Download - GCSE: Straight Line Equations Dr J Frost ([email protected]) Last modified: 3 rd September 2014
![Page 2: GCSE: Straight Line Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 3 rd September 2014](https://reader036.vdocuments.mx/reader036/viewer/2022062300/56649d0a5503460f949dce0c/html5/thumbnails/2.jpg)
GCSE specification: Understand that an equation of the form y = mx + c corresponds to a straight line graph
Plot straight line graphs from their equations
Plot and draw a graph of an equation in the form y = mx + c
Find the gradient of a straight line graph
Find the gradient of a straight line graph from its equation
Understand that a graph of an equation in the form y = mx + c has gradient of m and a y intercept of c (ie. crosses the y axis at c)
Understand how the gradient of a real life graph relates to the relationship between the two variables
Understand how the gradients of parallel lines are related
Understand how the gradients of perpendicular lines are related
Understand that if the gradient of a graph in the form y = mx + c is m, then the gradient of a line perpendicular to it will be -1/m
Generate equations of a line parallel or perpendicular to a straight line graph
![Page 3: GCSE: Straight Line Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 3 rd September 2014](https://reader036.vdocuments.mx/reader036/viewer/2022062300/56649d0a5503460f949dce0c/html5/thumbnails/3.jpg)
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?
π₯=2?β‘ βUnderstand that an equation corresponds to a line graph.β
The line represents all points which satisfies the equation.
![Page 4: GCSE: Straight Line Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 3 rd September 2014](https://reader036.vdocuments.mx/reader036/viewer/2022062300/56649d0a5503460f949dce0c/html5/thumbnails/4.jpg)
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
Starter
What is the equation of each line?
![Page 5: GCSE: Straight Line Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 3 rd September 2014](https://reader036.vdocuments.mx/reader036/viewer/2022062300/56649d0a5503460f949dce0c/html5/thumbnails/5.jpg)
Equation of a line
Understand that an equation of the form corresponds to a straight line graph
The equation of a straight line is
gradient y-intercept
![Page 6: GCSE: Straight Line Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 3 rd September 2014](https://reader036.vdocuments.mx/reader036/viewer/2022062300/56649d0a5503460f949dce0c/html5/thumbnails/6.jpg)
Gradient using two points
Given two points on a line, the gradient is:!
(1 ,4 )(3 ,10) π=3(5 ,7 )(8 ,1) π=β2
(2 ,2 )(β1 ,10) π=β83
?
?
?
![Page 7: GCSE: Straight Line Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 3 rd September 2014](https://reader036.vdocuments.mx/reader036/viewer/2022062300/56649d0a5503460f949dce0c/html5/thumbnails/7.jpg)
Gradient from an Equation Find the gradient of a straight line graph from its equation.
Putting in form :
Gradient is -2
Putting in form :
Gradient is ? ?
![Page 8: GCSE: Straight Line Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 3 rd September 2014](https://reader036.vdocuments.mx/reader036/viewer/2022062300/56649d0a5503460f949dce0c/html5/thumbnails/8.jpg)
Test Your Understanding
Find the gradient of the line with equation .
π π=πβπ?
![Page 9: GCSE: Straight Line Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 3 rd September 2014](https://reader036.vdocuments.mx/reader036/viewer/2022062300/56649d0a5503460f949dce0c/html5/thumbnails/9.jpg)
Exercise 1Determine the gradient of the lines which go through the following points.
Determine the gradient of the lines with the following equations:
A line goes through the points and . Line has the equation . Which has the greater gradient:
So has greater gradient.
? ? ? ? ? ? ? ?
? ? ?
? ? ?
?
?
1
2
3
a
b
c
d
efgh
abc
d
e
f
g
![Page 10: GCSE: Straight Line Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 3 rd September 2014](https://reader036.vdocuments.mx/reader036/viewer/2022062300/56649d0a5503460f949dce0c/html5/thumbnails/10.jpg)
x -5 -4 -3 -2 -1 0 1 2 3 4 5 6
y
4
3
2
1
-1
-2
-3
-4
Sketch the line with equation:
β‘ βPlot and draw a graph of an equation in the form y = mx + cβ
Drawing Straight Lines
Bro Tip: To sketch a line, just work out any two points on the line. Then join up. Using for one point and for the other makes things easy.
![Page 11: GCSE: Straight Line Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 3 rd September 2014](https://reader036.vdocuments.mx/reader036/viewer/2022062300/56649d0a5503460f949dce0c/html5/thumbnails/11.jpg)
x -5 -4 -3 -2 -1 0 1 2 3 4 5 6
y
4
3
2
1
-1
-2
-3
-4
Sketch the line with equation:
β‘ βPlot and draw a graph of an equation in the form y = mx + cβ
Test Your Understanding
![Page 12: GCSE: Straight Line Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 3 rd September 2014](https://reader036.vdocuments.mx/reader036/viewer/2022062300/56649d0a5503460f949dce0c/html5/thumbnails/12.jpg)
Finding intersection with the axis
Equation -axis -axisThe point where the line crosses the:
? ?
? ?
? ?
π₯
π¦When a line crosses the -axis:
When a line crosses the -axis:?
?
![Page 13: GCSE: Straight Line Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 3 rd September 2014](https://reader036.vdocuments.mx/reader036/viewer/2022062300/56649d0a5503460f949dce0c/html5/thumbnails/13.jpg)
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?
π=βπ π+ππ ?
![Page 14: GCSE: Straight Line Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 3 rd September 2014](https://reader036.vdocuments.mx/reader036/viewer/2022062300/56649d0a5503460f949dce0c/html5/thumbnails/14.jpg)
Test Your Understanding
Determine the equation of the line which has gradient 5 and goes through the point .
Determine the equation of the line which has gradient and goes through the point .
Find the equation of the line which is parallel to and goes through the point
?
?
?
1
2
3
![Page 15: GCSE: Straight Line Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 3 rd September 2014](https://reader036.vdocuments.mx/reader036/viewer/2022062300/56649d0a5503460f949dce0c/html5/thumbnails/15.jpg)
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:
?
?
![Page 16: GCSE: Straight Line Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 3 rd September 2014](https://reader036.vdocuments.mx/reader036/viewer/2022062300/56649d0a5503460f949dce0c/html5/thumbnails/16.jpg)
Exercise 2Determine the points where the following lines cross the and axis.
Using suitable axis, draw the line with equation .
A line has gradient 8 and goes through the point . Determine its equation.
A line has gradient and goes through the point . Determine its equation.
Determine the equation of the line parallel to and goes through the point .
Determine the equation of the line parallel to and goes through the point .
Determine the equation of the lines which go through the following pairs of points:
1
2
3
5
6
4
7
52
5
π₯
π¦
?
?
?
?
?
?
?
?
? ? ? ? ?
![Page 17: GCSE: Straight Line Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 3 rd September 2014](https://reader036.vdocuments.mx/reader036/viewer/2022062300/56649d0a5503460f949dce0c/html5/thumbnails/17.jpg)
x -5 -4 -3 -2 -1 0 1 2 3 4 5 6
y
4
3
2
1
-1
-2
-3
-4
Find the gradients of each pair of perpendicular lines. What do you notice?
m = -2
m = 1/2 m = -1/3
m = 3
?
? ?
?
![Page 18: GCSE: Straight Line Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 3 rd September 2014](https://reader036.vdocuments.mx/reader036/viewer/2022062300/56649d0a5503460f949dce0c/html5/thumbnails/18.jpg)
Perpendicular Lines
Gradient Gradient of Perpendicular Line
-4
?
?
?
?
?
?
If two lines are perpendicular, then the gradient of one is the negative reciprocal of the other.
Or:
!
![Page 19: GCSE: Straight Line Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 3 rd September 2014](https://reader036.vdocuments.mx/reader036/viewer/2022062300/56649d0a5503460f949dce0c/html5/thumbnails/19.jpg)
Example Problems
A line is goes through the point (9,10) and is perpendicular to another line with equation . What is the equation of the line?
A line goes through the points and . A second line is perpendicular to and passes through point B. Where does cross the x-axis?
Are the following lines parallel, perpendicular, or neither?
Neither. Gradients are and . But , not -1, so not perpendicular.
?
?
?
Q1
Q2
Q3
![Page 20: GCSE: Straight Line Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 3 rd September 2014](https://reader036.vdocuments.mx/reader036/viewer/2022062300/56649d0a5503460f949dce0c/html5/thumbnails/20.jpg)
Exercise 3A line goes through the indicated point and is perpendicular to another line . Determine the equation of in each case.
Find the equation of the line which passes through B, and is perpendicular to the line passing through both A and B.
Line has the equation . Line goes through the points and . Are the lines parallel, perpendicular, or neither?
so perpendicular.
π₯
π¦
π¦=β2π₯+4π
Determine the equation of the line .Known point on :
So equation of :
π₯
π¦=3π₯+5 π
Determine the equation of the line .
1
2
4
5
? ?
? ? ?
?
?
?
3
?
![Page 21: GCSE: Straight Line Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 3 rd September 2014](https://reader036.vdocuments.mx/reader036/viewer/2022062300/56649d0a5503460f949dce0c/html5/thumbnails/21.jpg)
GCSE specification: Understand that an equation of the form y = mx + c corresponds to a straight line graph
Plot straight line graphs from their equations
Plot and draw a graph of an equation in the form y = mx + c
Find the gradient of a straight line graph
Find the gradient of a straight line graph from its equation
Understand that a graph of an equation in the form y = mx + c has gradient of m and a y intercept of c (ie. crosses the y axis at c)
Understand how the gradient of a real life graph relates to the relationship between the two variables
Understand how the gradients of parallel lines are related
Understand how the gradients of perpendicular lines are related
Understand that if the gradient of a graph in the form y = mx + c is m, then the gradient of a line perpendicular to it will be -1/m
Generate equations of a line parallel or perpendicular to a straight line graph
![Page 22: GCSE: Straight Line Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 3 rd September 2014](https://reader036.vdocuments.mx/reader036/viewer/2022062300/56649d0a5503460f949dce0c/html5/thumbnails/22.jpg)
Two last thingsβ¦
Midpoint of two points Distance between two points
(3,6)
(5,9 )
(π ,π .π)? (3,6)
(7,9)5
4
3
Find change and change to form right-angled triangle.Then use Pythagoras.
?
Just find the average of and the average of .
![Page 23: GCSE: Straight Line Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 3 rd September 2014](https://reader036.vdocuments.mx/reader036/viewer/2022062300/56649d0a5503460f949dce0c/html5/thumbnails/23.jpg)
Past Exam QuestionsSee GCSEPastPaper_Solutions.pptxGCSERevision_StraightLineEquations.docx