ks3: straight lines
DESCRIPTION
KS3: Straight Lines. Dr J Frost ([email protected]) . Last modified: 27 th August 2013. y. What is the equation of this line? And more importantly, why is it that?. 4 3 2 1 -1 -2 -3 -4. x -5 - 4-3-2-10123456. ?. y. What and why?. 4 3 2 1 -1 -2 -3 -4. - PowerPoint PPT PresentationTRANSCRIPT
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KS3: Straight Lines
Dr J Frost ([email protected])www.drfrostmaths.com
Last modified: 14th October 2015
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Part 1Lines and their Equations
To print: Yr8StraightLines-Ex1LinesAndTheirEquations
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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.
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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?
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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?
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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.
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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?
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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.
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Exercise 1 – Example 2
Complete the table of values for .
? ? ? ?
If just sub it into the equation:
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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
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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
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Exercise 1 – Question 3
𝑦=4 𝑥−2
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Exercise 1 – Question 4
6 1.5 0? ? ?
Click to Reveal
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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.
? ?
? ?
? ?
? ?
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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 )
??
?
?
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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
?
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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 . ?
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Part 1bIntercepts with the axis
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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
?
?
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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.?
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Test Your Understanding
Equation -axis -axisThe point where the line crosses the:
? ?
? ?
? ?
Copy and complete this table.
? ?
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Part 2Gradient
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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?
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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?
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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?
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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’.
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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
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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
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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?
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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
?
?
?
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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
?
?
?
?
?
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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
? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ?
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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? ? ?
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Part 3
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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?
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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?)
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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.
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Test Your Understanding
A line has the equation . What is the -intercept of the line?
So -intercept is . ?
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Card Sort!
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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 .
?
?
?
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Part 4Parallel lines
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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.?
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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
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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
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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
? ?
? ? ? ? ?
?
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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.?
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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 :
?
?
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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
? ?
?
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𝐴𝐵𝐶𝐷
REVISIONVote with your diaries!
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−1153
The equation of a line is . What is the missing value of this point on the line?
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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
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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
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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
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𝑦=3 𝑥−3𝑦=−3𝑥+2𝑦=2 𝑥−3𝑦=−3
What is the equation of a line parallel to and goes through the point ?
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2−212− 12
What is the gradient of the line which goes through the points and ?
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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)?
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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 , ?
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1133− 13
What is the y-intercept of the line ?
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−31−3 𝑥3
What is the gradient of the line ?
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(2,0)(−2,0)(0,2 )(0 ,−2 )
Give the coordinate of the point where the line crosses the axis.
![Page 61: KS3: Straight Lines](https://reader033.vdocuments.site/reader033/viewer/2022061416/56816258550346895dd2aa72/html5/thumbnails/61.jpg)
(4,0)(−4,0)(0,4 )(0 ,−4 )
Give the coordinate of the point where the line crosses the axis.