ks3: straight lines

KS3: Straight Lines

Dr J Frost (jfrost@tiffin.kingston.sch.uk)www.drfrostmaths.com

Last modified: 14th October 2015

Part 1Lines and their Equations

To print: Yr8StraightLines-Ex1LinesAndTheirEquations

x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y

4

3

2

1

-1

-2

-3

-4

What is the equation of this line?And more importantly, why is it that?

𝑥=2? For any point we pick on the line, the value is always 2.

Lines and Equations of LinesA line consists of all points which satisfy some equation in terms

of and/or .

𝑦=3 𝑥+ 𝑦=2 𝑦=3 𝑥+1

(2,0 ) L J L

( 14 , 74 )(−1,3 ) J J

JLJL

? ? ?

? ? ?

? ? ?

On the line?

x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y

4

3

2

1

-1

-2

-3

-4𝑦=−1?

What and why?

x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y

4

3

2

1

-1

-2

-3

-4𝑦=𝑥?

What and why?

For any point we pick on the line, the value is always equal to the value.

x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y

4

3

2

1

-1

-2

-3

-4𝑦=−𝑥?

What and why?

x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y

8

6

4

2

-2

-4

-6

-8

Exercise 1 - ExampleUse the axis to sketch the line with equation

Pick two suitable values of suitable far apart (say -3 and 4)

Use the equation to work out what would be for each. Plot these points.

If you know the line is a straight line, we can just join them up.

Exercise 1 – Example 2

Complete the table of values for .

       ? ? ? ?

If just sub it into the equation:

x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y

8

6

4

2

-2

-4

-6

-8

Exercise 1 – Question 1𝑥+ 𝑦=2

x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y

8

6

4

2

-2

-4

-6

-8

Exercise 1 – Question 2

𝑦=− 12 𝑥+1

Exercise 1 – Question 3

𝑦=4 𝑥−2

Exercise 1 – Question 4

6 1.5 0? ? ?

Click to Reveal

Exercise 1 – Question 5

Put a tick or cross to determine whether each of the following points are on the line with the given equation.

? ?

? ?

? ?

? ?

Exercise 1 – Question 6

    Below the line

On the line

Above the line

For the given equation of a line and point, indicate whether the point is above the line, on the line or below the line. (Hint: Find out what is on the line for the given )

??

?

?

Exercise 1 – Question N1

The equation of a line is . If the value of some point on the line is , what is the full coordinate of the point, in terms of ?

If , then . Rearranging, .So coordinate is

?

Exercise 1 – Question N2

What is the area of the region enclosed between the line with equation , the axis, and the axis?

We can set to find where the lines cuts the axis:

Similarly when :

We have a triangle between the points .Area is . ?

Part 1bIntercepts with the axis

Intercepts

𝑦=2𝑥+6We want to find the

coordinates of the points where the line ‘intercepts’ the axes.

𝑥

𝑦What do we know about any point on the -axis?How then can we work out the coordinate of the -intercept?

So Point is

What do we know about any point on the -axis?How then can we work out the coordinate of the -intercept?

So Point is

?

?

One more example

Determine where the line crosses the:

a) -axis: Let .

b) -axis: Let

?

?

What mistakes do you think it’s easy to make?• Mixing up x/y: Putting answer as rather than .• Setting to find the -intercept, or to find the -intercept.?

Test Your Understanding

Equation -axis -axisThe point where the line crosses the:

? ?

? ?

? ?

Copy and complete this table.

? ?

Part 2Gradient

x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y

4

3

2

1

-1

-2

-3

-4-1 0 1 2

-3 -1 1 3

Sketch

? ? ? ?

Do you notice any connection between how increases each time and the equation?

x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y

4

3

2

1

-1

-2

-3

-4-1 0 1 2

3 2 1 0

Sketch

? ? ? ?

Do you notice any connection between how increases each time and the equation?

x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y

4

3

2

1

-1

-2

-3

-4-1 0 1 2

0.5 1 1.5 2

Sketch

? ? ? ?

Do you notice any connection between how increases each time and the equation?

The steepness of a line is known as the gradient.It tells us what changes by as increases by 1.

! ?

1

Gradient

The equation of a straight line is of the form:

The gradient is . is the ‘y-intercept’.

x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y4

3

2

1

-1

-2

-3

-4

On your printed sheet, identify the gradient of each line.

A

B

C

D

E

F

G

H

x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y4

3

2

1

-1

-2

-3

-4

A

B

C

D

E

F

G

H

x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y4

3

2

1

-1

-2

-3

-4

(−𝟏 ,−𝟐 )

(𝟑 ,𝟒 )Suppose we just had two points on the line and wanted to determine the gradient, but didn’t want to draw a grid.

has increased by 4.

has increased by 6.

So what does change by for each unit increase in ?

𝒎=𝟔𝟒=𝟏 .𝟓?

Gradient using two points

Given two points on a line, the gradient is:!

(1 ,4 )(3 ,10) 𝑚=3(5 ,7 )(8 ,1) 𝑚=−2

(2 ,2 )(−1 ,10) 𝑚=− 83

?

?

?

Gradient using the EquationWe can get the gradient of a line using just its equation.Rearrange into the form , and then the gradient is .

Examples Test Your Understanding

?

?

?

?

?

Equation Gradient

Exercise 2

Determine the gradient of the line with equation , in terms of the constants and .Rearranging: So the gradient is

By rearranging the equations into the form , determine the gradient of each line.

1 2

Point 1 Point 2 Gradient

Determine the gradient of the line which goes through the following points.

N1Write an equation that ensures that three points , and where , form a straight line (i.e. are “collinear”. We just require that the gradient between points 1 and 2, and points 2 and 3 are the same, i.e.

N2

? ? ? ? ? ? ? ? ? ? ?

? ? ? ? ? ? ? ? ? ? ? ?

? ?

Summary

The gradient of a line is the steepness: how much changes as increases by 1.We’ve seen 3 ways in which we can calculate the gradient:

a. Counting Squares

𝒎=−𝟑

b. Using the equation c. Using two points

𝑦=4− 32 𝑥

𝒎=−𝟑𝟐

(1 ,4 ) , (4 ,13 )

𝒎=𝟑? ? ?

Part 3

x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y4

3

2

1

-1

-2

-3

-4

A

B

C

D

E

F

G

H

RecapWhat was the gradient of these lines?

x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y4

3

2

1

-1

-2

-3

-4

A

B

C

D

E

F

G

H

y-intercept

The y-intercept is the point at which the line crosses the -axis.

It is the in (why?)

x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y4

3

2

1

-1

-2

-3

-4

A

B

C

D

E

F

G

H

Now determine the full equation of each line.

Test Your Understanding

A line has the equation . What is the -intercept of the line?

So -intercept is . ?

Card Sort!

Exercise 3

Gradient -intercept Equation

Copy and complete the following table. Gradient -intercept Equation

? ?

? ? ? ? ?

? ? ? ? ? ?

? ? ? ?

1 2

3 The equation of a line is . If the -intercept is 6, what is ?

The equation of a line is . If the -intercept is 8, what is ?4

N A line has equation . The area enclosed between this line, the -axis and the -axis is 1.Determine .Intercepts are and .

?

?

?

Part 4Parallel lines

Puzzle

(This was in a Year 8 End of Year exam)

𝐶 (0,5)𝐴(−1,5)

𝐵(2 ,−1)

The diagram shows three points and .A line is parallel to and passes through .

Find the equation of the line .

𝑦=−2𝑥+5?

𝑳

Preliminary Question: What will be the same about the equations of two lines if they are parallel?They have the same gradient.?

Test Your Understanding

𝐶 (0 ,4)

𝐴(−6 ,−2)

𝐵(4,3)

The diagram shows three points and .A line is parallel to and passes through .

Find the equation of the line .

𝑦=12 𝑥+4?

𝑳

Equation given a gradient and point

The gradient of a line is 3. It goes through the point (4, 10). What is the equation of the line?

Start with (where is to be determined)Substituting: Therefore

?

The gradient of a line is -2. It goes through the point (5, 10). What is the equation of the line?

𝒚=−𝟐 𝒙+𝟐𝟎 ?

E1

E2

Exercise 4Give the equation of a line which is parallel to .(where c can be any number)

Give the equation of a line which passes through and is parallel to another line which passes through the points and

Give the equation of a line which passes through the point (0, 6) and has the gradient -2.

Which line has the greater gradient, or ?First line rearranges to , second to So second line has the greater gradient.

1

2

3

4

?

?

?

A and B are straight lines. Line A has equation . Line B goes through the points and . Do lines A and B intersect?

Line A: so .Line B: .The gradients are different so the lines are not parallel, and therefore intersect.

N

Gradient Goes through Equation

a

b

c

d

e

f

4

? ?

? ? ? ? ?

?

Equation given two points

A straight line goes through the points (3, 6) and (5, 12). Determine the full equation of the line.

(3,6)

(5,12)

Gradient: 3

Equation:

?

A straight line goes through the points (5, -2) and (1, 0). Determine the full equation of the line.

(5, -2)

(1,0)Gradient: -0.5

Equation:

?

?

Choose one of the two points and then use the previous method we saw when we have a gradient and point.?

Test Your Understanding

A line passes through the points and . Find the equation of the line.

Using the point :

If you finish: A line passes through the points and . Give the coordinate of the point this line crosses the -axis.

If :

?

?

Exercise 5Work out the gradient given the points on the line.

Point 1 Point 2 Full Equation(0,0) (2,2) 𝑦=𝑥(-5,0) (0,-5) 𝑦=−𝑥−5(1,-3) (3,1) 𝑦=2 𝑥−5(-4,1) (4, 5) 𝑦=0.5 𝑥+3

Q1Q2Q3Q4

(-3,7) (2,2) 𝑦=−𝑥+4Q5(1,6) (3,-2) 𝑦=−4 𝑥+10Q6(-7,3) (5,-1) 𝑦=− 13 𝑥+

23

Q7

(4,9) (-3,10) 𝑦=− 17 𝑥+677

Q8

? ? ? ? ? ? ?

? A line goes through the points and . Determine the coordinate of the point the line crosses the -axis, in terms of .

A line goes through the point and .i) Find the equation of the line.

ii) Hence find the point at which this line intercepts the axis.

9 N

? ?

?

𝐴𝐵𝐶𝐷

REVISIONVote with your diaries!

−1153

The equation of a line is . What is the missing value of this point on the line?

x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y4

3

2

1

-1

-2

-3

-4𝑦=

12 𝑥+1𝑦=1 𝑥−2𝑦=−2𝑥+1𝑦=− 12 𝑥+1

x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y4

3

2

1

-1

-2

-3

-4𝑥=3𝑦=3𝑦=3 𝑥𝑦=𝑥+3

x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y4

3

2

1

-1

-2

-3

-4𝑦=𝑒𝜋 𝑖𝑦=−1𝑦=−𝑥𝑦=𝑥

𝑦=3 𝑥−3𝑦=−3𝑥+2𝑦=2 𝑥−3𝑦=−3

What is the equation of a line parallel to and goes through the point ?

2−212− 12

What is the gradient of the line which goes through the points and ?

𝑦=3 𝑥+5𝑦=3 𝑥−1𝑦=2 𝑥+5𝑦=5 𝑥−6

What is the full equation of a line which has gradient 3 and passes through the point (2,5)?

𝑦=12 𝑥+7𝑦=2 𝑥−3𝑦=2 𝑥+3𝑦=

12 𝑥+6

What is the full equation of the line which goes through the point , ?

1133− 13

What is the y-intercept of the line ?

−31−3 𝑥3

What is the gradient of the line ?

(2,0)(−2,0)(0,2 )(0 ,−2 )

Give the coordinate of the point where the line crosses the axis.

(4,0)(−4,0)(0,4 )(0 ,−4 )

Give the coordinate of the point where the line crosses the axis.