4:

f(x) 5 0

4x 2 4 5 0

x 5 1

The zero of f(x) 5 4x 2 4 is 1 .


Your Notes

Homework

Public Transit The percentage y of people in the U.S. that use public transit to commute to work can be modeled by the function y 5 20.045x 1 5.7 where x is the number of years since 1983. Find the zero of the function. Explain what the zero means in this situation.

Solution

Substitute for y in the equation of the and solve for x.

y 5 20.045x 1 5.7 Write the equation.

5 20.045x 1 5.7 Substitute for y.

Solve for x.

The zero of the function is about . The function has a slope, which means that the percentage of people using public transit to commute to work is . According to the model there will be no people who use public transit to commute to work years after , or in .

Example 3 Find the zero of a function

1. Baseball Salaries The table shows the average major league baseball player's salary (in thousands of dollars) from 1997 to 2001.

Year 1997 1999 2000 2001


Find an equation that models the average major league baseball player's salary (in thousands of dollars) as a function of the number of years since 1997. Approximate the average major league baseball player's salary in 1998, 2002, and 2003.



Your Notes

Homework

Public Transit The percentage y of people in the U.S. that use public transit to commute to work can be modeled by the function y 5 20.045x 1 5.7 where x is the number of years since 1983. Find the zero of the function. Explain what the zero means in this situation.

Solution

Substitute 0 for y in the equation of the best-fitting line and solve for x.

y 5 20.045x 1 5.7 Write the equation.

0 5 20.045x 1 5.7 Substitute 0 for y.

x ø 127 Solve for x.

The zero of the function is about 127 . The function has a negative slope, which means that the percentage of people using public transit to commute to work is decreasing . According to the model there will be no people who use public transit to commute to work 127 years after 1983 , or in 2110 .

Example 3 Find the zero of a function

1. Baseball Salaries The table shows the average major league baseball player's salary (in thousands of dollars) from 1997 to 2001.

Year 1997 1999 2000 2001


Find an equation that models the average major league baseball player's salary (in thousands of dollars) as a function of the number of years since 1997. Approximate the average major league baseball player's salary in 1998, 2002, and 2003.

Possible equation of the line of fit: y 5201x 1 1293

1998: 1494; 2002: 2298; 2003: 2499



Words to ReviewGive an example of the vocabulary word.

Point-slope form

Arithmetic sequence

Converse

Scatter plot

Line of fit

Common difference

Sequence

Perpendicular lines

Correlation

Best-fitting line

146 Words to Review • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Words to ReviewGive an example of the vocabulary word.

Point-slope form

y 2 y1 5 m(x 2 x1)

Arithmetic sequence

{1, 31, 61, 91, 121,... }

Converse

If two nonvertical lines have the same slope, then they are parallel.

If two nonvertical lines are parallel, then they have the same slope.

Scatter plot

x

y

2

6

2 6

Line of fit

A model used to represent the trend in data showing a positive or negative correlation.

Common difference

23 for { 7, 4, 1, –2, –5,... }

Sequence

{31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31}

Perpendicular lines

y 5 5x 2 3

y 5 2 1 } 5 x 1 2

Correlation

The scatter plot shows a negative correlation.

x

y

2

6

2 6

Best-fitting line

A line that best fits the data points on a scatter plot.

146 Words to Review • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Copyright © Holt McDougal. All rights reserved. Words to Review • Algebra 1 Notetaking Guide 147

Review your notes and Chapter 5 by using the Chapter Review on pages 352–355 of your textbook.

Interpolation

Zero of a function

Extrapolation

Copyright © Holt McDougal. All rights reserved. Words to Review • Algebra 1 Notetaking Guide 147

Review your notes and Chapter 5 by using the Chapter Review on pages 352–355 of your textbook.

InterpolationEstimating a value for 1997 when you are given values for 1996 and 1998.

Zero of a function

Extrapolation

Estimating a value for 2002 when you are given values for 1999 to 2001.

The zero of f (x) 5 4x 2 4 is 1.

5.1 write linear equations in slope-intercept form · write linear equations in slope-intercept...

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