Key Features of Linear Functions

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Key Ideas Easy 1 · Slope and y-int Easy 2 · x-int Easy 3 · Table Medium 1 · Slope Medium 2 · Intercepts Hard 1 · Horiz/Vert

What to Remember

Slope-intercept form

In y = mx + b, m is the slope and (0, b) is the y-intercept. A zero is the x-intercept, where y = 0.

From a table

m = y₂ − y₁x₂ − x₁. If x = 0 is in the table, that y-value is the y-intercept.

Standard form

For Ax + By = C, solve for y to read the slope. For intercepts, start with a vertical x, y table. Use 0 for x in one row and 0 for y in the other.

Horizontal and vertical lines

A horizontal line has slope 0 and equation y = c. A vertical line has an undefined slope and equation x = a. A vertical line that is not the y-axis has no y-intercept.

Easy 1

Slope and y-int (y = mx + b)

Read the slope and the y-intercept from slope-intercept form.

Example

For y = 3x + 2, name the slope and the y-intercept.

y = 3x + 2

y = mx + b

m = 3

b = 2

The y-intercept is (0, 2).

Answer: slope 3; (0, 2)

For y = 5x − 4, name the slope and the y-intercept.

Show Answer

y = 5x − 4

y = mx + b

m = 5

b = −4

The y-intercept is (0, −4).

Answer: slope 5; (0, −4)

For y = −2x + 7, name the slope and the y-intercept.

Show Answer

y = −2x + 7

y = mx + b

m = −2

b = 7

The y-intercept is (0, 7).

Answer: slope −2; (0, 7)

For y = 13x + 6, name the slope and the y-intercept.

Show Answer

y = 13x + 6

y = mx + b

m = 13

b = 6

The y-intercept is (0, 6).

Answer: slope 13; (0, 6)

For y = 4x, name the slope and the y-intercept.

Show Answer

y = 4x

y = mx + b

m = 4

b = 0

The y-intercept is (0, 0).

Answer: slope 4; (0, 0)

For y = −25x − 1, name the slope and the y-intercept.

Show Answer

y = −25x − 1

y = mx + b

m = −25

b = −1

The y-intercept is (0, −1).

Answer: slope −25; (0, −1)

Easy 2

x-int (y = mx + b)

Find the zero of an equation in slope-intercept form.

Example

Find the zero of y = 2x − 8.

Set y = 0.

0 = 2x − 8

8 = 2x

Divide by 2.

x = 4

The zero is (4, 0).

Answer: (4, 0)

Find the zero of y = 3x − 12.

Show Answer

Set y = 0.

0 = 3x − 12

12 = 3x

Divide by 3.

x = 4

The zero is (4, 0).

Answer: (4, 0)

Find the zero of y = −2x + 10.

Show Answer

Set y = 0.

0 = −2x + 10

−10 = −2x

Divide by −2.

x = 5

The zero is (5, 0).

Answer: (5, 0)

Find the zero of y = x + 6.

Show Answer

Set y = 0.

0 = x + 6

−6 = x

x = −6

The zero is (−6, 0).

Answer: (−6, 0)

Find the zero of y = 12x − 3.

Show Answer

Set y = 0.

0 = 12x − 3

3 = 12x

Multiply both sides by 2.

6 = x

The zero is (6, 0).

Answer: (6, 0)

Find the zero of y = −4x − 8.

Show Answer

Set y = 0.

0 = −4x − 8

8 = −4x

Divide by −4.

x = −2

The zero is (−2, 0).

Answer: (−2, 0)

Easy 3

Slope and y-int (table)

The y-intercept is one of the rows in the table.

Example

The table shows a linear function. Find the slope and the y-intercept.

x	0	1	2	3
y	4	7	10	13

m = y₂ − y₁x₂ − x₁

m = 3

When x = 0, y = 4.

The y-intercept is (0, 4).

Answer: slope 3; (0, 4)

The table shows a linear function. Find the slope and the y-intercept.

x	0	1	2	3
y	−2	1	4	7

Show Answer

m = y₂ − y₁x₂ − x₁

m = 3

When x = 0, y = −2.

The y-intercept is (0, −2).

Answer: slope 3; (0, −2)

The table shows a linear function. Find the slope and the y-intercept.

x	0	2	4	6
y	5	9	13	17

Show Answer

m = y₂ − y₁x₂ − x₁

m = 42

m = 2

When x = 0, y = 5.

The y-intercept is (0, 5).

Answer: slope 2; (0, 5)

The table shows a linear function. Find the slope and the y-intercept.

x	0	1	2	3
y	8	5	2	−1

Show Answer

m = y₂ − y₁x₂ − x₁

m = −3

When x = 0, y = 8.

The y-intercept is (0, 8).

Answer: slope −3; (0, 8)

The table shows a linear function. Find the slope and the y-intercept.

x	0	3	6	9
y	1	2	3	4

Show Answer

m = y₂ − y₁x₂ − x₁

m = 13

When x = 0, y = 1.

The y-intercept is (0, 1).

Answer: slope 13; (0, 1)

The table shows a linear function. Find the slope and the y-intercept.

x	−1	0	1	2
y	9	6	3	0

Show Answer

m = y₂ − y₁x₂ − x₁

m = −3

When x = 0, y = 6.

The y-intercept is (0, 6).

Answer: slope −3; (0, 6)

Medium 1

slope (Ax + By = C)

Solve for y, then read the slope.

Example

Find the slope of 4x + 2y = 10.

4x + 2y = 10

2y = −4x + 10

Divide by 2.

y = −2x + 5

m = −2

Answer: m = −2

Find the slope of x + 2y = 10.

Show Answer

x + 2y = 10

2y = −x + 10

Divide by 2.

y = −12x + 5

m = −12

Answer: m = −12

Find the slope of 3x − y = 6.

Show Answer

3x − y = 6

−y = −3x + 6

Divide by −1.

y = 3x − 6

m = 3

Answer: m = 3

Find the slope of 4x + 2y = 8.

Show Answer

4x + 2y = 8

2y = −4x + 8

Divide by 2.

y = −2x + 4

m = −2

Answer: m = −2

Find the slope of −2x + 5y = 15.

Show Answer

−2x + 5y = 15

5y = 2x + 15

Divide by 5.

y = 25x + 3

m = 25

Answer: m = 25

Find the slope of 6x − 3y = 9.

Show Answer

6x − 3y = 9

−3y = −6x + 9

Divide by −3.

y = 2x − 3

m = 2

Answer: m = 2

Medium 2

intercepts (Ax + By = C)

Start with a vertical table. Next to x = 0, find y. Next to y = 0, find x.

Example

Find the intercepts of 2x + 4y = 12.

x	y
0		Find y, when x = 0
	0	Find x, when y = 0

Find y, when x = 0.

4y = 12

Divide by 4.

y = 3

The y-intercept is (0, 3).

Find x, when y = 0.

2x = 12

Divide by 2.

x = 6

The x-intercept is (6, 0).

Answer: (6, 0) and (0, 3)

Find the intercepts of 3x + y = 9.

Show Answer

x	y
0		Find y, when x = 0
	0	Find x, when y = 0

Find y, when x = 0.

y = 9

The y-intercept is (0, 9).

Find x, when y = 0.

3x = 9

Divide by 3.

x = 3

The x-intercept is (3, 0).

Answer: (3, 0) and (0, 9)

Find the intercepts of x + 2y = 10.

Show Answer

x	y
0		Find y, when x = 0
	0	Find x, when y = 0

Find y, when x = 0.

2y = 10

Divide by 2.

y = 5

The y-intercept is (0, 5).

Find x, when y = 0.

x = 10

The x-intercept is (10, 0).

Answer: (10, 0) and (0, 5)

Find the intercepts of 2x − y = 8.

Show Answer

x	y
0		Find y, when x = 0
	0	Find x, when y = 0

Find y, when x = 0.

−y = 8

Multiply both sides by −1.

y = −8

The y-intercept is (0, −8).

Find x, when y = 0.

2x = 8

Divide by 2.

x = 4

The x-intercept is (4, 0).

Answer: (4, 0) and (0, −8)

Find the intercepts of 5x + 2y = 20.

Show Answer

x	y
0		Find y, when x = 0
	0	Find x, when y = 0

Find y, when x = 0.

2y = 20

Divide by 2.

y = 10

The y-intercept is (0, 10).

Find x, when y = 0.

5x = 20

Divide by 5.

x = 4

The x-intercept is (4, 0).

Answer: (4, 0) and (0, 10)

Find the intercepts of 4x − 3y = 12.

Show Answer

x	y
0		Find y, when x = 0
	0	Find x, when y = 0

Find y, when x = 0.

−3y = 12

Divide by −3.

y = −4

The y-intercept is (0, −4).

Find x, when y = 0.

4x = 12

Divide by 4.

x = 3

The x-intercept is (3, 0).

Answer: (3, 0) and (0, −4)

Hard 1

Horizontal and Vertical Lines

Read the slope and the y-intercept from the table.

Example

The table shows a line. Find the slope and the y-intercept.

x	−1	0	1	2
y	6	6	6	6

Every y-value is 6.

m = 0

When x = 0, y = 6.

The y-intercept is (0, 6).

The equation is y = 6.

Answer: slope 0; (0, 6)

Example

The table shows a line. Find the slope and the y-intercept.

x	2	2	2	2
y	−1	0	3	5

Every point has x = 2.

The run is 0, so the slope is undefined.

The line does not cross the y-axis, so there is no y-intercept.

The equation is x = 2.

Answer: undefined slope; no y-intercept; x = 2

The table shows a line. Find the slope and the y-intercept.

x	0	1	2	3
y	4	4	4	4

Show Answer

Every y-value is 4.

m = 0

When x = 0, y = 4.

The y-intercept is (0, 4).

The equation is y = 4.

Answer: slope 0; (0, 4)

The table shows a line. Find the slope and the y-intercept.

x	−2	0	2	4
y	−3	−3	−3	−3

Show Answer

Every y-value is −3.

m = 0

When x = 0, y = −3.

The y-intercept is (0, −3).

The equation is y = −3.

Answer: slope 0; (0, −3)

The table shows a line. Find the slope and the y-intercept.

x	5	5	5	5
y	−2	0	1	4

Show Answer

Every point has x = 5.

The run is 0, so the slope is undefined.

The line does not cross the y-axis, so there is no y-intercept.

The equation is x = 5.

Answer: undefined slope; no y-intercept; x = 5

The table shows a line. Find the slope and the y-intercept.

x	0	2	4
y	0	0	0

Show Answer

Every y-value is 0.

m = 0

When x = 0, y = 0.

The y-intercept is (0, 0).

The equation is y = 0.

Answer: slope 0; (0, 0)

The table shows a line. Find the slope and the y-intercept.

x	−4	−4	−4	−4
y	1	3	6	8

Show Answer

Every point has x = −4.

The run is 0, so the slope is undefined.

The line does not cross the y-axis, so there is no y-intercept.

The equation is x = −4.

Answer: undefined slope; no y-intercept; x = −4

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