PAPER · REASONING · PRACTICE

Linear functions: coefficients, graphs and determination

Relate slope and intercept to the graph, distinguish constant functions, and determine a line from two distinct inputs.

40 minutes · Year 8, lower volume · text pp. 59–66 · textbook lessons 30–33

Textbook reference: Year 8, lower volume · Chapter 19: Linear functions · lessons 30–33

Chapter core practice: use after the referenced textbook lessons, or for chapter review. See the lower-volume plan

One worked example, one guided attempt, eight core questions, four follow-ups and two extensions. Write on paper; the teacher controls the reveal.

Teacher reference Arrange in My Classroom

Lesson routePhase

4′Check prerequisites
5′Observe and predict
8′Worked example and guided attempt
15′Independent practice
5′Discuss and correct
3′Exit questions

Observe · explain

A question before practising

3 min: With b=1 fixed, predict the stationary point as k changes in y=kx+b. 8 min: In the existing one-line inquiry, save k=1,2,−1 with b=1. Then fix k=2 and save b=−2,0,3. Compare the saved cases and draw their lines on paper; the engine shows one line at a time. 5 min: Explain the common point (0,1) in the first set and parallel lines in the second. Use x=−1 to test “a larger k always gives a larger y”. Treat k=0 as a constant-function boundary; in this course a linear function requires k≠0. 2 min: Individually state the intercept and increasing/decreasing behaviour of y=−2x+3.

Organise the key idea

Here a linear function has $y=kx+b,\ k\neq0$ ; a direct-proportion function also has $b=0$ .

Open the original chapter Open the interactive inquiry

Use the lesson question here for paper discussion; the original activity opens separately.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: follow the full 18-minute investigation; record predictions before checking.

Worked example1

Find the linear function through $(0,2)$ and $(3,8)$ . Show the equations used to find k and b.

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Substitute both coordinate pairs into y=kx+b.
The solution is $k=2,\ b=2$ ; check both points.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Guided attempt2

Find the linear function through $(1,1)$ and $(3,5)$ . Show the equations used to find k and b.

First point: $k+b=\square$ .
Second point: $3k+b=\square$ .
Subtract: $2k=\square$ ; then $k=\square,\quad b=1-\square$ .

Substitute (1,1) and (3,5) into y=kx+b to obtain two equations.
Subtract the first equation from the second: $(3k+b)-(k+b)=5-1$ .
After finding k=2, put it into 1=k+b to find b, then check both points.

Substitute both coordinate pairs into y=kx+b.
The solution is $k=2,\ b=-1$ ; check both points.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite3

Is the given relation a direct-proportion function? State k and b.

y=3x

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Compare with y=kx+b and inspect the constant term.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite4

For the given relation, find y when $x=0$ .

y=-2x+4

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Substitute $x=0$ into the complete relation.
$-2\times(0)+(4)=4$ .

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice5

Find the x-intercept and explain why y is set to zero.

y=-3x+6

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Every point on the x-axis has y=0.
Solve $-3x+6=0$ to obtain $x=2$ .

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice6

Draw the graph of the given function. Mark the y-intercept and at least one more point.

y=2x-1

Use the stated conditions and the labelled scales.

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

At x=0, y=−1; at x=1, y=1.
Plot the two distinct points and join them with a straight line.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice7

Find the linear function through $(1,3)$ and $(3,1)$ . Show the equations used to find k and b.

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Substitute both coordinate pairs into y=kx+b.
The solution is $k=-1,\ b=4$ ; check both points.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice8

A linear function y=kx+3 passes through (2,7). Find k and check that the result is a linear function.

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

7=2k+3, hence 2k=4.
k=2 is nonzero.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question9

Under this chapter’s definition, is y=5 a linear function with nonzero k? Classify it accurately.

y=5

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

A horizontal line can still represent a function.
The extra nonzero-slope requirement is not met.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question10

Find the x-intercept and explain why y is set to zero.

y=3x-6

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Every point on the x-axis has y=0.
Solve $3x-6=0$ to obtain $x=2$ .

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice11

As x increases, does y increase or decrease? Explain using k.

y=-x+2

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Each increase of 1 in x changes y by −1.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice12

For the given relation, find y when $x=0$ .

y=2x-4

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Substitute $x=0$ into the complete relation.
$2\times(0)+(-4)=-4$ .

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice13

Find the linear function through $(0,0)$ and $(2,6)$ . Show the equations used to find k and b.

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Substitute both coordinate pairs into y=kx+b.
The solution is $k=3,\ b=0$ ; check both points.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice14

Find the linear function through $(-1,4)$ and $(2,-2)$ . Show the equations used to find k and b.

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Substitute both coordinate pairs into y=kx+b.
The solution is $k=-2,\ b=2$ ; check both points.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension15

The lines y=x+3, y=2x+3 and y=−x+3 have different slopes. Find their common point and explain why the coefficient of x disappears there.

Use the stated conditions and the labelled scales.

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

All three intercepts equal 3.
Substitute (0,3) into each original equation.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension16

A learner says every straight line is the graph of y=kx+b. Use the two listed points on x=2 to explain the missing exception.

x	y
2	0
2	3

Use the stated conditions and the labelled scales.

Read each ordered pair as input, output.
One output per input is required.
Repeated output values are allowed.

Check repeated inputs, not merely repeated outputs.
The repeated input has conflicting outputs.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Preparation and assessment notes

Prerequisites

Function representations, coordinates and simultaneous equations.

Look for in students’ work

Use the two prerequisite checks to choose support. Project the worked and guided tasks, then collect the eight paper core questions. Follow-ups and extensions are optional; accept other complete valid methods. This page does not save learners’ paper answers.

Collect the paper exit questions and note completion and common errors. This page does not grade or store paper answers.

Paper for the class

Download all worksheets (ZIP · 318 PDFs · 47.2 MB)

All 53 Year 7–8 upper- and lower-volume packs in Traditional Chinese and English. Includes teacher answers; student core, further practice and teacher papers are in separate folders.

Core student worksheet: two A4 pages. Further practice is separate. Teacher copies include reference reasoning.

Print edition

Textbook reference and authorship

Original chapter-core practice after the cited lower-volume textbook lessons. References identify skills; textbook exercises are not reproduced.

大湾区初中数学三维课堂（八年级下册）正文.pdf

Year 8, lower volume · text pp. 59–66 · textbook lessons 30–33

Original questions and explanations; the source textbook and answer PDFs are not reproduced here.

Linear functions: coefficients, graphs and determination

Year 8, lower volume · text pp. 59–66 · textbook lessons 30–33 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

Find the linear function through $(0,2)$ and $(3,8)$ . Show the equations used to find k and b.

Guided attempt2

Find the linear function through $(1,1)$ and $(3,5)$ . Show the equations used to find k and b.

First point: $k+b=\square$ .
Second point: $3k+b=\square$ .
Subtract: $2k=\square$ ; then $k=\square,\quad b=1-\square$ .

Prerequisite3

Is the given relation a direct-proportion function? State k and b.

y=3x

Prerequisite4

For the given relation, find y when $x=0$ .

y=-2x+4

Linear functions: coefficients, graphs and determination

Year 8, lower volume · text pp. 59–66 · textbook lessons 30–33 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

Find the x-intercept and explain why y is set to zero.

y=-3x+6

Independent practice6

Draw the graph of the given function. Mark the y-intercept and at least one more point.

y=2x-1

Use the stated conditions and the labelled scales.

Independent practice7

Find the linear function through $(1,3)$ and $(3,1)$ . Show the equations used to find k and b.

Independent practice8

A linear function y=kx+3 passes through (2,7). Find k and check that the result is a linear function.

Exit question9

Under this chapter’s definition, is y=5 a linear function with nonzero k? Classify it accurately.

y=5

Exit question10

Find the x-intercept and explain why y is set to zero.

y=3x-6

Linear functions: coefficients, graphs and determination — Further practice

Year 8, lower volume · text pp. 59–66 · textbook lessons 30–33

Further practice11

As x increases, does y increase or decrease? Explain using k.

y=-x+2

Further practice12

For the given relation, find y when $x=0$ .

y=2x-4

Further practice13

Find the linear function through $(0,0)$ and $(2,6)$ . Show the equations used to find k and b.

Further practice14

Find the linear function through $(-1,4)$ and $(2,-2)$ . Show the equations used to find k and b.

Extension15

The lines y=x+3, y=2x+3 and y=−x+3 have different slopes. Find their common point and explain why the coefficient of x disappears there.

Use the stated conditions and the labelled scales.

Extension16

A learner says every straight line is the graph of y=kx+b. Use the two listed points on x=2 to explain the missing exception.

x	y
2	0
2	3

Use the stated conditions and the labelled scales.

Linear functions: coefficients, graphs and determination — Teacher answers

Year 8, lower volume · text pp. 59–66 · textbook lessons 30–33

Ordinary route: prerequisites 4, short inquiry 5, worked/guided 8, independent 15, correction 5, exit 3 minutes. Advanced route: inquiry 18, organise 4, selected practice 13, exit 5 minutes.

Prerequisites: Function representations, coordinates and simultaneous equations.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: complete the full investigation.

3′ With b=1 fixed, predict the stationary point as k changes in y=kx+b.
8′ In the existing one-line inquiry, save k=1,2,−1 with b=1. Then fix k=2 and save b=−2,0,3. Compare the saved cases and draw their lines on paper; the engine shows one line at a time.
5′ Explain the common point (0,1) in the first set and parallel lines in the second. Use x=−1 to test “a larger k always gives a larger y”. Treat k=0 as a constant-function boundary; in this course a linear function requires k≠0.
2′ Individually state the intercept and increasing/decreasing behaviour of y=−2x+3.

Worked example1

Find the linear function through $(0,2)$ and $(3,8)$ . Show the equations used to find k and b.

Reference answer

$y=2x+2$

y=2x+2

Substitute both coordinate pairs into y=kx+b.
The solution is $k=2,\ b=2$ ; check both points.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Guided attempt2

Find the linear function through $(1,1)$ and $(3,5)$ . Show the equations used to find k and b.

First point: $k+b=\square$ .
Second point: $3k+b=\square$ .
Subtract: $2k=\square$ ; then $k=\square,\quad b=1-\square$ .

Reference answer

$y=2x-1$

y=2x-1

Substitute both coordinate pairs into y=kx+b.
The solution is $k=2,\ b=-1$ ; check both points.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite3

Is the given relation a direct-proportion function? State k and b.

y=3x

Reference answer

Yes; k=3≠0 and b=0.

Compare with y=kx+b and inspect the constant term.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite4

For the given relation, find y when $x=0$ .

y=-2x+4

Reference answer

$y=4$

y=4

Substitute $x=0$ into the complete relation.
$-2\times(0)+(4)=4$ .

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice5

Find the x-intercept and explain why y is set to zero.

y=-3x+6

Reference answer

$x=2$

x=2

Every point on the x-axis has y=0.
Solve $-3x+6=0$ to obtain $x=2$ .

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice6

Draw the graph of the given function. Mark the y-intercept and at least one more point.

y=2x-1

Use the stated conditions and the labelled scales.

Reference answer

The line passes through (0,−1) and (1,1), with slope 2.

At x=0, y=−1; at x=1, y=1.
Plot the two distinct points and join them with a straight line.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Use the stated conditions and the labelled scales.

Independent practice7

Find the linear function through $(1,3)$ and $(3,1)$ . Show the equations used to find k and b.

Reference answer

$y=-1x+4$

y=-1x+4

Substitute both coordinate pairs into y=kx+b.
The solution is $k=-1,\ b=4$ ; check both points.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice8

A linear function y=kx+3 passes through (2,7). Find k and check that the result is a linear function.

Reference answer

k=2≠0; the function is y=2x+3.

k=2

7=2k+3, hence 2k=4.
k=2 is nonzero.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question9

Under this chapter’s definition, is y=5 a linear function with nonzero k? Classify it accurately.

y=5

Reference answer

It is a constant function with k=0, so it is excluded from this definition of a linear function.

A horizontal line can still represent a function.
The extra nonzero-slope requirement is not met.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question10

Find the x-intercept and explain why y is set to zero.

y=3x-6

Reference answer

$x=2$

x=2

Every point on the x-axis has y=0.
Solve $3x-6=0$ to obtain $x=2$ .

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice11

As x increases, does y increase or decrease? Explain using k.

y=-x+2

Reference answer

It decreases because k=−1<0.

Each increase of 1 in x changes y by −1.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice12

For the given relation, find y when $x=0$ .

y=2x-4

Reference answer

$y=-4$

y=-4

Substitute $x=0$ into the complete relation.
$2\times(0)+(-4)=-4$ .

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice13

Find the linear function through $(0,0)$ and $(2,6)$ . Show the equations used to find k and b.

Reference answer

$y=3x+0$

y=3x+0

Substitute both coordinate pairs into y=kx+b.
The solution is $k=3,\ b=0$ ; check both points.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice14

Find the linear function through $(-1,4)$ and $(2,-2)$ . Show the equations used to find k and b.

Reference answer

$y=-2x+2$

y=-2x+2

Substitute both coordinate pairs into y=kx+b.
The solution is $k=-2,\ b=2$ ; check both points.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension15

The lines y=x+3, y=2x+3 and y=−x+3 have different slopes. Find their common point and explain why the coefficient of x disappears there.

Use the stated conditions and the labelled scales.

Reference answer

(0,3); setting x=0 leaves y=b=3 for every line.

All three intercepts equal 3.
Substitute (0,3) into each original equation.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Use the stated conditions and the labelled scales.

Extension16

A learner says every straight line is the graph of y=kx+b. Use the two listed points on x=2 to explain the missing exception.

x	y
2	0
2	3

Use the stated conditions and the labelled scales.

Reference answer

No. At least one input is paired with two different outputs.

Check repeated inputs, not merely repeated outputs.
The repeated input has conflicting outputs.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.